Linear ℤⁿ₂ -Manifolds and Linear Actions

We establish the representability of the general linear ℤⁿ₂-group and use the restricted functor of points – whose test category is the category of ℤⁿ₂-manifolds over a single topological point – to define its smooth linear actions on ℤⁿ₂-graded vector spaces and linear ℤⁿ₂-manifolds. Throughout the...

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Published in:	Symmetry, Integrability and Geometry: Methods and Applications
Date:	2021
Main Authors:	Bruce, Andrew James, Ibarguëngoytia, Eduardo, Poncin, Norbert
Format:	Article
Language:	English
Published:	Інститут математики НАН України 2021
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/211363
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Cite this:	Linear ℤⁿ₂ -Manifolds and Linear Actions. Andrew James Bruce, Eduardo Ibarguëngoytia and Norbert Poncin. SIGMA 17 (2021), 060, 58 pages

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author	Bruce, Andrew James Ibarguëngoytia, Eduardo Poncin, Norbert
author_facet	Bruce, Andrew James Ibarguëngoytia, Eduardo Poncin, Norbert
citation_txt	Linear ℤⁿ₂ -Manifolds and Linear Actions. Andrew James Bruce, Eduardo Ibarguëngoytia and Norbert Poncin. SIGMA 17 (2021), 060, 58 pages
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container_title	Symmetry, Integrability and Geometry: Methods and Applications
description	We establish the representability of the general linear ℤⁿ₂-group and use the restricted functor of points – whose test category is the category of ℤⁿ₂-manifolds over a single topological point – to define its smooth linear actions on ℤⁿ₂-graded vector spaces and linear ℤⁿ₂-manifolds. Throughout the paper, particular emphasis is placed on the full faithfulness and target category of the restricted functor of points of a number of categories that we are using.
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 060, 58 pages Linear Zn 2 -Manifolds and Linear Actions Andrew James BRUCE, Eduardo IBARGUËNGOYTIA and Norbert PONCIN Department of Mathematics, University of Luxembourg, Maison du Nombre, 6, avenue de la Fonte, L-4364 Esch-sur-Alzette, Luxembourg E-mail: andrewjamesbruce@googlemail.com, eduardo.ibarguengoytia@uni.lu, norbert.poncin@uni.lu Received November 05, 2020, in final form May 30, 2021; Published online June 16, 2021 https://doi.org/10.3842/SIGMA.2021.060 Abstract. We establish the representability of the general linear Zn 2 -group and use the restricted functor of points – whose test category is the category of Zn 2 -manifolds over a single topological point – to define its smooth linear actions on Zn 2 -graded vector spaces and linear Zn 2 -manifolds. Throughout the paper, particular emphasis is placed on the full faithfulness and target category of the restricted functor of points of a number of categories that we are using. Key words: supergeometry; ringed spaces; functors of points; linear group actions 2020 Mathematics Subject Classification: 58A50; 58C50; 14A22; 14L30; 13F25; 16L30; 17A70 1 Introduction In order to be able to deal with the technical details of vector bundles and related structures in the category of Zn2 -manifolds (for n = 1 see [6]), we need some foundational results on Zn2 -Lie groups and their smooth linear actions on linear Zn2 -manifolds. However, the proofs of some folklore results, i.e., results that we tended to accept somewhat hands-waving, are often not at all obvious in the Zn2 -context. The present paper, beyond its supposed applications, intrinsic interest and the beauty of some of its developments, raises the question of the scientific value of “results” that are partially based on speculations. Loosely speaking, Zn2 -manifolds (Zn2 = Z×n2 ) are “manifolds” for which the structure sheaf has a Zn2 -grading and the commutation rules for the local coordinates comes from the standard scalar product (see [11, 13, 14, 15, 18, 19, 20, 21, 37] for details). This is not just a trivial or straightforward generalization of the notion of a standard supermanifold, as one has to deal with formal coordinates that anticommute with other formal coordinates, but are themselves not nilpotent. Due to the presence of formal variables that are not nilpotent, formal power series are used rather than polynomials. Recall that for standard supermanifolds all functions are polynomial in the Grassmann odd variables. The theory of Zn2 -geometry is currently being developed and many foundational questions remain. For completeness, we include Appendix B in which the foundations of Zn2 -geometry are given. In this paper, we examine the relation between Zn2 -graded vector spaces and linear Zn2 -manifolds, and then we use this to define linear actions of Zn2 -Lie groups. In the literature on supergeometry, the symbol Rp\|q has two distinct, but related meanings. First, we have the notion of a Z2-graded, or super, vector space with p even and q odd dimensions, i.e., the real vector space Rp\|q = Rp ⊕ Rq. Secondly, we have the locally ringed space Rp\|q =( Rp, C∞Rp [ξ] ) , where ξi (i ∈ {1, . . . , q}) are the generators of a Grassmann algebra. The difference can be highlighted by identifying the points of these objects. The Z2-graded vector space has as its underlying topological space Rp+q (we just forget the “superstructure”), while for the locally ringed space the topological space is Rp. There are several ways of showing that these mailto:andrewjamesbruce@googlemail.com mailto:eduardo.ibarguengoytia@uni.lu mailto:norbert.poncin@uni.lu https://doi.org/10.3842/SIGMA.2021.060 2 A.J. Bruce, E. Ibarguëngoytia and N. Poncin two notions are deeply tied. In particular, the category of finite dimensional super vector spaces is equivalent to the category of “linear supermanifolds” (see [8, 32, 33, 34, 38]). In this paper, we will show that the categories of finite dimensional Zn2 -graded vector spaces V and linear Zn2 -manifolds V are isomorphic. We do this by explicitly constructing a “manifold- ification” functor M, which associates a linear Zn2 -manifold to every finite dimensional Zn2 -graded vector space, and a “vectorification” functor, which is the inverse of the previous functor. It turns out that working in a coordinate-independent way (V, V ) is much more complex than working with canonical coordinates ( Rp\|q,Rp\|q ) . Throughout this article, a special focus is placed on functors of points. The functor of points has been used informally in physics as from the very beginning. It is actually of importance in contexts where there is no good notion of point as in super- and Zn2 -geometry and in algebraic geometry. For instance, homotopical algebraic geometry [42, 43] and its generalisation that goes under the name of homotopical algebraic geometry over differential operators [25, 26], are completely based on the functor of points approach. In this paper, we are particularly interested in functors of Λ-points, i.e., functors of points from appropriate locally small categories C to a functor category whose source is not the category Cop but the category G of Zn2 -Grassmann algebras Λ. However, functors of points that are restricted to the very simple test category G are fully faithful only if we replace the target category of the functor category by a subcategory of the category of sets. More precisely, closely related to the above isomorphism of supervector spaces and linear supermanifolds is the so-called “even rules”. Loosely this means including extra odd parameters to render everything even and in doing so one removes copious sign factors (see for example [24, Section 1.7]). We will establish an analogue of the even rules in our higher graded setting which we will refer to as the “zero degree rules” (see Definition 2.1). To address this we will make ex- tensive use of Zn2 -Grassmann algebras Λ, Λ-points and the Schwarz–Voronov embedding, which is a fully faithful functor of points S from Zn2 -manifolds to a functor category with source G and the category of Fréchet manifolds (see for example [30]) over commutative Fréchet algebras as target (see [13]). We show that the zero degree rules functor F, understood as an assignment of a functor from G to the category of modules over commutative (Fréchet) algebras, given a (finite dimensional) Zn2 -graded vector space, is fully faithful (see Theorem 2.2 and Proposition 2.25). The “zero degree rules” allow one to identity a finite dimensional Zn2 -graded vector space, con- sidered as a functor, with the functor of points of its “manifoldification”. In other words, the composite S◦M and F can be viewed as functors between the same categories and are nat- urally isomorphic. This identification is fundamental when describing linear group actions on Zn2 -graded vector spaces and linear Zn2 -manifolds. Another important part of this work is the category of Zn2 -Lie groups and its fully faithful functor of points valued in a functor category with G as source category and Fréchet Lie groups over commutative Fréchet algebras as target category. We define the general linear Zn2 -group as a functor in this functor category and show that it is representable, i.e., is a genuine Zn2 - manifold (see Theorem 3.4). This leads to interesting insights into the computation of the inverse of an invertible degree zero Zn2 -graded square matrix of dimension p\|q with entries in a Zn2 -commutative algebra. Furthermore, the approach using Λ-points and the zero rules allows us to construct a canonical smooth linear action of the general linear Zn2 -group on Zn2 -graded vector spaces and linear Zn2 -manifolds. All these notions, in particular the equivalence between the definitions of a smooth linear action as natural transformation and as Zn2 -morphism, are carefully and explicitly explained in the main text. We remark that many of the statements in this paper are not surprising in themselves. However, due to the subtleties of Zn2 -geometry, many of the proofs are much more involved than the analogue statements in supergeometry. The main source of difficulty is that one has to deal with formal power series in non-zero degree coordinates, rather than polynomials as one does Linear Zn2 -Manifolds and Linear Actions 3 in supergeometry. This forces one to work with infinite dimensional objects and the J-adic topology (J is the ideal generated by non-zero degree elements). Many of the “categorical” proofs are significantly more involved than the proofs for supermanifolds. In general, there is a lot of work to establish the form of natural transformations as we have non-nilpotent elements of non-zero degree. While the ethos of the proofs may be standard, they are not, in general, simple or routine checks due to the aforementioned subtleties. Motivation from physics: Zn2 -gradings (n ≥ 2) can be found in the theory of parastatistics (see for example [27, 28, 29, 48]) and in relation to an alternative approach to supersymmetry [1, 2, 44]. “Higher graded” generalizations of the super Schrödinger algebra (see [3]) and the super Poincaré algebra (see [10]) have appeared in the literature. Furthermore, such gradings appear in the theory of mixed symmetry tensors as found in string theory and some formulations of supergravity (see [12]). It must also be pointed out that quaternions and more general Clifford algebras can be understood as Zn2 -graded Zn2 -commutative algebras [4, 5]. Finally, any “sign rule” can be interpreted in terms of a Zn2 -grading (see [18]). Background: For various sheaf-theoretical notions we will draw upon Hartshorne [31, Chap- ter II] and Tennison [41]. There are several good introductory books on the theory of supermani- folds including Bartocci, Bruzzo and Hernández Ruipérez [7], Carmeli, Caston and Fioresi [16], Deligne and Morgan [24] and Varadarajan [45]. For categorical notions we will be based on Mac Lane [35]. We will make extensive use of the constructions and statements found in our earlier publications [13, 14, 15]. 2 Zn 2 -graded vector spaces and Linear Zn 2 -manifolds 2.1 Zn 2 -graded vector spaces and the zero degree rules When dealing with linear superalgebra one encounters the so-called even rules (see [16, Sec- tion 1.8], [24, Section 1.7] and [45, pp. 123–124], for example). Very informally, the idea is to remove sign factors by allowing extra parameters to render the situation completely even. The idea has been applied in physics since the early days of supersymmetry. More precisely, let V (A) = (A⊗ V )0 be the even part of the extension of scalars in a (real) super vector space V , from the base field R to a supercommutative algebra A ∈ SAlg (in the even rules that we are about to describe, it actually suffices to use supercommutative Grassmann algebras A = R[θ1, . . . , θN ]: the θi are then the extra parameters mentioned before). The main result in even rules states, roughly, that defining a morphism φ : V ⊗ V → V is equivalent to defining it functorially on the even part of V after extension of scalars, i.e., is equivalent to defining a functorial family of morphisms φ(A) : V (A)× V (A)→ V (A) (indexed by A ∈ SAlg). More precisely, there is a 1 : 1 correspondence between parity respecting R-linear maps φ : V1 ⊗ · · · ⊗ Vn → V and functorial families φ(A) : V1(A)× · · · × Vn(A)→ V (A) (A ∈ SAlg) of A0-multilinear maps. We now proceed to generalise this theorem to the Zn2 -setting. We will work with the category Zn2GrAlg of Zn2 -Grassmann algebras rather than the category Zn2Alg of all Zn2 -commutative algebras. Let V = ⊕N i=0 Vγi be a (real) Zn2 -graded vector space, i.e., a (real) vector space with a direct sum decomposition over i ∈ {0, . . . , N} (we say that the vectors of Vγi are of degree γi ∈ Zn2 ). 4 A.J. Bruce, E. Ibarguëngoytia and N. Poncin The category of Zn2 -graded vector spaces (not necessarily finite dimensional) we denote as Zn2Vec. Morphisms in this category are degree preserving linear maps. We denote the category of modu- les over commutative algebras as AMod (see Appendix A). To V we associate a functor V (−) ∈ Fun0 ( Zn2Ptsop, AMod ) in the category of those functors whose value on any Zn2 -Grassmann algebra Λ ∈ Zn2Ptsop is a Λ0-module, and of those natural transformations that have Λ0-linear Λ-components. The functor V (−) is essentially the tensor product functor − ⊗ V . It is built in the following way. First, for every Zn2 -Grassmann algebra Λ, we define V (Λ) := (Λ⊗ V )0 ∈ Λ0Mod, where the tensor product is over R. Secondly, for any Zn2 -algebra morphism ϕ∗ : Λ → Λ′, we define V (ϕ∗) := ( ϕ∗ ⊗ 1V ) 0 , where the RHS is the restriction of ϕ∗ ⊗ 1V to the degree 0 part of Λ ⊗ V , so that V (ϕ∗) is an AMod-morphism V (ϕ∗) : V (Λ)→ V (Λ′), (2.1) whose associated algebra morphism is the restriction (ϕ∗)0 : Λ0 → Λ′0. It is clear that V (−) respects compositions and identities and is thus a functor, as announced. We thus get an assignment F: Zn2Vec 3 V 7→ F(V ) := V (−) ∈ Fun0 ( Zn2Ptsop, AMod ) . The map F is essentially − ⊗ • and is itself a functor. It associates to any grading respecting linear map φ : V →W and any Zn2 -Grassmann algebra Λ, a Λ0-linear map φΛ := (1Λ ⊗ φ)0 : V (Λ)→W (Λ). The family F(φ) := φ− is a natural transformation from F(V ) to F(W ). Since F respects compositions and identities, it is actually a functor valued in the restricted functor category Fun0 ( Zn2Ptsop, AMod ) . Definition 2.1. The functor F: Zn2Vec −→ Fun0 ( Zn2Ptsop, AMod ) is referred to as the zero degree rules functor. Theorem 2.2. The zero degree rules functor F: Zn2Vec −→ Fun0 ( Zn2Ptsop, AMod ) is fully faithful, i.e., for any pair of Zn2 -graded vector spaces V and W , the map FV,W : HomZn2 Vec(V,W ) −→ HomFun0(Zn2 Ptsop,AMod)(F(V ),F(W )), is a bijection. This result is the Zn2 -counterpart of the 1 : 1 correspondence mentioned above. Linear Zn2 -Manifolds and Linear Actions 5 Proof. We show first that the map FV,W is injective. Let φ, ψ : V →W be two degree preserving linear maps, and assume that F(φ) = φ− = ψ− = F(ψ), so that, for any Λ ∈ Zn2Ptsop and any λ⊗ v ∈ V (Λ), we have λ⊗ φ(v) = φΛ(λ⊗ v) = ψΛ(λ⊗ v) = λ⊗ ψ(v). (2.2) Notice now that V (Λ) = (Λ⊗ V )0 = N⊕ i=0 Λγi ⊗ Vγi and let Λ be the Grassmann algebra Λ1 := R[[θ1, . . . , θN ]] (2.3) that has exactly one generator θj in each non-zero degree γj ∈ Zn2 (N = 2n − 1). For any vj ∈ Vγj , equation (2.2) implies that θj ⊗ φ(vj) = θj ⊗ ψ(vj), so that φ and ψ coincide on Vγj , for all j ∈ {1, . . . , N}. For v0 ∈ V0 := Vγ0 and λ = 1, the same equation shows that φ and ψ coincide also on V0. To prove surjectivity, we consider an arbitrary natural transformation Φ− : V (−) → W (−) and will define a degree 0 linear map φ : V → W , such that F(φ) = φ− = Φ−, i.e., such that, for any Λ ∈ Zn2Ptsop, we have φΛ = ΦΛ on V (Λ). Since an element of V (Λ) (uniquely) decomposes into a sum over i ∈ {0, . . . , N} of (not uniquely defined) finite sums of decomposable tensors λi⊗vi, with (not uniquely defined) factors λi and vi of degree γi, it suffices to show that φΛ(λi ⊗ vi) = ΦΛ(λi ⊗ vi), (2.4) for all i ∈ {0, . . . , N}. Further, it suffices to prove condition (2.4) for Λ1 (see (2.3)) and for the tensors θi ⊗ vi (θ0 := 1, vi ∈ Vγi , i ∈ {0, . . . , N}). The observation follows from naturality of Φ. Indeed, assume that (2.4) is satisfied for Λ1 and the decomposable tensors just mentioned (assumption (?)). For any fixed i ∈ {1, . . . , N} (resp., i = 0), and for Λ, λi and vi as above, let ϕ∗ : Λ1 → Λ be the Zn2 -algebra map defined by ϕ∗(θi) = λi, ϕ ∗(θj) = 0 for j 6= i, j 6= 0, and ϕ∗(θ0) = ϕ∗(1) = 1 (resp., ϕ∗(θj) = 0 for all j 6= 0, and ϕ∗(θ0) = ϕ∗(1) = 1). For i ∈ {1, . . . , N}, when applying the naturality condition V (Λ1) W (Λ1) V (Λ) W (Λ) // ΦΛ1 �� V (ϕ∗) �� W (ϕ∗) // ΦΛ to θi ⊗ vi, we get clockwise W (ϕ∗)(ΦΛ1(θi ⊗ vi)) = W (ϕ∗)(θi ⊗ φ(vi)) = ϕ∗(θi)⊗ φ(vi) = λi ⊗ φ(vi) = φΛ(λi ⊗ vi), in view of (?), whereas anticlockwise we obtain ΦΛ(V (ϕ∗)(θi ⊗ vi)) = ΦΛ(λi ⊗ vi). 6 A.J. Bruce, E. Ibarguëngoytia and N. Poncin Hence, condition (2.4) holds for i ∈ {1, . . . , N}. For i = 0, the above naturality condition yields 1⊗φ(v0) = ΦΛ(1⊗ v0), when applied to 1⊗ v0. In view of the Λ0-linearity of the Λ-components of the natural transformations considered, we get now φΛ(λ0 ⊗ v0) = λ0 φΛ(1⊗ v0) = λ0(1⊗ φ(v0)) = ΦΛ(λ0 ⊗ v0). Finally, condition (2.4) holds for an arbitrary Λ, if it holds for Λ1. Surjectivity now reduces to constructing a Zn2Vec-morphism φ : V → W that satisfies (2.4) for Λ1 and decomposable tensors of the type θi ⊗ vi (i ∈ {0, . . . , N}). We first build φ(vj) ∈Wγj linearly in vj ∈ Vγj for an arbitrarily fixed j ∈ {0, . . . , N}. We set again θ0 = 1 ∈ Λ1,0. Since ΦΛ1(θj ⊗ vj) ∈ (Λ1 ⊗W )0, it reads ΦΛ1(θj ⊗ vj) = N∑ i=0 Mi∑ k=1 λki ⊗ wki , where Mi ∈ N, λki ∈ Λ1,γi and wki ∈Wγi . When setting Ai = { α ∈ N×N : N∑ `=1 α`γ` = γi } and λki = ∑ α∈Ai rkα,i θ α ( rkα,i ∈ R ) , where we used the standard multi-index notation, we get ΦΛ1(θj ⊗ vj) = N∑ i=0 ∑ α∈Ai θα ⊗ ( Mi∑ k=1 rkα,iw k i ) =: N∑ i=0 ∑ α∈Ai θα ⊗ wα,i (wα,i ∈Wγi). Denoting the canonical basis of RN by (e`)` and decomposing the RHS with respect to the values of \|α\| = α1 + · · ·+ αN ∈ N, we obtain ΦΛ1(θj ⊗ vj) = w0,0 + N∑ i=1 θi ⊗ wei,i + N∑ i=0 ∑ α∈Ai : \|α\|≥2 θα ⊗ wα,i. (2.5) Let now ϕ∗r0 (r0 ∈ R, r0 > 0 and r0 6= 1) be the Zn2 -algebra endomorphism of Λ1 that is defined by ϕ∗r0(θk) = r0θk if k 6= 0 and by ϕ∗r0(θ0) = 1. It follows from the naturality condition V (Λ1) W (Λ1) V (Λ1) W (Λ1) // ΦΛ1 �� V (ϕ∗r0) �� W (ϕ∗r0) // ΦΛ1 that W (ϕ∗r0)(ΦΛ1(θj ⊗ vj)) = w0,0 + N∑ i=1 θi ⊗ (r0wei,i) + N∑ i=0 ∑ α∈Ai : \|α\|≥2 θα ⊗ ( r \|α\| 0 wα,i ) Linear Zn2 -Manifolds and Linear Actions 7 and ΦΛ1 ( V (ϕ∗r0)(θj ⊗ vj) ) = r 1−δj0 0 w0,0 + N∑ i=1 θi ⊗ ( r 1−δj0 0 wei,i ) + N∑ i=0 ∑ α∈Ai : \|α\|≥2 θα ⊗ ( r 1−δj0 0 wα,i ) , where δj0 is the Kronecker symbol, coincide. As all the monomials in θ in the RHS-s of the two last equations are different, we get, (1) if j 6= 0: w0,0 = 0 and wα,i = 0, for all i ∈ {0, . . . , N} and all α ∈ Ai : \|α\| ≥ 2, and, (2) if j = 0: wei,i = 0, for all i ∈ {1, . . . , N}, and wα,i = 0, for all i ∈ {0, . . . , N} and all α ∈ Ai : \|α\| ≥ 2. Equation (2.5) thus yields ΦΛ1(θj ⊗ vj) = N∑ i=1 θi ⊗ wei,i (j 6= 0) and ΦΛ1(1⊗ v0) = w00. (2.6) If j 6= 0, a new application of naturality, now for the Zn2 -algebra endomorphism ϕ∗R0 (R0 ∈ R, R0 6= 1) of Λ1 that is defined by ϕ∗R0 (θi) = R0θi (i 6= 0, i 6= j), ϕ∗R0 (θj) = θj and ϕ∗R0 (θ0) = 1, leads to θj ⊗ wej ,j + ∑ i 6=j θi ⊗ (R0wei,i) = θj ⊗ wej ,j + ∑ i 6=j θi ⊗ wei,i, so that ΦΛ1(θj ⊗ vj) = θj ⊗ wej ,j (j 6= 0). (2.7) The vectors w00 ∈ W0 (see (2.6)) and wej ,j ∈ Wγj (j 6= 0) (see (2.7)) are well-defined and depend obviously linearly on v0 and vj , respectively. Hence, setting φ(v0) = w00 and φ(vj) = wej ,j (j 6= 0), we define a degree 0 linear map from V to W . Moreover, since (2.4) is clearly satisfied for Λ1 and the θi⊗ vi (i ∈ {0, . . . , N}), it is satisfied for any Λ, which completes the proof of surjectivity. � Since F: Zn2Vec→ Fun0 ( Zn2Ptsop, AMod ) is fully faithful, it is essentially injective, i.e., it is injective on objects up to isomorphism. It follows that Zn2Vec can be viewed as a full subcategory of the target category of F. The above considerations lead to the following definition. Definition 2.3. A functor V∈ Fun0 ( Zn2Ptsop, AMod ) is said to be representable, if there exists V ∈ Zn2Vec, such that F(V ) is naturally isomorphic to V. As F is essentially injective, a representing object V , if it exists, is unique up to isomorphism. We therefore refer sometimes to V as “the” representing object. 8 A.J. Bruce, E. Ibarguëngoytia and N. Poncin 2.2 Cartesian Zn 2 -graded vector spaces and Cartesian Zn 2 -manifolds In the literature, the space Rp\|q is viewed, either as the trivial Zn2 -manifold Rp\|q = ( Rp, C∞Rp [[ξ]] ) with canonical Zn2 -graded formal parameters ξ, or as the Cartesian Zn2 -graded vector space Rp\|q = Rp ⊕ N⊕ j=1 Rqj , where Rp (resp., Rqj ) is the term of degree γ0 = 0 ∈ Zn2 (resp., γj ∈ Zn2 ). Observe that we use the notation R• (resp., R•), when R• is viewed as a vector space (resp., as a manifold). It can happen that we write R• for both, the vector space and the manifold, however, in these cases, the meaning is clear from the context. Further, we set q0 = p, q = (q0, q1, . . . , qN ), and \|q\| = ∑ i qi. When embedding Rqi (i ∈ {0, . . . , N}) into Rp\|q, we identify each vector of the canonical basis of Rqi with the corresponding vector of the canonical basis of R\|q\|. We denote this basis by( eik ) i,k (i ∈ I = {0, . . . , N}, k ∈ Ki = {1, . . . , qi}) and assign of course the degree γi to every vector eik. We can now write Rp\|q = N⊕ i=0 Rqi = ⊕ (i,k)∈I×Ki R eik. The dual space of Rp\|q is defined by ( Rp\|q)∨ = Hom ( Rp\|q,R ) = N⊕ i=0 Homγi ( Rp\|q,R ) , where Hom is the internal Hom of Zn2Vec, i.e., the Zn2 -graded vector space of all linear maps, and where Homγi is the vector space of all degree γi linear maps. We sometimes write HomZn2 Vec instead of Hom. The dual basis of ( eik ) i,k is defined as usual by εki ( ej` ) = δji δ k ` , so that εki is a linear map of degree γi and( Rp\|q)∨ = ⊕ (i,k)∈I×Ki R εki . Let us finally mention that any Zn2 -vector x ∈ Rp\|q reads x = ∑ j,` x ` je j ` and that εki (x) = xki , (2.8) as usual. Notice now that if M is a smooth m-dimensional real manifold and (U,ϕ) is a chart of M , the coordinate map ϕ sends any point x ∈M to ϕ(x) = (x1, . . . , xm) ∈ Rm, so that ϕi(x) = xi. (2.9) Linear Zn2 -Manifolds and Linear Actions 9 Hence, what we refer to as coordinate function xi ∈ C∞(U) is actually the function ϕi. Equa- tions (2.8) and (2.9) suggest to associate to any Zn2 -graded vector space Rp\|q a Zn2 -manifold Rp\|q with coordinate functions εki . In other words, the associated p\|q-dimensional Zn2 -manifold will be the locally Zn2 -ringed space Rp\|q = ( Rp,ORp\|q ) = ( Rp, C∞Rp [[ε1 1, . . . , ε qN N ]] ) , where C∞Rp is the standard function sheaf of Rp, where the degree γj linear maps ε1 j , . . . , ε qj j (j ∈ {1, . . . , N}) are interpreted as coordinate functions or formal parameters of degree γj , and where the degree 0 linear maps ε1 0, . . . , ε p 0 are viewed as coordinates in Rp. We often set ξ`j := ε`j (j 6= 0) and x` := ε`0. (2.10) Remark 2.4. In the following, we denote the coordinates of Rp\|q by( x`, ξ`j ) = ( xa, ξA ) = (ua), if we wish to make a distinction between the coordinates of degree 0, γ1, . . . , γN , if we distinguish between zero degree coordinates and non-zero degree ones, or if we consider all coordinates together. We refer to the category of Zn2 -graded vector spaces Rp\|q (p, q1, . . . , qN ∈ N) and degree 0 linear maps, as the category Zn2CarVec of Cartesian Zn2 -vector spaces. As just mentioned, the interpretation of the dual basis as coordinates leads naturally to a map M: Zn2CarVec 3 Rp\|q 7→ Rp\|q ∈ Zn2Man, where Zn2Man is the category of Zn2 -manifolds and corresponding morphisms. This map can easily be extended to a functor. Indeed, if L : Rp\|q → Rr\|s is a morphism in Zn2CarVec ( it is canonically represented by a block diagonal matrix L ∈ gl ( r\|s×p\|q,R )) , its dual (Zn2 -transpose) L∨ : ( Rr\|s)∨ → ( Rp\|q)∨ (which is represented by the standard transpose tL ∈ gl ( p\|q × r\|s,R )) is also a degree 0 linear map. If we set L = ( L`iik ) , where i ∈ I, ` ∈ {1, . . . , si} label the row and i ∈ I, k ∈ {1, . . . , qi} label the column, we get L∨ ( ε′`i ) = qi∑ k=1 L`iikε k i , where ( ε′`i ) i,` is the basis of ( Rr\|s)∨. When using notation (2.10), we obtain L∗ ( x′` ) := L∨ ( x′` ) = p∑ k=1 L`00k x k ∈ O0 Rp\|q(R p) (` ∈ {1, . . . , r}) (2.11) and L∗ ( ξ′`j ) := L∨ ( ξ′`j ) = qj∑ k=1 L`jjk ξ k j ∈ O γj Rp\|q (Rp) (j 6= 0, ` ∈ {1, . . . , sj}). (2.12) These pullbacks define a Zn2 -morphism L : Rp\|q → Rr\|s. This is the searched Zn2 -morphism M(L) : M ( Rp\|q) → M ( Rr\|s). Since M(L) is defined interpreting the standard transpose tL as pullback (M(L))∗ of coordinates, we have (M(M ◦ L))∗ ' tL ◦ tM ' (M(L))∗ ◦ (M(M))∗ = (M(M) ◦M(L))∗, 10 A.J. Bruce, E. Ibarguëngoytia and N. Poncin so that M respects composition. Further, it obviously respects identities. Hence, we defined a functor M. The pullbacks (2.11) and (2.12) are actually linear homogeneous Zn2 -functions, i.e., homoge- neous Zn2 -functions in Olin Rp\|q(R p) := { p∑ k=1 rk x k + N∑ j=1 qj∑ k=1 rjk ξ k j : rk, r j k ∈ R } = ( Rp\|q)∨ ⊂ ORp\|q(R p), (2.13) where the last equality is obvious because of equation (2.10). Hence, the functor M is valued in the subcategory Zn2CarMan ⊂ Zn 2Man of Cartesian Zn2 -manifolds Rp\|q (p, q1, . . . , qN ∈ N) and Zn2 - morphisms whose coordinate pullbacks are global linear functions of the source manifold that have the appropriate degree: M: Zn2CarVec→ Zn2CarMan. The inverse “vectorification functor” V of this “manifoldification functor” M is readily defined: to a Cartesian Zn2 -manifold Rp\|q we associate the Cartesian Zn2 -vector space Rp\|q, and to a linear Zn2 -morphism we associate the degree 0 linear map that is defined by the transpose of the block diagonal matrix coming from the morphism’s linear pullbacks. It is obvious that V ◦ M = M◦ V= id. Proposition 2.5. We have an isomorphism of categories M: Zn2CarVec � Zn2CarMan : V (2.14) between the full subcategory Zn2CarVec ⊂ Zn2Vec of Cartesian Zn2 -vector spaces Rp\|q and the subcategory Zn2CarMan ⊂ Zn2Man of Cartesian Zn2 -manifolds Rp\|q and Zn2 -morphisms with linear coordinate pullbacks. Remark 2.6. Let us stress that the Zn2 -vector space of linear Zn2 -functions( Rp\|q)∨ ' Olin Rp\|q(R p) ⊂ ORp\|q(R p) is of course not an algebra. In the case p = 0, we get Olin R0\|q({?}) = Λlin ⊂ OR0\|q({?}) = Λ, where {?} denotes the 0-dimensional base manifold R0 of the Zn2 -point R0\|q, where Λ = R[[θ1 1, . . . , θqNN ]] is the Zn2 -Grassmann algebra that corresponds to R0\|q, and where Λlin is the Zn2 -vector space of homogeneous degree 1 polynomials in the θ1 1, . . . , θ qN N (with vanishing term Λlin 0 of Zn2 -degree zero). We close this subsection with some observations regarding the functor of points. The Yoneda functor of points of the category Zn2Man is the fully faithful embedding Y: Zn2Man→ Fun ( Zn2Manop, Set ) . In [13], we showed that Y remains fully faithful for appropriate restrictions of the source and target of the functor category, as well as of the resulting functor category. More precisely, we proved that the functor S: Zn2Man→ Fun0 ( Zn2Ptsop, A(N)FM ) (2.15) Linear Zn2 -Manifolds and Linear Actions 11 is fully faithful. The category A(N)FM is the category of (nuclear) Fréchet manifolds over a (nuc- lear) Fréchet algebra, and the functor category is the category of those functors that send a Zn2 - Grassmann algebra Λ to a (nuclear) Fréchet Λ0-manifold, and of those natural transformations that have Λ0-smooth Λ-components. For any M ∈ Zn2Man and any R0\|m ' Λ ∈ Zn2Ptsop, we have M(Λ) := S(M)(Λ) = Y(M)(Λ) = HomZn2 Man ( R0\|m,M ) . On the other hand, the Yoneda functor of points of the category Zn2CarVec is the fully faithful embedding • : Zn2CarVec 3 Rp\|q 7→ Rp\|q := HomZn2 Vec ( −,Rp\|q) ∈ Fun ( Zn2CarVecop, Set ) . The value of this functor on R0\|m ' R0\|m ' Λ, is the subset Rp\|q(Λ) = HomZn2 Vec ( R0\|m,Rp\|q) ' HomZn2 CarMan ( R0\|m,Rp\|q ) ⊂ Rp\|q(Λ) = S ( Rp\|q ) (Λ) = HomZn2 Man ( R0\|m,Rp\|q ) ' N⊕ i=0 qi⊕ k=1 OR0\|m,γi ({?}) = N⊕ i=0 qi⊕ k=1 Λγi = N⊕ i=0 qi⊕ k=1 Λγi ⊗Reik = N⊕ i=0 Λγi⊗ qi⊕ k=1 Reik = ( Λ⊗Rp\|q) 0 = F ( Rp\|q)(Λ) = Rp\|q(Λ) ∈ Λ0Mod. (2.16) More precisely, Rp\|q(Λ) = HomZn2 Vec ( R0\|m,Rp\|q) ' N⊕ i=0 qi⊕ k=1 Olin R0\|m,γi ({?}) = ( Λlin ⊗Rp\|q) 0 ∈ Set. (2.17) Remark that, if we denote the coordinates of Rp\|q compactly by (ua), the bijection in equa- tion (2.17) sends a degree 0 linear map L to the linear pullbacks L∗(ua) of the corresponding Zn2 -morphism L = M(L). Remark 2.7. If we restrict the functor Rp\|q (resp., the functor F ( Rp\|q)) from Zn2CarVecop ' Zn2CarManop (resp., from Zn2Ptsop) to the joint subcategory Zn2CarPtsop of Zn2 -points and Zn2 - morphisms with linear coordinate pullbacks, the restricted Hom functor Rp\|q is actually a sub- functor of the restricted tensor product functor F ( Rp\|q). This observation clarifies the rela- tionship between the fully faithful “functor of points” F(•)(−) = •(−) of the full subcategory Zn2CarVec ⊂ Zn2Vec and its standard fully faithful Yoneda functor of points •(−). Indeed, we observed already that the values of the Hom functor on Zn2 -points are subsets of the values of the tensor product functor. Further, on morphisms, the values of Hom ( −,Rp\|q) are restrictions of the values of ( −⊗Rp\|q) 0 . Indeed, if HomZn2 CarPts ( R0\|n,R0\|m) 3 L ' V(L) = L ∈ HomZn2 Vec ( R0\|n,R0\|m), the morphisms Rp\|q(L) and F ( Rp\|q)(L) are defined on Rp\|q(Λ) and its supset F ( Rp\|q)(Λ), respectively. When interpreting an element K of the first as an element of the second, we use the identifications K ' K ' (K∗(ua))a ∈ N⊕ i=0 qi⊕ k=1 Λγi . 12 A.J. Bruce, E. Ibarguëngoytia and N. Poncin Similar identifications are of course required when Λ is replaced by Λ′. We thus get Rp\|q(L)(K) = HomZn2 Vec ( L,Rp\|q)(K) = K ◦ L ' K ◦ L ' ( L∗(K∗(ua)) ) a . On the other hand, we have F ( Rp\|q)(L∗)((K∗(ua))a) = ( L∗(K∗(ua)) ) a , since ( L∗ ⊗ 1 Rp\|q ) 0 ' N⊕ i=0 qi⊕ k=1 L∗, when read through the isomorphism N⊕ i=0 Λγi⊗ qi⊕ k=1 R eik ' N⊕ i=0 qi⊕ k=1 Λγi . This completes the proof of the subfunctor-statement. 2.3 Finite dimensional Zn 2 -graded vector spaces and linear Zn 2 -manifolds In this subsection, we extend equivalence (2.14) in a coordinate-free way. 2.3.1 Finite dimensional Zn 2 -graded vector spaces We focus on the full subcategory Zn2FinVec ⊂ Zn2Vec of finite dimensional Zn2 -graded vector spaces, i.e., of Zn2 -vector spaces V of finite dimension p\|q ( p ∈ N, q = (q1, . . . , qN ) ∈ N×N ) . Clearly Zn2CarVec ⊂ Zn2FinVec is a full subcategory. Above, we already used the canonical basis of Rp\|q, i.e., the basis eik = t(0 . . . 0; . . . ; 0 . . . 1 . . . 0; . . . ; 0 . . . 0), where 1 sits in position k of block i. If (bik)i,k (i ∈ I = {0, . . . , N}, k ∈ Ki = {1, . . . , qi}, deg(bik) = γi ∈ Zn2 ) is a basis of V , the degree respecting linear map b : V 3 v = ∑ i,k vki b i k 7→ ∑ i,k vki e i k = t ( v1 0, . . . , v qN N ) =: vI ∈ Rp\|q maps a basis to a basis and is thus an isomorphism of Zn2 -vector spaces. We already discussed extensively the functor of points F= F(•)(−) = •(−) of Zn2Vec. Since Zn2FinVec is a full subcategory of Zn2Vec, the functor F remains fully faithful when restricted to Zn2FinVec: Proposition 2.8. The functor of points F: Zn2FinVec → Fun0 ( Zn2Ptsop, AMod ) of the category Zn2FinVec is fully faithful. Remark 2.9. Later on, we consider linear Zn2 -manifolds and denote them sometimes using the same letter V as for Zn2 -vector spaces. We often disambiguate the concept considered by writing V in the vector space case. Linear Zn2 -Manifolds and Linear Actions 13 2.3.2 Linear Zn 2 -manifolds In this subsection, we investigate the category of linear Zn2 -manifolds, linear Zn2 -functions of its objects, as well as its functor of points. Linear Zn 2 -manifolds and their morphisms. A Zn2 -manifold of dimension p\|q is a locally Zn2 -ringed space M := ( \|M \|,OM ) that is locally isomorphic to Rp\|q. Definition 2.10. A linear Zn2 -manifold of dimension p\|q is a locally Zn2 -ringed space L =( \|L\|,OL ) that is globally isomorphic to Rp\|q, i.e., it is a Zn2 -manifold such that there exists a Zn2 -diffeomorphism h: L −→ Rp\|q. The diffeomorphism h is referred to as a linear coordinate map or a linear one-chart-atlas. We now mimic classical differential geometry and say that two linear one-chart-atlases are linearly compatible, if their union is a “linear two-chart-atlas”. In other words: Definition 2.11. Two linear coordinate maps h1, h2 : L→ Rp\|q are said to be linearly compatible, if the Zn2 -morphisms h2 ◦ h−1 1 , h1 ◦ h−1 2 : Rp\|q −→ Rp\|q have linear coordinate pullbacks, i.e., if they are Zn2CarMan-morphisms. Linear compatibility is an equivalence relation on linear one-chart-atlases. There is a 1 : 1 correspondence between equivalence classes of linear one-chart-atlases and maximal linear atla- ses, i.e., the unions of all linear one-chart-atlases of an equivalence class. For simplicity, we refer to a maximal linear atlas as a linear atlas. Just as a classical smooth manifold is a set that admits an atlas, or, better, a set endowed with an equivalence class of atlases, a linear Zn2 -manifold is a locally Zn2 -ringed space L equipped with a linear atlas (L,hα)α. We continue working in analogy with differential geometry and define a linear Zn2 -morphism between linear Zn2 -manifolds as a locally Zn2 -ringed space morphism, or, equivalently, a Zn2 - morphism, with linear coordinate form: Definition 2.12. Let L and L′ be two linear Zn2 -manifolds of dimension p\|q and r\|s, respectively. A Zn2 -morphism φ : L→ L′ is linear, if there exist linear coordinate maps h: L→ Rp\|q and k: L′ → Rr\|s in the linear atlases of L and L′, such that the Zn2 -morphism k ◦ φ ◦ h−1 : Rp\|q → Rr\|s has linear coordinate pullbacks. It follows that any linear coordinate map h of the linear atlas of a linear Zn2 -manifold L, is a linear Zn2 -morphism between the linear Zn2 -manifolds L and Rp\|q. This justifies the name “linear coordinate map”. Further, the inverse h−1 of h is a linear Zn2 -morphism. Proposition 2.13. If φ : L→ L′ is a linear Zn2 -morphism, then, for any linear coordinate maps (L,h′) and (L′, k′) of the linear atlases of L and L′, respectively, the Zn2 -morphism k′ ◦ φ ◦ h′−1 has linear coordinate pullbacks. 14 A.J. Bruce, E. Ibarguëngoytia and N. Poncin Proof. We use the notations of Definition 2.12 and Proposition 2.13. Since k′ ◦ φ ◦ h′−1 = ( k′ ◦ k−1 ) ◦ ( k ◦ φ ◦ h−1 ) ◦ ( h ◦ h′−1 ) and each parenthesis of the RHS has linear pullbacks, their composite has linear pullbacks as well. � Proposition 2.14. Linear Zn2 -manifolds and linear Zn2 -morphisms form a subcategory Zn2LinMan ⊂ Zn2Man of the category of Zn2 -manifolds. Further, Cartesian Zn2 -manifolds and Zn2 -morphisms with linear coordinate pullbacks form a full subcategory Zn2CarMan ⊂ Zn2LinMan. Proof. If φ : L → L′ and ψ : L′ → L′′ are linear Zn2 -morphisms, the composite Zn2 -morphism is linear as well. Indeed, if k◦φ◦h−1 and q◦ψ◦p−1 have linear pullbacks, then q◦ψ◦k−1 has linear pullbacks and so has q ◦ (ψ ◦ φ) ◦ h−1. Further, for any linear Zn2 -manifold L, the Zn2 -identity map idL is linear, as for any linear coordinate map h, we have h ◦ idL ◦h−1 = idRp\|q . The second statement is obvious. � Sheaf of linear Zn 2 -functions. Definition 2.15. Let L ∈ Zn2LinMan be of dimension p\|q and let \|U \| ⊂ \|L\| be open. A Zn2 - function f ∈ OL(\|U \|) is a linear Zn2 -function, if there exists a linear coordinate map h: L→ Rp\|q, such that (h∗)−1(f) ∈ Olin Rp\|q ( \|h\|(\|U \|) ) . We denote the subset of all linear Zn2 -functions of OL(\|U \|) by Olin L (\|U \|). The subset Olin Rp\|q ( \|h\|(\|U \|) ) is defined in the obvious way. If f ∈ Olin L (\|U \|), then for any chart (L,h′) of the linear atlas of L, we have (h′∗)−1(f) ∈ Olin Rp\|q ( \|h′\|(\|U \|) ) . This follows from the equation (h′∗)−1(f) = ( h ◦ h′−1 )∗( (h∗)−1(f) ) and the compatibility of the two charts. As Olin L (\|U \|) ⊂ OL(\|U \|) is visibly closed for linear combinations, it is a vector subspace of OL(\|U \|). Hence, the intersection Olin L,γi(\|U \|) := Olin L (\|U \|) ∩ OL,γi(\|U \|) ⊂ OL(\|U \|) is also a vector subspace. We thus get vector subspaces Olin L,γi (\|U \|) ⊂ Olin L (\|U \|), so their direct sum over i is a vector subspace as well. Since any f ∈ Olin L (\|U \|) reads uniquely as f = N∑ i=0 fi ( fi ∈ OL,γi(\|U \|) ) , we get (h∗)−1f0 + ∑ j (h∗)−1fj = (h∗)−1f = ∑ ` r` x ` + ∑ j ∑ ` rj` ξ ` j . As (h∗)−1 is Zn2 -degree preserving, we find that fi ∈ Olin L,γi (\|U \|), so that Olin L (\|U \|) = ⊕ i Olin L,γi(\|U \|) ∈ Zn2Vec. Linear Zn2 -Manifolds and Linear Actions 15 Remark 2.16. Observe that: (i) For any open subset \|U \| ⊂ \|L\| and any linear coordinate map h: L→ Rp\|q, the map h∗ : Olin Rp\|q(\|h\|(\|U \|))→ Olin L (\|U \|) is an isomorphism of Zn2 -vector spaces of dimension p\|q. (ii) The restriction maps and the gluing property of OL endow Olin L with a sheaf of Zn2 -vector spaces structure. (iii) A Zn2 -morphism φ : L→ L′ between linear Zn2 -manifolds is itself linear, if and only if φ∗ is a degree respecting linear map φ∗ : Olin L′ (\|L′\|)→ Olin L (\|L\|). It is straightforward to check the first two statements. For the third one, let L (resp., L′) be of dimension p\|q (resp., r\|s) and denote the coordinates of the corresponding Cartesian Zn2 - manifold by ua = ( xa, ξA ) ( resp., vb = ( yb, ηB )) . The morphism φ is linear, if and only if there exist linear coordinate maps (L,h) and (L′, k), such that k ◦ φ ◦ h−1 has linear coordinate pullbacks, i.e., such that( (h∗)−1 ◦ φ∗ ◦ k∗ ) (vb) ∈ Olin Rp\|q(R p). (2.18) On the other hand, in view of the first item of the previous remark, the condition φ∗ ( Olin L′ (\|L′\|) ) ⊂ Olin L (\|L\|) of the third item is equivalent to asking that (φ∗ ◦ k∗) (∑ b rbv b ) ∈ h∗ ( Olin Rp\|q(R p) ) . (2.19) The conditions (2.18) and (2.19) are visibly equivalent. Functor of points of Zn 2 LinMan. We start with the following Proposition 2.17. For any linear Zn2 -manifold L (of dimension p\|q) and any Zn2 -Grassmann algebra Λ ' R0\|m, the set L(Λ) := HomZn2 Man ( R0\|m, L ) ' HomZn2 Alg(OL(\|L\|),Λ) of Λ-points of L admits a unique Fréchet Λ0-module structure, such that, for any chart h: L→ Rp\|q of the linear atlas of L, the induced map hΛ : L(Λ) 3 x∗ 7→ x∗ ◦ h∗ ∈ Rp\|q(Λ) is a Fréchet Λ0-module isomorphism. The definition of the category FAMod of Fréchet modules over Fréchet algebras can be found in Appendix A. In the preceding proposition, it is implicit that the (unital) Fréchet algebra morphism that is associated to hΛ is idΛ0 . 16 A.J. Bruce, E. Ibarguëngoytia and N. Poncin Proof. Let Λ ∈ Zn2GrAlg. In view of the fundamental theorem of Zn2 -morphisms, there is a 1 : 1 correspondence between the Λ-points x∗ of Rp\|q and the (p+ \|q\|)-tuples x∗ ' ( xaΛ, ξ A Λ ) ∈ Λ×p0 × Λ×q1γ1 × · · · × Λ×qNγN (we used this correspondence already in equation (2.16)). Indeed, the algebra Λ is the Zn2 - commutative nuclear Fréchet R-algebra of global Zn2 -functions of some R0\|m (in particular, the degree zero term Λ0 of Λ is a commutative nuclear Fréchet algebra). Hence, all its homogeneous subspaces Λγi (i ∈ {0, . . . , N}, γ0 = 0) are nuclear Fréchet vector spaces. Since any product (resp., any countable product) of nuclear (resp., Fréchet) spaces is nuclear (resp., Fréchet), the set Rp\|q(Λ) of Λ-points of Rp\|q is a nuclear Fréchet space. The latter statements can be found in [14]. The Fréchet Λ0-module structure on Rp\|q(Λ) is then defined by / : Λ0 × Rp\|q(Λ) 3 (a, x∗) 7→ a / x∗ := ( a · xaΛ, a · ξAΛ ) ∈ Rp\|q(Λ). (2.20) Since this action (which is compatible with addition in Λ0 and addition in Rp\|q(Λ)) is defined using the continuous associative multiplication · : Λγi × Λγj → Λγi+γj of the Fréchet algebra Λ, it is (jointly) continuous. We now define the Λ0-module structure on L(Λ). Observe first that, for any chart map h: L � Rp\|q : h−1 of the linear atlas of L, the induced maps hΛ : L(Λ) � Rp\|q(Λ) : ( h−1 ) Λ are inverse maps: ( h−1 ) Λ = (hΛ)−1 =: h−1 Λ . For K ∈ N \ {0}, k ∈ {1, . . . ,K}, ak ∈ Λ0, and y∗k ∈ L(Λ), we set∑ k ak ? y∗k := h−1 Λ (∑ k ak / hΛ(y∗k) ) ∈ L(Λ). This defines a Λ0-module structure on L(Λ) that makes hΛ a Λ0-module isomorphism. The Λ0-module structures L(Λ)h and L(Λ)k that are implemented by h and another chart k of the linear atlas, respectively, are related by the Λ0-module isomorphism k−1 Λ ◦ hΛ : L(Λ)h → L(Λ)k. Hence, the Λ0-module structure on L(Λ) is well-defined. In order to get a Fréchet structure on the real vector space L(Λ) that we just defined, we need a countable and separating family of seminorms (pn)n∈N, such that any sequence in L(Λ) that is Cauchy for every pn, converges for every pn to a fixed vector (i.e., a vector that does not depend on n). We define this family (of course) by transferring to L(Λ) the analogous family (ρn)n∈N of the Fréchet vector space Rp\|q(Λ) (see [14, Theorem 14]). In other words, for each y∗ ∈ L(Λ), we set pn(y∗) := ρn(hΛ(y∗)) ∈ R+. It is straightforwardly checked that (pn)n∈N is a countable family of seminorms that has the required properties. Moreover, the vector space isomorphism hΛ is an isomorphism of Fréchet vector spaces, i.e., a continuous linear map with a continuous inverse. We show that hΛ is continuous for the seminorm topologies implemented by the pn and the ρn, i.e., that, for all n ∈ N, there exist m ∈ N and C > 0, such that ρn(hΛ(y∗)) ≤ C pm(y∗), for all y∗ ∈ L(Λ). This requirement is of course satisfied. Hence, the composite k−1 Λ ◦ hΛ of isomorphisms of Fréchet spaces is an isomorphism of Fréchet spaces, so that the Fréchet space structure on L(Λ) is well-defined. Linear Zn2 -Manifolds and Linear Actions 17 The Λ0-module structure and the Fréchet vector space structure on L(Λ) combine into a Fréchet Λ0-module structure, if they are compatible, i.e., if the Λ0-action ? : Λ0 × L(Λ) 3 (a, y∗) 7→ h−1 Λ (a / hΛ(y∗)) ∈ L(Λ) (2.21) is continuous. The condition is obviously satisfied as this action is the composite of the continu- ous maps id×hΛ, / and h−1 Λ . Further, the map hΛ is clearly a Fréchet Λ0-module isomorphism, for any h in the linear atlas of L. There is obviously no other Fréchet Λ0-module structure on L(Λ) with that property. Indeed, if there were, it would be isomorphic to the Fréchet Λ0-module structure on Rp\|q(Λ), hence isomorphic to the Fréchet Λ0-module structure that we just constructed. � In the following, we denote the Λ0-action ? on L(Λ) by simple juxtaposition, i.e., we write ay∗ instead of a ? y∗. To proceed, we need some preparation. Let Fun0 ( Zn2Ptsop, FAMod ) be the category of functors F , whose values F (Λ) are Fréchet Λ0- modules, and of natural transformations β, whose Λ-components βΛ are continuous Λ0-linear maps. We already used above the category Fun0 ( Zn2Ptsop, AFM ) of functors, whose values are Fréchet Λ0-manifolds, and of natural transformations, whose components are Λ0-smooth maps. Proposition 2.18. The category Fun0 ( Zn2Ptsop, FAMod ) is a subcategory of the category Fun0 ( Zn2Ptsop, AFM ) . Proof. Observe first that composition of natural transformations (resp., identities of functors) is (resp., are) induced by composition (resp., identities) in the target category of the functors con- sidered, which is (resp., are) in both target categories the standard set-theoretical composition (resp., identities). Hence, composition and identities are the same in both functor categories. However, we still have to show that objects (resp., morphisms) of the first functor category are objects (resp., morphisms) of the second. Let F be a functor with target FAMod. Since a Fréchet Λ0-module (i.e., a Fréchet vector space with a (compatible) continuous Λ0-action) is clearly a Fréchet Λ0-manifold, the functor F sends Zn2 -Grassmann algebras Λ to Fréchet Λ0-manifolds F (Λ). Let now ϕ∗ : Λ → Λ′ be a morphism of Zn2 -algebras. As F (ϕ∗) : F (Λ) → F (Λ′) is a morphism between Fréchet modules over the Fréchet algebras Λ0 and Λ′0, respectively, it is continuous and it has an associated continuous (unital, R-) algebra morphism ψ : Λ0 → Λ′0, such that F (ϕ∗)(av + a′v′) = ψ(a)F (ϕ∗)(v) + ψ(a′)F (ϕ∗)(v′), (2.22) for all a, a′ ∈ Λ0 and all v, v′ ∈ F (Λ). We must show that F (ϕ∗) is a morphism between Fréchet manifolds over Λ0 and Λ′0, respectively, i.e., we must show that F (ϕ∗) is smooth and has first order derivatives that are linear in the sense of (2.22) (see [13]). Since, for any t ∈ R, we have ψ(t) = tψ(1) = t, it follows from (2.22) that dxF (ϕ∗)(v) := lim t→0 1 t (F (ϕ∗)(x + tv)− F (ϕ∗)(x)) = F (ϕ∗)(v) and dk+1 x F (ϕ∗)(v1, . . . , vk+1) = 0, for any x, v, v1, . . . , vk+1 ∈ F (Λ) and any k ≥ 1. Hence, all derivatives exist everywhere and are (jointly) continuous. This implies that F (ϕ∗) has the required properties, so that F is a functor with target AFM. 18 A.J. Bruce, E. Ibarguëngoytia and N. Poncin As for morphisms, let η : F → G be a natural transformation between functors valued in FAMod. Its Λ-components ηΛ : F (Λ) → G(Λ) are continuous and Λ0-linear maps. Repeat- ing the proof given in the preceding paragraph for F (ϕ∗), we obtain that ηΛ is Λ0-smooth, i.e., is smooth and has Λ0-linear first order derivatives. Therefore, the morphism η of the functor category with target FAMod is a morphism of the functor category with target AFM. � Since Zn2LinMan ⊂ Zn2Man and Fun0 ( Zn2Ptsop, FAMod ) ⊂ Fun0 ( Zn2Ptsop, AFM ) are subcategories, we expect that: Proposition 2.19. The functor S: Zn2Man→ Fun0 ( Zn2Ptsop, AFM ) (see equation (2.15)) restricts to a functor S: Zn2LinMan→ Fun0 ( Zn2Ptsop, FAMod ) . Proof. We have to explain why S sends linear Zn2 -manifolds and linear Zn2 -morphisms to objects and morphisms, respectively, of the target subcategory. Let L ∈ Zn2LinMan. The functor S(L) is an object of the functor category with target AFM. Since composition and identities are the same in both target categories, it suffices to show that, for any Zn2 -Grassmann algebra Λ, the value S(L)(Λ) = L(Λ) is a Fréchet Λ0-module and that, for any Zn2 -algebra morphism ϕ∗ : Λ→ Λ′, the morphism L(ϕ∗) : L(Λ) 3 y∗ 7→ ϕ∗ ◦ y∗ ∈ L(Λ′) is a morphism of the category FAMod. The first of the preceding conditions holds in view of Pro- position 2.17. We start proving the second condition for L = Rp\|q. Since Rp\|q(ϕ∗) is a morphism of AFM, it is smooth, hence, continuous. Further, omitting the summation symbols and using our standard notation, we get Rp\|q(ϕ∗) ( ak / x∗k ) = Rp\|q(ϕ∗) ( ak · xaΛ,k, ak · ξAΛ,k ) = ( ϕ∗ ( ak ) · ϕ∗(xaΛ,k), ϕ∗ ( ak ) · ϕ∗ ( ξAΛ,k )) = ϕ∗ ( ak ) / Rp\|q(ϕ∗)(x∗k). It now suffices to recall that the Zn2 -algebra morphism ϕ∗ is the pullback ϕ∗? over the whole base manifold {?} of a Zn2 -morphism ϕ : R0\|m′ → R0\|m, and that all pullbacks of Zn2 -morphisms are continuous, so that the restriction ϕ∗ : Λ0 → Λ′0 is a continuous algebra morphism. We are now able to prove that the second condition holds also for an arbitrary linear Zn2 -manifold L. Indeed, since ϕ∗ : Λ → Λ′ is a morphism of Zn2 -algebras, the map L(ϕ∗) : L(Λ) → L(Λ′) is a morphism of AFM, hence, it is continuous. Recall now that any chart h: L→ Rp\|q is a Zn2 -morphism, so that S(h) : L(−) → Rp\|q(−) is a natural transformation h− with Λ-components hΛ : L(Λ) → Rp\|q(Λ) that are Fréchet Λ0-module isomorphisms in view of Proposition 2.17. Naturality of h− implies that hΛ′ ◦ L(ϕ∗) = Rp\|q(ϕ∗) ◦ hΛ, and, due to invertibility, that L(ϕ∗) = h−1 Λ′ ◦ R p\|q(ϕ∗) ◦ hΛ. Linear Zn2 -Manifolds and Linear Actions 19 Definition (2.21) yields L(ϕ∗) ( ak ? y∗k ) = ( h−1 Λ′ ◦ R p\|q(ϕ∗) ◦ hΛ )( h−1 Λ ( ak / hΛ(y∗k) )) = ϕ∗ ( ak ) ? L(ϕ∗)(y∗k) (we used our standard notation). Hence, the functor S(L) is an object of the functor category with target FAMod. As for morphisms, we consider a linear Zn2 -morphism φ : L→ L′ and will prove that S(φ), which is a natural transformation φ− of the functor category with target AFM, i.e., a natural transformation with Λ0-smooth Λ-components φΛ, has actually continuous (but this results from Λ0-smoothness) Λ0-linear components. Let p\|q (resp., r\|s) be the dimension of L (resp., of L′). We first discuss the case of a linear Zn2 -morphism Φ: Rp\|q → Rr\|s between the corresponding Cartesian Zn2 -manifolds with canonical coordinates ( xa, ξA ) and( yb, ηB ) , respectively. We know from [13] that, if the Zn2 -morphism (resp., the linear Zn2 - morphism) Φ reads Φ∗ ( yb ) = ∑ \|α\|≥0 Φb α(x) ξα ( resp., = ∑ a Lbax a ) , (2.23a) Φ∗ ( ηB ) = ∑ \|α\|>0 ΦB α (x) ξα ( resp., = ∑ A LBAξ A ) (2.23b) (where the right-hand sides have the appropriate degree and where the coefficients L∗∗ are real numbers), then the Λ-component ΦΛ associates to the Λ-point x∗ ' ( xaΛ; ξAΛ ) = ( xa\|\|, x̊ a Λ; ξAΛ ) of Rp\|q(Λ), the Λ-point x∗ ◦ Φ∗ ' ( ybΛ; ηBΛ ) of Rr\|s(Λ) that is given by ybΛ = ∑ \|α\|≥0 ∑ \|β\|≥0 1 β! ( ∂βxΦb α )( x\|\| ) x̊βΛ ξ α Λ ( resp., = ∑ a Lbax a Λ ) , (2.24a) ηBΛ = ∑ \|α\|>0 ∑ \|β\|≥0 1 β! ( ∂βxΦB α )( x\|\| ) x̊βΛ ξ α Λ ( resp., = ∑ A LBAξ A Λ ) . (2.24b) Here, we used the obvious decomposition Λ = R × Λ̊ and wrote xaΛ = (xa\|\|, x̊ a Λ). The particular linear versions of equations (2.24a) and (2.24b) (in parentheses), show that the component ΦΛ is Λ0-linear, as needed. In the general case of a linear Zn2 -morphism φ : L → L′, the Zn2 -morphism Φ := k ◦ φ ◦ h−1 : Rp\|q → Rr\|s has linear coordinate pullbacks Φ∗ ( yb ) and Φ∗ ( ηB ) (and is thus a linear Zn2 - morphism), for any charts h and k of L and L′, respectively. Since φ = k−1 ◦ Φ ◦ h, we have φΛ = k−1 Λ ◦ ΦΛ ◦ hΛ and, in view of Proposition 2.17 and the result of the preceding paragraph, all three factors of the RHS are Λ0-linear. Finally, the natural transformation S(φ) is a natural transformation of the functor category with target FAMod. � Theorem 2.20. The functor of points S: Zn2LinMan→ Fun0 ( Zn2Ptsop, FAMod ) of the category Zn2LinMan is fully faithful. 20 A.J. Bruce, E. Ibarguëngoytia and N. Poncin Proof. We need to prove that the map SL,L′ : HomZn2 LinMan(L, L ′) 3 φ 7→ φ− ∈ HomFun0(Zn2 Ptsop,FAMod)(L(−), L′(−)) is bijective, for all linear Zn2 -manifolds L, L′. Since S is the restriction of the fully faithful functor S: Zn2Man → Fun0 ( Zn2Ptsop, AFM ) , the map SL,L′ is injective. To prove that SL,L′ is also surjective, it actually suffices to show that the property holds for Cartesian Zn2 -manifolds. Indeed, in this case, if η : L(−) → L′(−) is a natural transformation of Fun0 ( Zn2Ptsop, FAMod ) , then k− ◦ η ◦ h−1 − is a natural transformation in the same category from Rp\|q(−) to Rr\|s(−), and this transformation is implemented by a linear Zn2 -morphism ϕ : Rp\|q → Rr\|s. It follows that η = k−1 − ◦ ϕ− ◦ h− = (k−1 ◦ ϕ ◦ h)−, where the latter composite is a linear Zn2 -morphism. Let now H : Rp\|q(−)→ Rr\|s(−) be a natural transformation of Fun0 ( Zn2Ptsop, FAMod ) , hence, a natural transformation of Fun0 ( Zn2Ptsop, AFM ) . We know from [13] that H is implemented by a Zn2 -morphism Φ: Rp\|q → Rr\|s, but we still have to prove that this morphism is linear. It follows from equations (2.24a) and (2.24b) that HΛ = ΦΛ is given by ybΛ = ∑ \|α\|≥0 ∑ \|β\|≥0 F bαβ ( x\|\| ) x̊βΛ ξ α Λ, (2.25a) ηBΛ = ∑ \|α\|>0 ∑ \|β\|≥0 FBαβ ( x\|\| ) x̊βΛ ξ α Λ, (2.25b) where we set F ∗αβ(x) := 1 β! ∂βxΦ∗α ∈ C∞(Rp) (2.26) (the Φ∗α ∈ C∞(Rp) are the coefficients of the coordinate pullbacks by Φ, see equations (2.23a) and (2.23b)), and where the RHS-s have of course the same Zn2 -degree as the corresponding coordinates of Rr\|s. Since HΛ is Λ0-linear, we have∑ α ∑ β F ∗αβ ( r x\|\| ) r\|α\|+\|β\|x̊βΛ ξ α Λ = r ∑ α ∑ β F ∗αβ ( x\|\| ) x̊βΛ ξ α Λ, i.e., r\|α\|+\|β\|F ∗αβ ( r x\|\| ) = r F ∗αβ ( x\|\| ) , for any r ∈ R>0 ⊂ Λ0, any α, β and for any x\|\| ∈ Rp. When deriving with respect to r, we obtain r\|α\|+\|β\|−1 ( (\|α\|+ \|β\|)F ∗αβ ( rx\|\| ) + r p∑ a=1 xa\|\| ( ∂xa\|\|F ∗ αβ )( rx\|\| )) = F ∗αβ ( x\|\| ) , so that setting r = 1 yields p∑ a=1 xa\|\|∂xa\|\|F ∗ αβ = (1− n)F ∗αβ ( x\|\| ) (n := \|α\|+ \|β\| ∈ N), (2.27) again for all α, β and all x\|\| ∈ Rp. Linear Zn2 -Manifolds and Linear Actions 21 Recall now that Euler’s homogeneous function theorem states that, if F ∈ C1(Rp\{0}), then, for any ν ∈ R, we have p∑ a=1 xa∂xaF = νF (x), ∀x ∈ Rp \ {0} is equivalent to F (rx) = rνF (x), ∀r > 0, ∀x ∈ Rp \ {0}. In view of (2.27), we thus get F ∗αβ ( rx\|\| ) = r1−nF ∗αβ ( x\|\| ) , ∀r > 0, ∀x\|\| ∈ Rp, (2.28) where we could extend the equality from Rp \ {0} to Rp due to continuity. If r tends to 0+, the limit of the LHS is F ∗αβ(0) ∈ R and, for n = 0 (resp., n = 1; resp., n ≥ 2), the limit of the RHS is 0 ( resp., F ∗αβ ( x\|\| ) ; resp., +∞ · F ∗αβ ( x\|\| )) . In the case n ≥ 2, we conclude that F ∗αβ ( x\|\| ) = 0, ∀x\|\| ∈ Rp, ∀α, β : \|α\|+ \|β\| ≥ 2. (2.29) For n = 0, we get F ∗00(0) = 0. Observe that α = β = 0 is only possible in equation (2.25a). Differentiating (2.28), in the case n = 0, with respect to any component xa\|\| of x\|\| and simplifying by r, we obtain( ∂xa\|\|F b 00 )( rx\|\| ) = ∂xa\|\|F b 00 ( x\|\| ) , and taking the limit r → 0+, we get ∂xa\|\|F b 00 ( x\|\| ) = ∂xa\|\|F b 00(0) =: Lba ∈ R. Integration yields F b00 ( x\|\| ) = ∑ a Lbax a \|\|, ∀x\|\| ∈ Rp, ∀b, (2.30) as F b00(0) = 0. In the remaining case n = \|α\| + \|β\| = 1, we have necessarily α = 0 and β = ea, or α = eA and β = 0 (the e∗ are of course the vectors of the canonical basis of Rp and R\|q\|, respectively). For Zn2 -degree reasons, the first (resp., second) possibility is incompatible with equation (2.25b) (resp., equation (2.25a)). Hence, the only terms in (2.25a) that still need being investigated are the terms (α, β) = (0, ea). It follows from equation (2.28) and its limit r → 0+ (see above) that F b0 ea ( x\|\| ) = Kb a, where we set Kb a := F b0 ea(0) ∈ R. However, equations (2.26) and (2.30) imply that Kb a = F b0 ea ( x\|\| ) = ∂xa\|\|F b 00 ( x\|\| ) = Lba, so that F b0 ea ( x\|\| ) = Lba ∈ R, ∀x\|\| ∈ Rp, ∀a, b. (2.31) 22 A.J. Bruce, E. Ibarguëngoytia and N. Poncin In equation (2.25b), the only terms that still need being investigated are the terms (α, β) = (eA, 0). Using again the limit r → 0+ of equation (2.28), we find FBeA0 ( x\|\| ) = LBA , ∀x\|\| ∈ Rp, ∀A,B, (2.32) where we wrote LBA instead of FBeA0(0). When combining now the results of equations (2.29), (2.30), (2.31), and (2.32), we see that equations (2.25a) and (2.25b) reduce to ybΛ = ∑ a Lba ( xa\|\| + x̊aΛ ) and ηBΛ = ∑ A LBA ξ A Λ and that the Zn2 -morphism Φ that induces the natural transformation H is defined by the coordinate pullbacks Φ∗ ( yb ) = ∑ a Lba x a and Φ∗ ( ηB ) = ∑ A LBA ξ A, i.e., that Φ is linear (see (2.23a), (2.23b), (2.24a), and (2.24b)). � 2.3.3 Isomorphism between finite dimensional Zn 2 -graded vector spaces and linear Zn 2 -manifolds In this subsection, we extend the isomorphism M: Zn2CarVec Zn2CarMan : V of Proposition 2.5 between the full subcategories Zn2CarVec ⊂ Zn2FinVec and Zn2CarMan ⊂ Zn2LinMan, to an isomorphism M: Zn2FinVec Zn2LinMan : V. Zn 2 -symmetric tensor algebra. We start with some remarks on tensor and Zn2 -symmetric tensor algebras over a (finite dimensional) Zn2 -vector space (see [36] and [9]). Let V = N⊕ i=0 Vi := N⊕ i=0 Vγi ∈ Zn2FinVec be of dimension p\|q. The Zn2 -symmetric tensor algebra of V is defined exactly as in the non- graded case, as the quotient of the Zn2 -graded associative unital tensor algebra of V by the homogeneous ideal Ī = ( vi ⊗ vj − (−1)〈γi,γj〉vj ⊗ vi : vi ∈ Vi, vj ∈ Vj ) . More precisely, for k ≥ 2, we have V ⊗k = N⊕ i1,...,ik=0 Vi1 ⊗ · · · ⊗ Vik = ⊕ i1≤···≤ik Vi1,...,ik := ⊕ i1≤···≤ik ( ⊕ σ∈Perm Vσi1 ⊗ · · · ⊗ Vσik ) , where Perm is the set of all permutations of i1 ≤ · · · ≤ ik. For instance, if n = 1, i.e., in the standard super case, the space V ⊗3 is the direct sum of the tensor products whose three factors Linear Zn2 -Manifolds and Linear Actions 23 have the subscripts 000, 001, 010, 011, 100, 101, 110, 111. The notation we just introduced means that we write V ⊗3 = V000 ⊕ V001 ⊕ V011 ⊕ V111, where we used the lexicographical order and where V000 = V0 ⊗ V0 ⊗ V0, V001 = V0 ⊗ V0 ⊗ V1 ⊕ V0 ⊗ V1 ⊗ V0 ⊕ V1 ⊗ V0 ⊗ V0, et cetera. Further, as we are dealing with formal power series in this paper, we define the Zn2 -graded tensor algebra of V by TV := ΠkV ⊗k, where Πk means that we consider not only finite sums of tensors of different tensor degrees, but full sequences of such tensors. The vector space structure on such sequences is obvious and the algebra structure is defined exactly as in the standard case. Indeed, for T k ∈ V ⊗k and U ` ∈ V ⊗`, we have T k⊗U ` ∈ V ⊗(k+`) and we just extend this tensor product by linearity. In other words, if T = ∞∑ k=0 T k ∈ TV and U = ∞∑ `=0 U ` ∈ TV, we set T ⊗ U = ∑ k ∑ ` T k ⊗ U ` = ∑ m ∑ k+`=m T k ⊗ U ` ∈ TV. (2.33) It is clear that the just defined tensor multiplication endows TV with a Zn2 -graded algebra structure. Indeed, since V ⊗k = ⊕ i1≤···≤ik Vi1,...,ik = N⊕ p=0 ⊕ i1≤···≤ik∑ j γij=γp Vi1,...,ik =: N⊕ p=0 ( V ⊗k ) p is visibly a Zn2 -graded vector space, the space TV is itself Zn2 -graded: TV = Πk N⊕ p=0 ( V ⊗k ) p = N⊕ p=0 Πk ( V ⊗k ) p =: N⊕ p=0 ( TV ) p . Now, if T ∈ ( TV ) p and U ∈ ( TV ) q , we have T k ∈ ( V ⊗k ) p and U ` ∈ ( V ⊗` ) q , so that T ⊗ U ∈( TV ) p+q (where p+ q means γp + γq), which shows that TV is a Zn2 -graded (associative unital) algebra (over R), as announced. The ideal Ī is homogeneous with respect to the decomposition TV = Πk ⊕ i1≤···≤ik Vi1,...,ik , i.e., Ī = Πk(≥2) ⊕ i1≤···≤ik ( Vi1,...,ik ∩ Ī ) . Therefore, the Zn2 -symmetric tensor algebra of V is given by S̄V = Πk ⊕ i1≤···≤ik Vi1,...,ik/ ( Vi1,...,ik ∩ Ī ) =: Πk ⊕ i1≤···≤ik Vi1 � · · · � Vik = N⊕ p=0 Πk ⊕ i1≤···≤ik∑ j γij=γp Vi1 � · · · � Vik , (2.34) 24 A.J. Bruce, E. Ibarguëngoytia and N. Poncin see [9]. We denote by � the Zn2 -commutative multiplication that is induced on S̄V by the multiplication ⊗ of TV . By definition, we have, for [T ] ∈ (S̄V )γi and [U ] ∈ (S̄V )γj (obvious notation), [T ]� [U ] = [T ⊗ U ] = (−1)〈γi,γj〉[U ]� [T ]. For instance, if vi ∈ Vi ⊂ (S̄V )γi , vj ∈ Vj ⊂ (S̄V )γj and if i ≤ j, we get vi � vj = [vi ⊗ vj ] = [ (−1)〈γi,γj〉vj ⊗ vi ] = (−1)〈γi,γj〉vj � vi ∈ Vi � Vj . (2.35) Notice further that, if i < j, the linear map ι : Vi ⊗ Vj 3 T 7→ [T ] ∈ Vi � Vj (ι : Vi ⊗ Vj 3 vi ⊗ vj 7→ vi � vj ∈ Vi � Vj) (2.36) is a vector space isomorphism. Indeed, if [T ] = 0, the representative T is a vector in (Vi ⊗ Vj ⊕ Vj ⊗ Vi) ∩ Ī and is therefore a finite sum of generators of Ī: (−1)〈γi,γj〉 ∑ k vkj ⊗ vki = ∑ k vki ⊗ vkj − T ∈ (Vi ⊗ Vj) ∩ (Vj ⊗ Vi) = {0}. (2.37) It follows that the LHS of equation (2.37) vanishes; hence, the first term of the RHS vanishes, due to the isomorphism Vi ⊗ Vj ' Vj ⊗ Vi, and thus T vanishes as well. In order to show that ι is also surjective, consider an arbitrary vector in Vi � Vj . It reads [T ] = [∑ k vki ⊗ vkj + ∑ ` w`j ⊗ w`i ] . The image by ι of∑ k vki ⊗ vkj + (−1)〈γi,γj〉 ∑ ` w`i ⊗ w`j ∈ Vi ⊗ Vj is the corresponding class. This class coincides with [T ], since the difference of the representatives is a vector of Ī. It follows that, for n = 2 for instance, we have in particular V00 � V00 � V01 � V10 � V10 � V10 � V11 ' �2V00 ⊗ V01 ⊗�3V10 ⊗ V11 ' ∨2V00 ⊗ V01 ⊗ ∧3V10 ⊗ V11, (2.38) where ∨ (resp., ∧) is the symmetric (resp., antisymmetric) tensor product. Moreover, if the (finite dimensional) vector space V has dimension q0\|q1, q2, q3, we denote the vectors of its basis (in accordance with the notation we adopted earlier in this text) by bij , where i ∈ {0, 1, 2, 3} refers to the degrees 00, 01, 10, 11 and where j ∈ {1, . . . , qi}. The basis of the Zn2 -symmetric tensor product (2.38) is then made of the tensors b0j1 ∨ b 0 j2 ⊗ b 1 j3 ⊗ b 2 j4 ∧ b 2 j5 ∧ b 2 j6 ⊗ b 3 j7 (j1 ≤ j2 and j4 < j5 < j6), which can also be written b0j1 � b 0 j2 � b 1 j3 � b 2 j4 � b 2 j5 � b 2 j6 � b 3 j7 (j1 ≤ j2 and j4 < j5 < j6) (see (2.36)). More generally, the basis of Vi1�· · ·�Vik (i1 ≤ · · · ≤ ik) is made of the tensors bi1j1 � · · · � b ik jk (2.39) Linear Zn2 -Manifolds and Linear Actions 25 (j` ≤ j`+1 (resp., <), if i` = i`+1 and 〈γi` , γi`+1 〉 even (resp., odd)). To refer to the previous condition regarding the j-s, we write in the following j1 � · · ·� jk. Observe also that SkV = ⊕ i1≤···≤ik Vi1 � · · · � Vik = Sk ⊕ i Vi = (⊗ i SVi )k , as well as that, in order to define a linear map on Vi1 � · · ·�Vik (see (2.38)), it suffices to define a k-linear map on Vi1 × · · · × Vik that is Zn2 -commutative in the variables i` = · · · = im. Manifoldification functor. If V is a Zn2 -graded vector space, its dual V ∨ is defined by V ∨ := Hom(V,R) = N⊕ i=0 Homγi(V,R) = N⊕ i=0 Hom(Vi,R) = N⊕ i=0 (Vi) ∨ ∈ Zn2Vec. More explicitly, we consider the space of R-linear maps from V to R of any Zn2 -degree. It is clear that the linear maps of degree γi are the linear maps from Vi to R (that vanish in any other degree). Hence,( V ∨ ) i = (Vi) ∨ =: V ∨i . It follows that, if V is finite dimensional of dimension p\|q, its dual V ∨ has the same dimension. Moreover, any basis (bik)i,k (i ∈ {0, . . . , N} and k ∈ {1, . . . , qi}, where we set q0 := p) of V defines a dual basis (βki )i,k of V ∨. Let now V ∈ Zn2FinVec be of dimension p\|q. We set V ∨∗ := N⊕ j=1 V ∨j ∈ Zn2FinVec ( dim ( V ∨∗ ) = 0\|q ) . Proposition 2.21. If V is a Zn2 -graded vector space of dimension p\|q, there is a non-canonical isomorphism of Zn2 -commutative associative unital R-algebras [ : S̄ ( V ∨∗ ) ∼−→ R[[ξ]], where R[[ξ]] is the global function algebra of R0\|q. Proof. As usual, we ordered the Zn2 -degrees lexicographically, so that the ξ`j-s are ordered unambiguously. We have R[[ξ]] = ΠαR ξα, where the multi-index α has components α`j ∈ N (resp., α`j ∈ {0, 1}), if 〈γj , γj〉 is even (resp., odd). On the other hand, it follows from equations (2.34) and (2.39) that, choosing a basis ( bj` ) j,` of V∗ (defined similarly as V ∨∗ ) and denoting its dual basis by (β`j)j,`, leads to S̄ ( V ∨∗ ) = Πk ⊕ j1≤···≤jk ⊕ `1�···�`k Rβ`1j1 � · · · � β `k jk = Πk ⊕ \|α\|=k Rβα = ΠαRβα, (2.40) where α`j ∈ N (resp., α`j ∈ {0, 1}), if 〈γj , γj〉 is even (resp., odd). In view of (2.33) and (2.35), the multiplications of R[[ξ]] and S̄ ( V ∨∗ ) are exactly the same, so that the two Zn2 -commutative algebras are canonically isomorphic, once a basis of V∗ has been chosen. � 26 A.J. Bruce, E. Ibarguëngoytia and N. Poncin Remark 2.22. We denoted the isomorphism by [ to remind us of its dependence on the ba- sis ( bj` ) j,` . We are now prepared to define the linear Zn2 -manifold associated to a finite dimensional Zn2 - vector space. From here we denote the vector space by V instead of V and reserve the notation V for the manifold V := M(V). Hence, let V ∈ Zn2FinVec be of dimension p\|q. The p-dimensional vector space V0 of degree 0 is of course a smooth manifold of dimension p, as well as a linear Zn2 -manifold V0 of dimension p\|0. On the other hand, the algebra S̄ ( V∨∗ ) is a sheaf of Zn2 -commutative associative unital R- algebras over {?}, i.e., it is a Zn2 -ringed space with underlying topological space {?}, and, in view of Proposition 2.21, this space is (non-canonically) globally isomorphic to R0\|q = ({?},R[[ξ]]). Hence, the space ( {?}, S̄ ( V∨∗ )) is a linear Zn2 -manifold V> of dimension 0\|q. Finally, the product V = V0 × V> is a Zn2 -manifold of dimension p\|q, with base manifold V0 × {?} ' V0 and function sheaf OV that is, for any open subset Ω ⊂ V0 ' Rp, given by OV (Ω) = OV0×V>(Ω× {?}) = C∞V0(Ω)⊗̂R OV>({?}) ' C∞(Ω)⊗̂RR[[ξ]] = C∞(Ω)[[ξ]] = ORp\|q(Ω) (2.41) (since Ω and {?} are Zn2 -chart domains; for more information about the problem with the function sheaf of product Zn2 -manifolds, we refer the reader to [15]). In particular, the Zn2 -algebras OV (V0) and ORp\|q(R p) are isomorphic (see also Definition 13 of product Zn2 -manifolds in [15]), so that the Zn2 -manifolds V and Rp\|q are diffeomorphic (given what has been said above, the diffeomorphism is implemented by the choice of a basis of V). Finally V ∈ Zn2LinMan ( dimV = p\|q ) . We define the manifoldification functor M on objects by M(V) = V. We now define M on morphisms. A degree zero linear map L : V → W between finite dimensional vector spaces (of dimensions p\|q and r\|s, respectively) is a family of linear maps Li : Vi →Wi (i ∈ {0, . . . , N}). We denote the transpose maps by tLi : W∨ i → V∨i . The linear map L0 : V0 → W0 is of course a smooth map L0 : V0 → W0, where V0,W0 are the vector spaces V0,W0 viewed as smooth manifolds. The map L0 can also be interpreted as Zn2 -morphism L0 : V0 → W0 between the Zn2 -manifolds V0,W0 (which are of dimension zero in all non-zero degrees). The base morphism of L0 is L0 itself and, for any open subset Ω ⊂ W0, the pullback (L0)∗Ω is the (unital) algebra morphism −◦L0\|ω : C∞(Ω)→ C∞(ω) ( ω := L−1 0 (Ω) ) that extends the transpose tL0(−) = − ◦ L0. The linear maps tLj : W∨ j → V∨j (j ∈ {1, . . . , N}) define a linear map S̄ (t L ) : S̄ ( W∨ ∗ ) → S̄ ( V∨∗ ) . Observe first that to define such a map, it suffices to define a linear map in each tensor degree k, hence, it suffices to define a linear map(t L )�k j1...jk : W∨ j1 � · · · �W∨ jk → V∨j1 � · · · �V∨jk , for any j1 ≤ · · · ≤ jk (ja ∈ {1, . . . , N}). Since the k-linear maps(t L )×k j1...jk : W∨ j1 × · · · ×W∨ jk 3 ( ω1 j1 , . . . , ω k jk ) 7→ tLj1 ( ω1 j1 ) � · · · � tLjk ( ωkjk ) ∈ V∨j1 � · · · �V∨jk are Zn2 -commutative in the variables j` = · · · = jm, they define the degree zero linear maps(t L )�k j1...jk ( we set (t L )� 0 = idR ) and thus the degree zero linear map S̄ (t L ) that we are looking Linear Zn2 -Manifolds and Linear Actions 27 for. In view of our definitions, the latter is a (unital) Zn2 -algebra morphism between the global function algebras of the Zn2 -manifolds W> and V>, and it therefore defines a unique Zn2 -morphism L> : V> →W>. The base morphism of L> is the identity c : {?} → {?}. We thus get a Zn2 -morphism M(L) := L := L0 × L> : M(V) = V = V0 × V> →M(W) = W = W0 ×W>, (2.42) with base map L0 × c ' L0 and pullback (Ω open subset of W0, ω := L−1 0 (Ω)) L∗Ω : OW (Ω) = C∞W0 (Ω)⊗̂R S̄ ( W∨ ∗ ) → OV (ω) = C∞V0(ω)⊗̂R S̄ ( V∨∗ ) , (2.43) which is fully defined by (− ◦ L0\|ω)⊗ S̄ (t L ) . We must now prove that the Zn2 -morphism M(L) = L is a morphism of Zn2LinMan, i.e., that in linear coordinates it has linear coordinate pullbacks. As said above, the linear coordinate map k: W → Rr\|s is the product of the linear coordinate maps k0 : W0 → Rr\|0 and k> : W> → R0\|s. The first of these coordinate maps is implemented by a basis bW of W0 and its global pullback b∗W : C∞(Rr)→ C∞W0 (W0) sends a coordinate function y` ∈ C∞(Rr) to b∗W ( y` ) = y` ◦ bW = β`W ∈ C∞W0 (W0), where βW is the dual basis (observe that b∗W extends the transpose of bW viewed as vector space isomorphism). Similarly, it is clear from Proposition 2.21 that the global pullback [−1 W of the second coordinate map sends a coordinate function η`j ∈ R[[η]] to [−1 W ( η`j ) = β`j ∈ S̄ ( W∨ ∗ ) , where ( β`j ) j,` is the dual of a basis of W∗. Based on what we just said and on the state- ment (2.43), we get that the coordinate pullbacks in the linear coordinate expression of L are ( b∗V )−1(( b∗W ( y` )) ◦ L0 ) = ( b∗V )−1(t L0 ( β`W )) = ( b∗V )−1 (∑ k (L0)`kβ k V ) = ∑ k (L0)`kx k and [V (t Lj ( [−1 W ( η`j ))) = [V (t Lj ( β`j )) = [V (∑ k (Lj) ` kβ k j ) = ∑ k (Lj) ` kξ k j , where the notations are self-explanatory. Hence, M(L) : M(V) → M(W) is a morphism of Zn2LinMan. Since M(L) is essentially the transpose of L, we have defined a functor M: Zn2FinVec→ Zn2LinMan and this functor coincides on Zn2CarVec with the functor M that we defined earlier. We already mentioned that the Zn2 -diffeomorphism, say h, between V = M(V) and Rp\|q is implemented by a basis (bik)i,k of V. Now we can explain this observation in more detail. Indeed, the basis chosen provides a Zn2 -vector space isomorphism b : V→ Rp\|q, hence, the image M(b) : M(V) → M ( Rp\|q) is a Zn2 -diffeomorphism (it is even an isomorphism of Zn2LinMan), say b : V → Rp\|q. The diffeomorphism b = M(b) is a special case of the map L = M(L) of Zn2LinMan, whose construction has been described above. It is almost obvious from the penultimate paragraph that the diffeomorphism h coincides with the diffeomorphism b. Indeed, the diffeomorphism h is the product of two Zn2 -diffeomorphisms h0 : V0 → Rp\|0 and h> : V> → R0\|q (see k in the penultimate paragraph). The same holds for b, which is defined as b = b0× b>, 28 A.J. Bruce, E. Ibarguëngoytia and N. Poncin where b0 : V0 → Rp\|0 and b> : V> → R0\|q (see (2.42) and (2.43)). The map h0 is canonically induced by the basis (b0k)k of V0, and so is b0 ; hence h0 = b0. The Zn2 -diffeomorphism b> is defined by the corresponding Zn2 -algebra isomorphism S̄(tb) : S̄ (( R0\|q)∨)→ S̄ ( V∨∗ ) , where the source algebra is ΠαR εα = R[[ξ]]. As seen above, this algebra morphism is fully defined by the transposes tbj : (Rqj )∨ → V∨j and their action on the basis (ε`j)`. The action is tbj ( ε`j ) = ε`j ◦ bj = β`j , since the image of any vj = ∑ k v k j b j k ∈ Vj by the two maps is v`j . It follows that S̄ (t b ) = [−1. (2.44) This yields b> = h>. Finally, we get h = b = M(b). (2.45) Vectorification functor. In this subsection, we define the vectorification functor V: Zn2LinMan→ Zn2FinVec. If L ∈ Zn2LinMan has dimension p\|q, we set V(L) := L := ( Olin L (\|L\|) )∨ = ⊕ i ( Olin L,γi(\|L\|) )∨ =: ⊕ i Li ∈ Zn2FinVec, (2.46) where Li has dimension qi (q0 = p). Further, in view of item (iii) of Remark 2.16, if φ : L→ L′ is a morphism of Zn2LinMan, then tφ∗ is a degree preserving linear map V(φ) := Φ := tφ∗ : V(L) = ( Olin L (\|L\|) )∨ → ( Olin L′ (\|L′\|) )∨ = V(L′). The definition of V(φ) implies that V is a functor. Compositions of the manifoldification and the vectorification functors. (i) We first turn our attention to V◦M. If V ∈ Zn2FinVec ( dim V = p\|q ) , its image M(V) = V = V0 × V> ∈ Zn2LinMan ( dimV = p\|q ) is the product of the linear Zn2 -manifolds V0 and V>. Let ( bi` ) i,` be a basis of V with dual ( β`i ) i,` and induced Zn2 -vector space isomorphism b : V→ Rp\|q (we denote the induced diffeomorphism from V0 to Rp by b0). As explained above, it defines a linear coordinate map h = M(b) : V → Rp\|q (2.47) with pullback morphism h∗ = (− ◦ b0)⊗̂R [ −1 : C∞(Rp)⊗̂RR[[ξ]]→ C∞V0(V0)⊗̂R S̄ ( V∨∗ ) Linear Zn2 -Manifolds and Linear Actions 29 (see (2.45), (2.43) and (2.44)). Using equation (2.13), denoting the basis of ( Rp\|q)∨ as usual by ( ε`i ) i,` , and remembering the identifications (2.10), we thus get V(M(V)) = ( Olin V (V0) )∨ = ( h∗Olin Rp\|q(R p) )∨ = ( h∗ (⊕ ` R ε`0 ⊕ ⊕ j,` R ε`j ))∨ = (⊕ ` R ( ε`0 ◦ b0 ) ⊕ ⊕ j,` R [−1(ξ`j) )∨ = (⊕ ` Rβ`0 ⊕ ⊕ j,` Rβ`j )∨ = V. (ii) Regarding M◦ V, recall that if L ∈ Zn2LinMan ( dim L = p\|q ) . Definition (2.46) yields V(L) = L = ( Olin L (\|L\|) )∨ (notice that L denotes a vector space here, and not a linear map) and Definition (2.41) leads to M(L) := L := (L0,OL), where L0 is L0 = ( Olin L,γ0 (\|L\|) )∨ viewed as smooth manifold, and where OL(ω) (ω ⊂ L0 open) is OL(ω) = C∞L0 (ω)⊗̂R S̄ ( L∨∗ ) (see (2.41)). If we choose a basis ( β`j ) j,` of L∨∗ , we have S̄ ( L∨∗ ) = Πk ⊕ j1≤···≤jk ⊕ `1�···�`k Rβ`1j1 � · · · � β `k jk = ΠαRβα, where αkj ∈ N (resp., αkj ∈ {0, 1}), if 〈γj , γj〉 is even (resp., odd) (see (2.40)). Just as C∞Rp(Ω)⊗̂R ΠαR ξα = C∞Rp(Ω)⊗̂RR[[ξ]] = C∞Rp(Ω)[[ξ]] = ΠαC ∞ Rp(Ω) ξα (Ω ⊂ Rp open) (see [15]), we have OL(ω) = ΠαC ∞ L0 (ω)βα = Πk ⊕ j1≤···≤jk ⊕ `1�···�`k C∞L0 (ω)β`1j1 � · · · � β `k jk . (2.48) Remark 2.23. Let us mention that L and L denote a priori different linear Zn2 -manifolds and that our goal is to show that they do coincide. Recall first that, for any Zn2 -manifold M , there is a projection εM : OM → C∞\|M \| of \|M \|-sheaves of Zn2 -algebras and that εM commutes with pullbacks. In particular, if h : L→ Rp\|q is a linear coordinate map of L (a (linear) Zn2 -diffeomorphism), its pullback is, for any open subset \|U \| ⊂ \|L\|, a Zn2 -algebra isomorphism h∗ : ORp\|q(\|h\|(\|U \|))→ OL(\|U \|) and it restricts to a Zn2 -vector space isomorphism h∗ : Olin Rp\|q(\|h\|(\|U \|))→ Olin L (\|U \|). Further, as just said, we have εL ◦ h∗ = h∗ ◦ εRp\|q = (− ◦ \|h\|) ◦ εRp\|q 30 A.J. Bruce, E. Ibarguëngoytia and N. Poncin on ORp\|q(\|h\|(\|U \|)). Taking \|U \| = \|L\| and restricting the equality to degree zero linear functions Olin Rp\|q ,γ0 (Rp) = (Rp)∨ (see (2.13)), we obtain εL ◦ h∗ = − ◦ \|h\|, (2.49) or, equivalently, εL = (− ◦ \|h\|) ◦ (h∗)−1, (2.50) where (h∗)−1 is a vector space isomorphism from (L0)∨ = Olin L,γ0 (\|L\|) to (Rp)∨ and where −◦\|h\| is an algebra isomorphism from C∞(Rp) to C∞(\|L\|). In view of the diffeomorphism \|h\| : \|L\| → Rp, the smooth manifold \|L\| is linear. Hence, it is a finite dimensional vector space also denoted \|L\| and \|h\| is a vector space isomorphism, whose dual t\|h\| = − ◦ \|h\| is a vector space isomorphism from (Rp)∨ ⊂ C∞(Rp) to \|L\|∨. It follows (see also equation (2.50)) that the canonical map εL is a vector space isomorphism from (L0)∨ to \|L\|∨. When identifying these vector spaces, we get εL = id and \|L\| = L0, hence the corresponding linear manifolds do also coincide: \|L\| = L0. To prove that the linear Zn2 -manifolds L and L coincide, it now suffices to show that their function sheaves coincide. The pullback of h is an isomorphism h∗ : ORp\|q → OL of sheaves of Zn2 -algebras. Since h∗ is a Zn2 -vector space isomorphism h∗ : ( R0\|q)∨ = Olin R0\|q(Rp)→ Olin L,∗(\|L\|) = L∨∗ , the images ( h∗ ( ε`j )) j,` are a basis ( β`j ) j,` of L∨∗ . Moreover, we know that \|h\| = ( . . . , εL ( h∗ ( ε`0 )) , . . . ) = ( . . . ,h∗ ( ε`0 ) , . . . ) , as εL = id on (L0)∨. Therefore, if f(x) ∈ C∞(Rp), we get h∗(f(x)) = f(h∗(x)) = f ◦ ( . . . ,h∗ ( ε`0 ) , . . . ) = f ◦ \|h\| ∈ C∞(\|L\|). (2.51) Equation (2.51) (which generalizes equation (2.49)) shows that h∗ is an algebra isomorphism h∗ : C∞(Rp) → C∞(\|L\|). Similarly, if ω ⊂ \|L\| is open, Ω := \|h\|(ω) ⊂ Rp and f(x) ∈ C∞(Ω), we have h∗(f(x)) = f ◦ \|h\|\|ω ∈ C∞(ω), so that h∗ : C∞(Ω)→ C∞(ω) (2.52) is also an algebra isomorphism. Finally, the Zn2 -algebra isomorphism h∗ : ΠαC ∞(Ω) ξα → OL(ω) (2.53) sends any series ∑ α fα(x)ξα to∑ α h∗(fα(x)) ( . . . ,h∗ ( ξ`j ) , . . . )α = ∑ α h∗(fα(x)) ( . . . ,h∗ ( ε`j ) , . . . )α = ∑ α h∗(fα(x))βα ∈ OL(ω) Linear Zn2 -Manifolds and Linear Actions 31 (see (2.48)). The Zn2 -algebra morphism h∗ : ΠαC ∞(Ω)ξα → OL(ω) (2.54) we get this way (notice that the targets of the arrows (2.53) and (2.54) are different) is visibly an isomorphism. Indeed, it is obviously injective, and it is surjective due to (2.52). It follows from (2.53) and (2.54) that OL(ω) = OL(ω), for any open subset ω ⊂ \|L\|. Since h∗ commutes with restrictions, the sheaves OL and OL coincide and M(V(L)) = L. An alternative way of saying what we just said is to observe that in view of (2.53) every element of OL(ω) is the image by h∗ of a unique series ∑ α fα(x)ξα and therefore belongs to OL(ω). Conversely, in view of (2.52) every element ∑ α gαβ α (gα ∈ C∞(ω)) of OL(ω) uniquely reads ∑ α h∗(fα(x))βα, is therefore the image by h∗ of ∑ α fα(x)ξα and so belongs to OL(ω). (iii) We leave it to the reader to check that both functors, V◦M and M◦ V, coincide also on morphisms with the identity functors. Theorem 2.24. The functors M: Zn2FinVec � Zn2LinMan : V are an isomorphism of categories. Comparison of the functors of points. Since Zn2FinVec ' Zn2LinMan, the fully faithful functors of points F (see Proposition 2.8) and S (see Theorem 2.20) of these categories should coincide. However, up till now, the functor F is valued in Fun0 ( Zn2Ptsop, AMod ) , whereas the functor S is valued in Fun0 ( Zn2Ptsop, FAMod ) . Since FAMod is a subcategory of AMod, the latter functor category is a subcategory of the former. Hence, if we show that the image F(V) of any object V of Zn2FinVec is a functor of Fun0 ( Zn2Ptsop, FAMod ) (?) and that the image F(φ) of any morphism φ : V→W of Zn2FinVec is a natural transforation of Fun0 ( Zn2Ptsop, FAMod ) (∗), we can conclude that F is a functor F: Zn2FinVec→ Fun0 ( Zn2Ptsop, FAMod ) . We start proving (?). Since FAMod is a subcategory of AMod, we just have to show that the image F(V)(Λ) = V(Λ) = (Λ⊗V)0 of any object Λ of Zn2Ptsop is a Fréchet Λ0-module (•) and that the image F(V)(ϕ∗) = V(ϕ∗) = (ϕ∗ ⊗ 1V)0 of any morphism ϕ∗ : Λ→ Λ′ of Zn2Alg is a morphism of FAMod (◦). To prove (•), we consider a basis of V (dim V = p\|q), i.e., an isomorphism b : V � Rp\|q : b−1 of Zn2 -vector spaces. Since F(b) = b− is a natural isomorphism of Fun0 ( Zn2Ptsop, AMod ) , any of its Λ-components is an isomorphism bΛ : V(Λ) � Rp\|q(Λ) : b−1 Λ of Λ0-modules. We use this isomorphism to transfer to V(Λ) the Fréchet vector space structure of Rp\|q(Λ) = ( Λ⊗Rp\|q) 0 = ⊕ i ⊕ k Λγi = ΠiΠkΛγi = Λ×p0 × Λ×q1γ1 × · · · × Λ×qNγN (2.55) (see proof of Proposition 2.17 and equation (2.16)), thus obtaining a well-defined Fréchet struc- ture and making bΛ a Fréchet vector space isomorphism, i.e., a continuous linear map with 32 A.J. Bruce, E. Ibarguëngoytia and N. Poncin continuous inverse. Since bΛ is Λ0-linear, the action · of Λ0 on V(Λ) is related to its action / on Rp\|q(Λ) by a · v = b−1 Λ (a / bΛ(v)), for any a ∈ Λ0 and any v ∈ V(Λ). The action · is thus the composite of the continuous maps id×bΛ, /, and b−1 Λ , hence, it is itself continuous. The Λ0-module and the Fréchet vector space structures on V(Λ) therefore define a Fréchet Λ0-module structure on V(Λ) and bΛ becomes an isomorphism of Fréchet Λ0-modules (for any basis b of V). As concerns (◦), recall that V(ϕ∗) is a (ϕ∗)0-linear map, where the algebra morphism (ϕ∗)0 : Λ0 → Λ′0 is the restriction of ϕ∗. Observe now that, in view of (2.55), we have Rp\|q(ϕ∗) = (ϕ∗ ⊗ 1)0 = ΠiΠk ϕ ∗, so that Rp\|q(ϕ∗) is continuous as product of continuous maps (indeed, the Zn2Alg-morphism ϕ∗ is continuous as pullback of the associated Zn2 -morphism). As b− is a natural transformation of Fun0 ( Zn2Ptsop, AMod ) , we have V(ϕ∗) = b−1 Λ′ ◦Rp\|q(ϕ∗) ◦ bΛ, so that V(ϕ∗) is continuous (and (ϕ∗)0-linear), hence, is a morphism of FAMod. It remains to show that (∗) holds. We know that F(φ) = φ− is a natural transformation of Fun0 ( Zn2Ptsop, AMod ) , i.e., its Λ-components φΛ are Λ0-linear maps and the naturality con- dition is satisfied. It thus suffices to explain that φΛ = (1 ⊗ φ)0 is continuous. Since Λ is a Fréchet algebra, it is a locally convex topological vector space (LCTVS) and 1 : Λ → Λ is a degree zero continuous linear map. Further, since V and W are finite dimensional Zn2 -vector spaces, the degree zero linear map φ : V → W is automatically continuous for the canonical LCTVS structures on its source and target. It follows that 1 ⊗ φ and (1 ⊗ φ)0 are continuous linear maps. Proposition 2.25. The functor F: Zn2FinVec→ Fun0 ( Zn2Ptsop, FAMod ) is fully faithful. Proof. The result is obvious in view of Proposition 2.8, since Fun0 ( Zn2Ptsop, FAMod ) is a sub- category of Fun0 ( Zn2Ptsop, AMod ) . � We are now ready to refine the idea expressed at the beginning of this subsection that the (fully faithful) functors of points F: Zn2FinVec→ Fun0 ( Zn2Ptsop, FAMod ) (see Proposition 2.25) and S: Zn2LinMan→ Fun0 ( Zn2Ptsop, FAMod ) (see Theorem 2.20) of the isomorphic categories M: Zn2FinVec � Zn2LinMan : V should coincide. Linear Zn2 -Manifolds and Linear Actions 33 Theorem 2.26. The functors S◦M, F: Zn2FinVec→ Fun0 ( Zn2Ptsop, FAMod ) are naturally isomorphic. We first prove the theorem in the Cartesian case M: Zn2CarVec � Zn2CarMan : V (see Proposition 2.5). More precisely, it follows from Proposition 2.25 and Theorem 2.20 that the functors F and S are (fully faithful) functors F: Zn2CarVec→ Fun0 ( Zn2Ptsop, FAMod ) and S: Zn2CarMan→ Fun0 ( Zn2Ptsop, FAMod ) . Actually: Proposition 2.27. The functors S◦M, F: Zn2CarVec→ Fun0 ( Zn2Ptsop, FAMod ) are naturally isomorphic. Proof. In order to construct a natural isomorphism I : S◦M→ F, we must define, for any Rp\|q, a natural isomorphism I Rp\|q : S ( Rp\|q ) → F ( Rp\|q) of Fun0 ( Zn2Ptsop, FAMod ) that is natural in Rp\|q. To build I Rp\|q , we have to define, for each Λ, an isomorphism I Rp\|q ,Λ : S ( Rp\|q ) (Λ)→ F ( Rp\|q)(Λ) of Fréchet Λ0-modules that is natural in Λ. Recalling that the source and target of this arrow are Rp\|q(Λ) = HomZn2 Man ( R0\|m,Rp\|q ) ( R0\|m ' Λ ) and Rp\|q(Λ) = ( Λ⊗Rp\|q) 0 = N⊕ i=0 qi⊕ k=1 Λγi ⊗Reik = N⊕ i=0 qi⊕ k=1 Λγi = Λ×p0 × Λ×q1γ1 × · · · × Λ×qNγN , respectively, we set I Rp\|q ,Λ : x 7→ ∑ i,k x∗ ( uki ) ⊗ eik = ( x∗ ( uki )) = ( x∗ ( xk ) , x∗ ( ξkj )) =: ( xkΛ, ξ k j,Λ ) , where ( uki ) = ( xk, ξkj ) are the coordinates of Rp\|q and where ( eik ) i,k is the canonical basis of Rp\|q. Since we actually used this 1 : 1 correspondence to transfer the Fréchet Λ0-module structure from Rp\|q(Λ) to Rp\|q(Λ) (see (2.20)), the bijection I Rp\|q ,Λ is an isomorphism of Fréchet Λ0-modules. 34 A.J. Bruce, E. Ibarguëngoytia and N. Poncin This isomorphism is natural with respect to Λ. Indeed, if ϕ∗ : Λ→ Λ′ is a Zn2 -algebra map (with corresponding Zn2 -morphism ϕ), we have I Rp\|q ,Λ′ ( Rp\|q(ϕ∗)(x) ) = I Rp\|q ,Λ′ (x ◦ ϕ) = ( ϕ∗ ( x∗ ( xk )) , ϕ∗ ( x∗ ( ξkj ))) = (ϕ∗ ⊗ 1)0 ( I Rp\|q ,Λ (x) ) . It now suffices to check that I Rp\|q is natural with respect to Rp\|q . Hence, let L : Rp\|q → Rr\|s be a degree zero linear map and let L : Rp\|q → Rr\|s be the corresponding linear Zn2 -morphism M(L). In order to prove that IRr\|s ◦ S(L) = F(L) ◦ I Rp\|q , (2.56) we have to show that the Λ-components of these natural transformations coincide. To find that these Fréchet Λ0-module morphisms coincide, we must explain that they associate the same image to every x ∈ Rp\|q(Λ). When denoting the coordinates of Rr\|s by ( u′`i ) = ( x′`, ξ′`j ) , we obtain IRr\|s,Λ ( S(L)Λ(x) ) = IRr\|s,Λ(L ◦ x) = ( x∗ ( L∗ ( x′` )) , x∗ ( L∗ ( ξ′`j ))) = ( x∗ ( L∗ ( u′`i ))) , where L∗(u′`i ) = qi∑ k=1 L`iik u k i , in view of (2.11) and (2.12). It follows that ( x∗ ( L∗ ( u′`i ))) = ( qi∑ k=1 L`iik x∗ ( uki )) = ∑ i,` (∑ k L`iik x∗ ( uki )) ⊗ e′i` = ∑ i,k x∗ ( uki ) ⊗ (∑ ` L`iike ′i ` ) = ∑ i,k x∗ ( uki ) ⊗ L ( eik ) = (1⊗ L)0 (∑ i,k x∗ ( uki ) ⊗ eik ) = F(L)Λ ( I Rp\|q ,Λ (x) ) , where ( e′i` ) i,` is the basis of Rr\|s. � We are now able to prove Theorem 2.26. Proof. For simplicity, we set T := Fun0 ( Zn2Ptsop, FAMod ) . In order to build a natural isomorphism I: S◦M→ F, we must define, for any V ∈ Zn2FinVec, a natural isomorphism IV : S(V )→ F(V) of T that is natural in V. Set dim V = p\|q and let b be a basis of V, or, equivalently, a Zn2 -vector space isomorphism b : V→ Rp\|q. In view of (2.47), the morphism M(b) : V → Rp\|q is a linear Zn2 -diffeomorphism. Using Proposition 2.25 and Theorem 2.20, we obtain that F(b) : F(V)→ F ( Rp\|q) Linear Zn2 -Manifolds and Linear Actions 35 and S(M(b)) : S(V )→ S ( Rp\|q ) are natural isomorphisms of T. As I Rp\|q : S ( Rp\|q ) → F ( Rp\|q) is a natural isomorphism of T as well, the transformation IV := F ( b−1 ) ◦ I Rp\|q ◦ S(M(b)) is a natural isomorphism IV : S(V )→ F(V) as requested. In view of equation (2.56), the transformation IV is well-defined, i.e., is indepen- dent of the basis chosen. It remains to show that IV is natural in V, i.e., that, for any degree zero linear map φ : V→ W (dim W = r\|s) and for any basis b (resp., c) of V (resp., W), we have F(φ) ◦F ( b−1 ) ◦ I Rp\|q ◦ S(M(b)) = F ( c−1 ) ◦ IRr\|s ◦ S(M(c)) ◦ S(M(φ)), or, equivalently, IRr\|s ◦ S ( M ( c ◦ φ ◦ b−1 )) = F ( c ◦ φ ◦ b−1 ) ◦ I Rp\|q . Since L := c ◦ φ ◦ b−1 is a degree zero linear map L : Rp\|q → Rr\|s, equation (2.56) allows once more to conclude. � Internal Homs. A topological property is a property of topological spaces that is invariant under homeomorphisms (isomorphisms of topological spaces). More intuitively, a “topological property” is a property that only depends on the topological structure, or, equivalently, that can be expressed by means of open subsets. Similarly, equivalences of categories (“isomorphisms” of categories) preserve all “categorical properties and concepts”. Hence, an equivalence should preserve products. It turns out that this statement is actually correct. More precisely, if E: S→ T is part of an equivalence of categories, then a functor D: I→ S has limit s if and only if the functor E◦ D: I → T has limit E(s). Applying the statement to the discrete index category I with two objects {1, 2} and setting D(i) = si (i ∈ {1, 2}), we get that s1 and s2 have product s if and only if E(s1) and E(s2) have product E(s). Now, the category Zn2FinVec has the obvious binary product ×. It follows that, for any vector spaces V,W ∈ Zn2FinVec, the manifolds M(V), M(W) ∈ Zn2LinMan have product M(V)×M(W) = M(V ×W). If L, L′ ∈ Zn2LinMan, the categorical isomorphism implies that L = M(V(L)) and similarly for L′, so that the product L× L′ exists and is L× L′ = M(V(L)× V(L′)). (2.57) Hence, the category Zn2LinMan has finite products. Equation (2.57) shows that we got the product of Zn2LinMan by transferring to T := Zn2LinMan the product of S := Zn2FinVec. We can similarly transfer to T the closed symmetric monoidal structure of S. Indeed, the category Zn2Vec is closed symmetric monoidal for the standard 36 A.J. Bruce, E. Ibarguëngoytia and N. Poncin tensor product −⊗Zn2 Vec − of Zn2 -vector spaces and the standard internal Hom HomZn2 Vec(−,−) of Zn2 -vector spaces, which is defined, on objects for instance, by HomZn2 Vec(V,W) := ⊕ i HomZn2 Vec,γi(V,W) ∈ Zn2Vec, for any V,W ∈ Zn2Vec. Of course, if V,W ∈ S, then HomZn2 Vec(V,W) ∈ S, and the same holds for V ⊗Zn2 Vec W. It follows that S = Zn2FinVec is also a closed symmetric monoidal category. If we set now L⊗T L ′ := M(V(L)⊗Zn2 Vec V(L′)), HomT(L, L ′) := M ( HomZn2 Vec(V(L), V(L′)) ) , (2.58) and similarly for morphisms, we get a closed symmetric monoidal structure on T = Zn2LinMan: Proposition 2.28. The category Zn2LinMan is closed symmetric monoidal for the struc- ture (2.58). Alternatively, we could have defined HomT(L, L ′) ∈ T using the fully faithful functor of points S: T 3 L 7→ HomZn2 Man(−, L) =: L(−) ∈ Fun0 ( Zn2Ptsop, FAMod ) , i.e., defining first a functor FL,L′(−) in the target category, and then showing that this functor is representable by some HomT(L, L ′) ∈ T: FL,L′(−) = HomZn2 Man(−,HomT(L, L ′)) = HomT(L, L ′)(−). This “functor of points approach” is often easier. To shed some light on our more abstract definition above, we now compute HomT ( Rp\|q,Rr\|s ) (Λ) (�) assuming some familiarity with Zn2 -graded matrices gl(r\|s×p\|q,Λ) with entries in Λ ∈ Zn2Alg. Details can be found in Section 3.1 which we leave in its natural place. However, we highly recommend reading it before working though the end of this section. We observe first that HomZn2 Vec,γk ( Rp\|q,Rr\|s) = glγk ( r\|s× p\|q, R ) ∈ Vec. In order to understand the gist here, we consider the case n = 2, so that a matrix X ∈ glγk ( r\|s× p\|q, R ) has the block format X =  X00 X01 X02 X03 X10 X11 X12 X13 X20 X21 X22 X23 X30 X31 X32 X33 , (2.59) where the degree xij of the block Xij is xij = γi + γj + γk. (2.60) Since the entries of the Xij are real numbers and so of degree γ0, all the blocks with non-vani- shing xij do vanish. For instance, if γk = 01 ∈ Zn2 (resp., γk = 11) (do not confuse with the row- column index 01 in X01 (resp., 11 in X11)), the degree xij = 0 if and only if ij ∈ {01, 10, 23, 32} (resp., ij ∈ {03, 12, 21, 30}) (as in most of the other cases in this text, the Zn2 -degrees are lexicographically ordered), so that only these Xij do not vanish. It follows that HomZn2 Vec ( Rp\|q,Rr\|s) = gl ( r\|s× p\|q,R ) ∈ Zn2FinVec Linear Zn2 -Manifolds and Linear Actions 37 is made of the matrices (2.59), where no block Xij vanishes a priori. The canonical basis of this Zn2 -vector space are the obvious matrices Eik,j` (i, j ∈ {0, . . . , N}, k ∈ {1, . . . , si}, ` ∈ {1, . . . , qj}) with all entries equal to 0 except the entry kl in Xij which is 1. In view of equation (2.60), the vectors of this basis have the degrees γi + γj . We can of course identify (up to renumbering) this Zn2 -vector space with Rt\|u, where un (n ∈ {0, . . . , N}) is equal to un = ∑ i,j : γi+γj=γn siqj (2.61) (we set s0 = r, q0 = p, u0 := t). Hence: HomZn2 Vec ( Rp\|q,Rr\|s) = gl ( r\|s× p\|q,R ) = Rt\|u ∈ Zn2CarVec. (2.62) Combining (2.58) and (2.62), we get HomZn2 LinMan ( Rp\|q,Rr\|s ) = M ( HomZn2 Vec ( Rp\|q,Rr\|s)) = Rt\|u ∈ Zn2CarMan. (2.63) We now come back to (�). Setting as usual R0\|m ' Λ, we get the isomorphism HomZn2 LinMan ( Rp\|q,Rr\|s ) (Λ) = Rt\|u(Λ) ' ΠN n=0 Λ×unγn of Fréchet Λ0-modules. On the other hand, the vector space gl0 ( r\|s × p\|q,Λ ) is a Λ0-module and this module “coincides” obviously with gl0 ( r\|s× p\|q,Λ ) = ΠN n=0 Λ×unγn . By transferring the Fréchet structure, we get an “equality” of Fréchet Λ0-modules. Hence, the Fréchet Λ0-module isomorphism HomZn2 LinMan ( Rp\|q,Rr\|s ) (Λ) = Rt\|u(Λ) ' ΠN n=0 Λ×unγn = gl0 ( r\|s× p\|q,Λ ) ∈ FΛ0Mod. (2.64) There is a natural upgrade that is independent of the internal Homs and makes G := gl0(r\|s× p\|q,−) a functor G ∈ Fun0 ( Zn2Ptsop, FAMod ) . Indeed, it suffices to define G on a Zn2Alg-morphism ϕ∗ : Λ→ Λ′ as G(ϕ∗) : G(Λ) 3 X 7→ ϕ∗(X) ∈ G(Λ′), where ϕ∗(X) is defined entry-wise. The morphism G(ϕ∗) is clearly (ϕ∗)0-linear. It is also continuous, as it can be viewed as a product of copies of ϕ∗. Since G respects compositions and identities it is actually a functor of the functor category mentioned. The functors G and Rt\|u(−) = S(Rt\|u) are of course naturally isomorphic. Since S is a fully faithful functor S: Zn2LinMan→ Fun0 ( Zn2Ptsop, FAMod ) , the functor G can be viewed as represented by the linear Zn2 -manifold Rt\|u. Proposition 2.29. The functor gl0(r\|s×p\|q,−) is representable and the Cartesian Zn2 -manifold gl0 ( r\|s× p\|q ) := Rt\|u, with dimension t\|u defined in equation (2.61), is “its” representing object. Example 2.30. For n = 2, we find that gl0(1\|1, 1, 1) = R4\|4,4,4. 38 A.J. Bruce, E. Ibarguëngoytia and N. Poncin 3 Zn 2 -Lie groups and linear actions 3.1 Zn 2 -matrices We will consider matrices that are valued in some Zn2 -Grassmann algebra Λ, though everything we say generalizes to arbitrary Zn2 -commutative associative unital R-algebras. A homogeneous matrix X ∈ glx ( r\|s× p\|q,Λ ) of degree x ∈ Zn2 is understood to be a block matrix X = X00 . . . X0N ... . . . ... XN0 . . . XNN , with the entries of each block Xij being elements of the Zn2 -Grassmann algebra Λ. Here the degree xij ∈ Zn2 of Xij is xij = γi + γj + x and the dimension of Xij is dim(Xij) = si × qj (setting s0 = r and q0 = p as usual). Addition of such matrices and multiplication by reals are defined in the obvious way and they endow glx ( r\|s× p\|q,Λ ) with a vector space structure. We set gl ( r\|s× p\|q,Λ ) := ⊕ x∈Zn2 glx ( r\|s× p\|q,Λ ) ∈ Zn2Vec. Multiplication by an element of Λ requires an extra sign factor given by the row of the matrix, i.e., for any homogeneous λ ∈ Λγk , we have that (λ X)ij = (−1)〈γk,γi〉λ Xij . We thus obtain on gl ( r\|s × p\|q,Λ ) a Zn2 -graded module structure over the Zn2 -commutative algebra Λ. If r\|s = p\|q, we write gl ( p\|q,Λ ) := gl(p\|q × p\|q,Λ). Multiplication of matrices in gl(p\|q,Λ) is via standard matrix multiplication – now taking care that the entries are from a Zn2 -commutative algebra. Equipped with this multiplication, the Zn2 - graded Λ-module gl ( p\|q,Λ ) is a Zn2 -graded associative unital R-algebra. In particular, the degree zero matrices gl0 ( p\|q,Λ ) form an associative unital R-algebra. Since multiplication of matrices only uses multiplication and addition in Λ, we can replace Λ not only, as said above, by any Zn2 -commutative associative unital R-algebra, but also by any Zn2 -commutative ring R and then get a ring gl0(p\|q,R). We denote by GL(p\|q,R) the group of invertible matrices in gl0(p\|q,R). For further details the reader may consult [23]. 3.2 Invertibility of Zn 2 -matrices Let R be a Zn2 -commutative ring which is Hausdorff-complete in the J-adic topology, where J is the (proper) homogeneous ideal of R that is generated by the elements of non-zero degree γj ∈ Zn2 , j ∈ {1, . . . , N}. The Zn2 -graded ring morphism ε : R→ R/J , where R/J = ⊕ i Ri/(Ri ∩ J) = R0/(R0 ∩ J) Linear Zn2 -Manifolds and Linear Actions 39 vanishes in all non-zero degrees, induces a ring morphism ε̃ : gl0 ( p\|q,R ) 3 X 7→ ε̃(X) ∈ Diag ( p\|q,R/J ) , where ε̃(X) is the block-diagonal matrix with diagonal blocks ε̃(Xii) (with commuting entries). The following proposition appeared as Proposition 5.1 in [22]: Proposition 3.1. Let R be a J-adically Hausdorff-complete Zn2 -commutative ring and let X ∈ gl0 ( p\|q,R ) be a degree zero p\|q × p\|q matrix with entries in R, written in the standard block format X =  X00 . . . X0N ... . . . ... XN0 . . . XNN . We have: X ∈ GL(p\|q,R)⇔ Xii ∈ GL(qi, R), ∀i ⇔ ε̃(X) ∈ GL(p\|q,R/J)⇔ ε̃(Xii) ∈ GL(qi, R/J), ∀i. In this work, we are of course mainly interested in the case R := Λ = R ⊕ Λ̊ and J = Λ̊, so that R/J = R. 3.3 Zn 2 -Lie groups and their functor of points Groups, or, better, group objects can easily be defined in any category with finite products, i.e., any category C with terminal object 1 and binary categorical products c× c′ (c, c′ ∈ C). If C is a concrete category, the definition of a group object is very simple. For instance, if C is the concrete category AFM of Fréchet manifolds over a Fréchet algebra A, a group object G in C is just an object G∈ C that is group whose structure maps µ : G× G→ G and inv : G→ G are C-morphisms, i.e., A-smooth maps. We refer of course to a group object in AFM as a Fréchet A-Lie group. If C is the category Zn2Man of Zn2 -manifolds, the definition of a group object is similar, but all the (natural) requirements (above) have to be expressed in terms of arrows (since there are no points here). More precisely, a group object G in C is an object G ∈ C that comes equipped with C-morphisms µ : G×G→ G, inv : G→ G and e : 1→ G (the terminal object 1 is here the Zn2 -manifold R0\|0 = ({?},R)), which are called multiplication, inverse and unit, and satisfy the standard group properties (expressed by means of arrows): µ is associative, inv is a two-sided inverse of µ and e is a two-sided unit of µ. To understand the arrow expressions of these properties, we need the following notations. We denote by ∆: G→ G×G the canonical diagonal C-morphism and we denote by eG : G → G the composite of the unique C-morphism 1G : G → 1 and the unit C-morphism e : 1 → G. The left inverse condition now reads µ ◦ (inv× idG) ◦∆ = eG and the left unit condition reads µ ◦ (eG × idG) ◦∆ = idG 40 A.J. Bruce, E. Ibarguëngoytia and N. Poncin (and similarly for the right conditions). The associativity of µ is of course encoded by µ ◦ (µ× idG) = µ ◦ (idG×µ). (3.1) We refer to a group object in Zn2Man as a Zn2 -Lie group. A morphism F : G→ G′ of Fréchet A-Lie groups is of course defined as an A-smooth map that is a group morphism. Analogously, a morphism F : G→ G′ from a Fréchet A-Lie group to a Fréchet A′-Lie group is a morphism of AFM that is also a group morphism. We denote the category of Fréchet A-Lie groups by AFLg and we write AFLg for the category of Fréchet Lie groups over any Fréchet algebra. Further, a morphism Φ: G→ G′ of Zn2 -Lie groups is a Zn2 -morphism that respects the multi- plications, the inverses and the units (obvious arrow definitions). The category of Zn2 -Lie groups we denote by Zn2Lg. The functor of points of Zn2 -manifolds S: Zn2Man→ Fun0 ( Zn2Ptsop, AFM ) (3.2) induces a fully faithful functor of points of Zn2 -Lie groups: Theorem 3.2. The functor S: Zn2Lg→ Fun0 ( Zn2Ptsop, AFLg ) (3.3) is fully faithful. Moreover, if M ∈ Zn2Man and S(M) = M(−) ∈ Fun0 ( Zn2Ptsop, AFLg ) , then M ∈ Zn2Lg. This theorem was announced as [13, Theorem 3.30] without proper explanation or proof. Proof. It is clear that we have subcategories AFLg ⊂ AFM, Fun0 ( Zn2Ptsop, AFLg ) ⊂ Fun0 ( Zn2Ptsop, AFM ) and Zn2Lg ⊂ Zn2Man. Therefore, in order to prove that the functor (3.2) restricts to a functor (3.3), it suffices to show that S sends objects G and morphisms Φ of Zn2Lg to objects and morphisms of the functor category with target AFLg. Observe first that, for any M,N ∈ Zn2Man, we have the functor equality S(M ×N) = (M ×N)(−) = M(−)×N(−) = S(M)× S(N), (3.4) in view of the universal property of M × N . Further, if φ : M → M ′ and ψ : N → N ′ are two Zn2 -morphisms, the natural transformation S(φ× ψ) = (φ× ψ)− : (M ×N)(−)→ (M ′ ×N ′)(−) becomes φ− × ψ−, if we read it through the identification (3.4). Now, if G ∈ Zn2Lg with structure Zn2 -morphisms µ, inv (and e), then the AFM-valued functor S(G) = G(−) is actually AFLg-valued. This means that it sends any Zn2 -Grassmann algebra Λ and any Zn2Alg-morphism ϕ∗ : Λ→ Λ′ to an object G(Λ) and a morphism G(ϕ∗) of AFLg. For G(Λ) ∈ Λ0FM, notice that the natural transformations S(µ) = µ−, S(inv) = inv− (and S(e) = e−) have Λ0-smooth Λ-components µΛ : G(Λ)×G(Λ)→ G(Λ), invΛ : G(Λ)→ G(Λ) (and eΛ : 1(Λ)→ G(Λ)) Linear Zn2 -Manifolds and Linear Actions 41 (the Fréchet Λ0-manifold 1(Λ) is the singleton that consists of the Zn2Alg-morphism ιΛ that sends any real number to itself viewed as an element of Λ) that define a group structure on G(Λ) (with unit 1Λ := eΛ(ιΛ)), which is therefore a Fréchet Λ0-Lie group. The group properties of these structure maps are consequences of the group properties of the structure maps of G. For instance, when we apply S to the associativity equation (3.1) and then take the Λ-component of the resulting natural transformation, we get µΛ ◦ ( µΛ × idG(Λ) ) = µΛ ◦ ( idG(Λ)×µΛ ) . As for G(ϕ∗) : G(Λ)→ G(Λ′), we know that it is an AFM-morphism and have to show that it respects the multiplications µΛ and µΛ′ , i.e., that µΛ′ ◦ (G(ϕ∗)×G(ϕ∗)) = G(ϕ∗) ◦ µΛ. (3.5) However, this equality is nothing other than the naturalness property of µ−. Finally, let Φ: G → G′ be a Zn2Lg-morphism and denote the multiplications of the source and target by µ and µ′, respectively. In order to prove that the natural transformation S(Φ) = Φ− : G(−) → G′(−) of the functor category with target AFM is a natural transformation of the functor category with target AFLg, it suffices to show that ΦΛ is a morphism of AFLg, which results from the application of the functor S to the commutative diagram µ′ ◦ (Φ× Φ) = Φ ◦ µ. (3.6) The next task is to show that the functor (3.3) is fully faithful, i.e., that the map SG,G′ : HomZn2 Lg(G,G ′) 3 Φ 7→ Φ− ∈ HomFun0(Zn2 Ptsop, AFLg)(G(−), G′(−)) (3.7) is a 1 : 1 correspondence, for any Zn2 -Lie groups G,G′. Since the functor (3.2) is fully faithful, any natural transformation in the target set of (3.7) is implemented by a unique Zn2 -morphism φ : G → G′ and it suffices to show that φ respects the group operations, for instance, that is satisfies equation (3.6). However, equation (3.6) is satisfied if and only if µ′Λ ◦ (φΛ × φΛ) = φΛ ◦ µΛ, for all Λ. The latter condition holds, since φΛ is, by assumption, a group morphism. We must still prove the last statement of Theorem 3.2. The assumption implies that, for any Zn2 -Grassmann algebra Λ and any Zn2 -algebra morphism ϕ∗ : Λ → Λ′, we get a Fréchet Λ0-Lie group M(Λ) and a (ϕ∗)0-smooth group morphism M(ϕ∗) : M(Λ) → M(Λ′). We denote by 1Λ (resp., µΛ, invΛ) the unit element (resp., the Λ0-smooth multiplication, the Λ0-smooth inverse) of the group structure on the Fréchet Λ0-manifold M(Λ). We have already observed (see (3.5)) that the fact that M(ϕ∗) respects the multiplications µΛ and µΛ′ is equivalent to that of µ− being natural. The natural transformation µ− : (M ×M)(−)→M(−) is implemented by a unique Zn2 -morphism µ : M ×M → M . We obtain similarly a Zn2 -morphism inv : M → M . As for e : 1→M , we notice that the maps eΛ : 1(Λ) 3 ιΛ 7→ 1Λ ∈M(Λ) (Λ ∈ Zn2GrAlg) define visibly a natural transformation with Λ0-smooth Λ-components. Hence, it is implemented by a unique Zn2 -morphism e : 1 → M . We leave it to the reader to check that µ, inv and e satisfy (3.1) and the other group properties. � 42 A.J. Bruce, E. Ibarguëngoytia and N. Poncin 3.4 The general linear Zn 2 -group We want to define the general linear Zn2 -group of order p\|q so that it is a Zn2 -Lie group GL ( p\|q ) . In view of Theorem 3.2, it suffices to define a functor GL ( p\|q ) (−) ∈ Fun0 ( Zn2Ptsop, AFLg ) that is represented by a Zn2 -manifold GL ( p\|q ) . Definition 3.3. The general linear Zn2 -group GL ( p\|q ) is defined, for any Λ ∈ Zn2GrAlg, by GL ( p\|q ) (Λ) := GL ( p\|q,Λ ) = { X ∈ gl0(p\|q,Λ): X is invertible } , and, for any Zn2Alg-morphism ϕ∗ : Λ→ Λ′ and any X ∈ GL ( p\|q ) (Λ), by GL ( p\|q ) (ϕ∗)(X) := ϕ̃∗X, where ϕ̃∗ is ϕ∗ acting on X entry-by-entry. Theorem 3.4. The maps GL ( p\|q ) (−) of Definition 3.3 define a representable functor. We refer to the representing object GL ( p\|q ) ∈ Zn2Lg as the general linear Zn2 -group of dimension p\|q. Proof. Recall that: 1. It follows from equation (2.64) that gl0 ( p\|q,Λ ) = ΠN n=0Λ×unγn = Λ×t0 ×ΠN j=1Λ ×uj γj ' Rt\|u(Λ), where un is given by (2.61) (t = u0). 2. It follows from Proposition 3.1 that X ∈ gl0 ( p\|q,Λ ) is invertible if and only if ε̃(X) ∈ GL ( p\|q,R ) , if and only if ε̃(Xii) ∈ GL(qi,R), for all i ∈ {0, . . . , N}, if and only if Xii ∈ GL(qi,Λ), for all i ∈ {0, . . . , N}. In particular, a matrix X ∈ gl0 ( p\|q,R ) = Rt = Rp 2+ ∑ j q 2 j = Diag ( p\|q,R ) is invertible if and only if Xii ∈ GL(qi,R), for all i. It follows that Ut := GL ( p\|q ) (R) = ΠN i=0 GL(qi,R) ⊂ Rt. (3.8) As Ut ⊂ Rt is open, we can consider the Zn2 -domain Ut\|u := (Ut,ORt\|u \|Ut), (3.9) as well as its functor of points Ut\|u(−) ∈ Fun0 ( Zn2Ptsop, AFM ) , with value on Λ Ut\|u(Λ) ' Ut × Λ̊×t0 ×ΠN j=1Λ ×uj γj (see [13]). Linear Zn2 -Manifolds and Linear Actions 43 On the other hand, we get GL ( p\|q ) (Λ) = { X ∈ Rt × Λ̊×t0 ×ΠN j=1Λ ×uj γj : (. . . , ε̃(Xii), . . . ) ∈ ΠN i=0 GL(qi,R) } = Ut × Λ̊×t0 ×ΠN j=1Λ ×uj γj , so that Ut\|u(−) and GL ( p\|q ) (−) “coincide” on objects Λ: if we denote the coordinates of Rt\|u as usually by (ua) = ( xa, ξA ) , this “equality” reads Ut\|u(Λ) 3 x∗ ' (x∗(ua))a ∈ GL ( p\|q ) (Λ). Moreover, Ut\|u(−) and GL ( p\|q ) (−) coincide on morphisms ϕ∗ : Λ → Λ′. Indeed, the map GL ( p\|q ) (ϕ∗) acts on a matrix (x∗(ua))a ∈ GL ( p\|q ) (Λ) ⊂ Λ×t0 ×ΠN j=1Λ ×uj γj by acting on all its entries x∗(ua) by ϕ∗, whereas the map Ut\|u(ϕ∗) acts on a Zn2Alg-morphism x∗ ∈ Ut\|u(Λ) by left composition ϕ∗◦x∗; if we identify x∗ with the tuple (x∗(ua))a, then Ut\|u(ϕ∗) acts by acting on each x∗(ua) by ϕ∗, which proves the claim. It follows that GL ( p\|q ) (−) is a functor GL ( p\|q ) (−) ∈ Fun0 ( Zn2Ptsop, AFM ) that is represented by GL ( p\|q ) := Ut\|u ∈ Zn2Man, (3.10) so that it now suffices to prove that this functor is valued in AFLg, i.e., it suffices to show that GL ( p\|q ) (Λ) ∈ Λ0FLg and that GL ( p\|q ) (ϕ∗) is an AFLg-morphism. Recall that gl0 ( p\|q,Λ ) is an associative unital R-algebra for the standard matrix multipli- cation · (standard matrix addition, standard matrix multiplication by reals and standard unit matrix I) (see Section 3.1). It is clear that the subset GL ( p\|q ) (Λ) ⊂ gl0 ( p\|q,Λ ) is closed under ·: µΛ : GL ( p\|q ) (Λ)×GL ( p\|q ) (Λ) 3 (X,Y ) 7→ X · Y ∈ GL ( p\|q ) (Λ) (3.11) is an associative unital multiplication on GL ( p\|q ) (Λ). Therefore, µΛ and invΛ : GL ( p\|q ) (Λ) 3 X 7→ X−1 ∈ GL ( p\|q ) (Λ) (3.12) endow GL ( p\|q ) (Λ) with a group structure (with unit I). Finally, the Fréchet Λ0-manifold GL ( p\|q ) (Λ) together with its group structure µΛ, invΛ (and I) is a Fréchet Λ0-Lie group, if its structure maps µΛ and invΛ are Λ0-smooth. This condition is actually satisfied (see below). As for GL ( p\|q ) (ϕ∗), we know that it is an AFM-morphism and need to show that it respects the multiplications µΛ, µΛ′ . This condition is clearly met because GL ( p\|q ) (ϕ∗) acts entry-wise by the Zn2Alg-morphism ϕ∗. It remains to explain why µΛ and invΛ are Λ0-smooth. Notice first that the source of the multiplication (3.11) is the open subset Ω(Λ) := Ut\|u(Λ)× Ut\|u(Λ) of the Fréchet space F (Λ) := Rt\|u(Λ) × Rt\|u(Λ) (see [13]) and that we can choose the Fréchet vector space (and Fréchet Λ0-module) Rt\|u(Λ) as its target. Since Λ is the (Zn2 - commutative nuclear) Fréchet R-algebra of global Zn2 -functions of some Zn2 -point R0\|m, its ad- dition and internal multiplication (its multiplication by reals and subtraction) are continuous maps. It follows that each component function of the standard matrix multiplication µΛ is 44 A.J. Bruce, E. Ibarguëngoytia and N. Poncin continuous, so that µΛ is itself continuous. We must now explain why all directional deriva- tives of µΛ exist everywhere and are continuous, and why the first derivative is Λ0-linear. Let (X,Y ) ∈ Ω(Λ) and (V,W ) ∈ F (Λ). We get d(X,Y )µΛ (V,W ) = lim t→0 (X + tV ) · (Y + tW )−X · Y t = X ·W + V · Y. Hence, the first derivative exists everywhere, is continuous and Λ0-linear. Indeed, for any a ∈ Λ0, we have d(X,Y )µΛ (a · V, a ·W ) = a · d(X,Y )µΛ (V,W ). It is easily checked that d2 (X,Y )µΛ(V1,W1, V2,W2) = V2 ·W1 + V1 ·W2 and k≥3 (X,Y )µΛ(V1,W1, . . . , Vk,Wk) = 0, so that µΛ is actually Λ0-smooth. As for invΛ : Ut\|u(Λ) ⊂ Rt\|u(Λ)→ Rt\|u(Λ), we start computing the directional derivative of IΛ := µΛ ◦ (invΛ× idΛ) ◦∆Λ : Ut\|u(Λ) ⊂ Rt\|u(Λ) 3 X 7→ X−1 ·X = I ∈ Rt\|u(Λ) (∆Λ is the diagonal map), assuming continuity of invΛ, for the time being. For any V ∈ Rt\|u(Λ), we have dXIΛ(V ) = lim t→0 (X + tV )−1 · (X + tV )−X−1 ·X t = lim t→0 (fXV (t) ·X + gXV (t) · V ) = 0, where fXV (t) = (X + tV )−1 −X−1 t and gXV (t) = (X + tV )−1. It follows that dX invΛ(V ) = lim t→0 fXV (t) = lim t→0 ( (fXV (t) ·X + gXV (t) · V ) ·X−1 − gXV (t) · V ·X−1 ) = −X−1 · V ·X−1, so that the first derivative is defined everywhere, is continuous, as well as Λ0-linear. Also the higher order derivatives exist everywhere and are continuous. For instance, the second order derivative is given by d2 X invΛ(V,W ) = − lim t→0 ( fXW (t) · V ·X−1 + gXW (t) · V · fXW (t) ) = X−1 ·W ·X−1 · V ·X−1 +X−1 · V ·X−1 ·W ·X−1. Finally, the inverse map invΛ is Λ0-smooth, provided we prove its still pending continuity. We will show that the continuity of (3.12) boils down to the continuity of the inverse map ιΛ : Λ× 3 λ 7→ λ−1 ∈ Λ× in Λ. Here Λ× ⊂ Λ is the group of invertible elements of Λ. Since Λ is a (unital) Fréchet R-algebra, its inverse map ιΛ is continuous if and only if Λ× is a Gδ-set, i.e., if and only if it is a countable intersection of open subsets of Λ [47]. We will show that Λ× is Linear Zn2 -Manifolds and Linear Actions 45 actually open in the specific Fréchet R-algebra Λ considered. In view of Equation (16) in [14], the topology of Λ = R[[θ]] (Λ ' R0\|m) is induced by the countable family of seminorms ρβ(λ) = 1 β! \|ε(∂βθ λ)\| = \|λβ\| ( β ∈ N×\|m\|, λ = ∑ α λαθ α ∈ Λ ) , where ε is the projection ε : Λ → R. This means that the topology is made of the unions of finite intersections of the open semiballs Bβ(ν, ε) = {λ ∈ Λ: ρβ(λ− ν) < ε} = {λ ∈ Λ: \|λβ − νβ\| < ε} = {λ ∈ Λ: λβ ∈ b(νβ, ε)}( β ∈ N×\|m\|, ν = ∑ α ναθ α ∈ Λ, ε > 0 and b(νβ, ε) is the open ball in R with center νβ and radius ε ) . Since Λ× = {λ ∈ Λ: λ0 ∈ R \ {0}} and R \ {0} = ⋃ r∈R\{0} b(r, εr) (for some εr > 0), we get Λ× = ⋃ r∈R\{0} {λ ∈ Λ: λ0 ∈ b(r, εr)} = ⋃ r∈R\{0} B0(r, εr), which implies that Λ× is open and that ιΛ is continuous, as announced. Before we are able to deduce from this that invΛ is continuous, we need an inversion formula for X ∈ GL ( p\|q ) (Λ). Notice first that, in view of [23, Proposition 4.7], an invertible 2× 2 block matrix X = ( A B C D ) with square diagonal blocks A and D and entries (of all blocks) in a ring, has a block UDL decomposition if and only if D is invertible. In this case, the UDL decomposition is( A B C D ) = ( I BD−1 0 I )( A−BD−1C 0 0 D )( I 0 D−1C I ) . As upper and lower unitriangular matrices are obviously invertible, it follows that the diagonal matrix is invertible, hence that A − BD−1C is invertible. Similarly, the invertible matrix X has a block LDU decomposition if and only if A is invertible and in this case D − CA−1B is invertible. Moreover, in view of Proposition 3.1, a matrix X ∈ gl0 ( p\|q,Λ ) is invertible if and only if all its diagonal blocks Xii are invertible. Let now X = ( A B C D ) be a 2 × 2 block decomposition of X ∈ gl0 ( p\|q,Λ ) that respects the (N + 1) × (N + 1) block decomposition X =  X00 . . . X0N ... . . . ... XN0 . . . XNN . Since A (resp., D) is invertible if and only if Ã = ( A 0 0 I ) ( resp., D̃ = ( I 0 0 D )) 46 A.J. Bruce, E. Ibarguëngoytia and N. Poncin is invertible, hence, if and only if the Xkk on the diagonal of A (resp., D) are invertible, we get that X is invertible if and only if A and D are invertible. If we combine everything we have said so far in this paragraph, we find that if X ∈ GL ( p\|q ) (Λ), then A,D,A−BD−1C,D − CA−1B are all invertible. Therefore, we can use the formula X−1 = ( ( A−BD−1C )−1 −A−1B ( D − CA−1B )−1 −D−1C ( A−BD−1C )−1 ( D − CA−1B )−1 ) , (3.13) for any X ∈ GL ( p\|q ) (Λ). In order to simplify proper understanding, we consider for instance the case n = 2, p\|q = p\|q1, q2, q3 = 1\|2, 1, 1 and X = ( A B C D ) =  a b c d e f g h i j k l m n p q r s t u v w x y z  ∈ GL(1\|2, 1, 1)(Λ), where A =  a b c f g h k l m , and so on. We focus for instance on the first of the four block matrices in X−1, i.e., on ( A − BD−1C )−1 . The matrix D is a 2× 2 invertible matrix with square diagonal blocks and entries in Λ. Since the four diagonal block matrices in X are invertible, it follows from what we have said above that the inverse D−1 is given by equation (3.13) with A = t ∈ Λ, B = u ∈ Λ, C = y ∈ Λ and D = z ∈ Λ. Hence all entries of D−1 are composites of the addition, the subtraction, the multiplication and the inverse in Λ, and so are all entries in the invertible 2× 2 block matrix A−BD−1C =  α β γ δ ε ζ η θ ξ  (3.14) with square diagonal blocks (which are invertible) and with entries in Λ (the square diagonal blocks have entries in Λ0). Hence, the inverse (A−BD−1C)−1 can again be computed by (3.13). We focus on its entry κ := ( α− (β γ) ( ε ζ θ ξ )−1( δ η ))−1 ∈ Λ. (3.15) Notice that here we cannot conclude that ε and ξ are invertible and apply (3.13) to compute the internal inverse. However, this inverse is the inverse of a square matrix with entries in the commutative ring Λ0, for which the standard inversion formula holds (recall that a square matrix with entries in a commutative ring is invertible if and only if its determinant is invertible):( ε ζ θ ξ )−1 = (εξ − ζθ)−1 ( ξ −ζ −θ ε ) . (3.16) Since all the entries of (3.14) are composites of the addition, subtraction, multiplication and inverse in Λ, it follows from (3.15) and (3.16) that the same is true for the entry κ of X−1. More precisely the entry κ corresponds to a map κ̃ that is a composite of the inclusion of GL ( p\|q ) (Λ) into its topological supspace Λ×(t+\|u\|) (continuous), the projection of Λ×(t+\|u\|) onto Λ×v (v ≤ t+\|u\|) (continuous) and of products of the identity map id of Λ (continuous), the diagonal map ∆ Linear Zn2 -Manifolds and Linear Actions 47 of Λ (continuous), the switching map σ of Λ×Λ (continuous), the addition a of Λ (continuous), the scalar multiplication e of Λ (continuous), its subtraction s (continuous), multiplication m (continuous) and its inverse ι (continuous). Indeed, it is for instance easily seen that the map Λ×4 3 (t, u, y, z) 7→ −z−1y ( t− uz−1y )−1 ∈ Λ is a (continuous) composite of products of these continuous maps. We thus understand that the entry κ of X−1 corresponds to a continuous map κ̃ : GL ( p\|q ) (Λ) → Λ. The same holds of course also for all the other entries of X−1. Finally, the inverse map invΛ : GL ( p\|q ) (Λ) 3 X 7→ X−1 ∈ Λ×(t+\|u\|) is continuous and it remains continuous when view as valued in the subspace GL ( p\|q ) (Λ). � Example 3.5. In view of equations (3.10), (3.9) and (3.8), the general linear Z2 2-group of order 1\|1, 1, 1 is GL(1\|1, 1, 1) ' ( (R×)4,OR4\|4,4,4 \|(R×)4 ) , where R× = R \ {0}. 3.5 Smooth linear actions In this section we define linear actions of Zn2 -Lie groups G on finite dimensional Zn2 -vector spaces V ' V (we identify the isomorphic categories Zn2FinVec and Zn2LinMan). The definition can be given in the category of Zn2 -manifolds, but it is slightly more straightforward if we use the functor of points. Notice that the functors of points of G ∈ Zn2Lg ⊂ Zn2Man and V ∈ Zn2LinMan ⊂ Zn2Man are functors S(G) = G(−) ∈ Fun0 ( Zn2Ptsop, AFLg ) ⊂ Fun0 ( Zn2Ptsop, AFM ) and S(V ) = V (−) ∈ Fun0 ( Zn2Ptsop, FAMod ) ⊂ Fun0 ( Zn2Ptsop, AFM ) . Definition 3.6. Let G ∈ Zn2Lg and V ∈ Zn2LinMan. A smooth linear action of G on V is a natural transformation σ− : (G× V )(−) = G(−)× V (−)→ V (−) in Fun0 ( Zn2Ptsop, AFM ) (natural transformation with Λ0-smooth Λ-components) that satisfies the following conditions: (i) Identity: for all vΛ ∈ V (Λ), we have σΛ(1Λ, vΛ) = vΛ, where 1Λ is the unit of G(Λ). (ii) Compatibility: for all gΛ, g ′ Λ ∈ G(Λ) and all vΛ ∈ V (Λ), we have σΛ ( gΛ, σΛ(g′Λ, vΛ) ) = σΛ ( µΛ(gΛ, g ′ Λ), vΛ ) , where µΛ is the multiplication of G(Λ). (iii) Λ0-linearity: for all gΛ ∈ G(Λ), all vΛ, v ′ Λ ∈ V (Λ) and all a ∈ Λ0, we have 48 A.J. Bruce, E. Ibarguëngoytia and N. Poncin (a) σΛ(gΛ, vΛ + v′Λ) = σΛ(gΛ, vΛ) + σΛ(gΛ, v ′ Λ), (b) σΛ(gΛ, a · vΛ) = a · σΛ(gΛ, vΛ), where · is the action of Λ0 on V (Λ). Since S: Zn2Man→ Fun0 ( Zn2Ptsop, AFM ) (3.17) is fully faithful (for more details, see [13, 14, 15]), there is a 1 : 1 correspondence between natural transformations σ− as above and Zn2 -morphisms σ : G× V → V. This correspondence implies in particular that condition (ii) is equivalent to the equality σ ◦ (idG×σ) = σ ◦ (µ× idV ) (3.18) of Zn2 -morphisms from G×G× V → V (µ : G×G → G is the multiplication of G). The same holds for condition (i) and the equality σ ◦ (e× idV ) = idV (3.19) of Zn2 -morphisms from V ' 1× V → V (e : 1→ G is the two-sided unit of µ). 3.5.1 Canonical action of the general linear group We will now define the canonical action of the general linear Zn2 -group GL ( p\|q ) = Ut\|u ∈ Zn2Lg on the Cartesian Zn2 -manifold Rp\|q ∈ Zn2LinMan. To do this, we use both, the fully faithful functor (3.17) and the fully faithful functor Y: Zn2Man 3M 7→ HomZn2 Man(−,M) ∈ Fun ( Zn2Manop, Set ) . (3.20) We start defining a natural transformation σ− of Fun ( Zn2Manop, Set ) from Ut\|u(−)×Rp\|q(−) to Rp\|q(−). We will denote the coordinates of Ut\|u (resp., Rp\|q) here by Xa b (resp., xc), where a, b ∈ {1, . . . , p + \|q\|} (resp., where c ∈ {1, . . . , p + \|q\|}). For this, we must associate to any S ∈ Zn2Man, a set-theoretical map σS that assigns to any (X,φ) ∈ Ut\|u(S)× Rp\|q(S) = HomZn2 Man ( S, Ut\|u ) ×HomZn2 Man ( S,Rp\|q ) , i.e., to any (appropriate) coordinate pullbacks( Xa S,b,x c S ) := (X∗(Xa b ), φ∗(xc)) ∈ O(S)×(p+\|q\|)2 × O(S)×(p+\|q\|), (3.21) a unique element σS(X,φ) ∈ Rp\|q(S), i.e., unique (appropriate) coordinate pullbacks σS ( Xa S,b,x c S ) ∈ O(S)×(p+\|q\|). Since ( xc S ) c is viewed as a tuple (horizontal row), the natural definition of this image (horizontal row) is σS ( Xa S,b,x c S ) = ( xb S Xa S,b ) a , (3.22) where the sum and products are taken in the global Zn2 -function algebra O(S) of S. It is clear that the elements of this target-tuple have the required degrees, as the same holds for the elements Linear Zn2 -Manifolds and Linear Actions 49 of the source-tuple. The transformation σ− we just defined is clearly natural. Indeed, for any Zn2 - morphism ψ : S′ → S, the induced set-theoretical mapping between the Hom-sets with source S and the corresponding ones with source S′ is −◦ψ, so that the induced set-theoretical mapping between the tuples of global Zn2 -functions of S and S′ is ψ∗. The naturalness of σ− follows now from the fact that ψ∗ is a Zn2Alg-morphism. Since (3.20) is fully faithful, the natural transformation σ− is implemented by a unique Zn2 - morphism σ : GL ( p\|q ) × Rp\|q → Rp\|q, (3.23) which in turn implements, via (3.17), a unique natural transformation in Fun0 ( Zn2Ptsop, AFM ) between the same functors, but restricted to Zn2Ptsop. Since this transformation is the restriction of σ− to Zn2Ptsop, we use this symbol for both transformations (provided that any confusion can be excluded). It is easily seen that σΛ ( Xa Λ,b,x c Λ ) = ( xb Λ Xa Λ,b ) a , with sum and products in Λ, has the properties (i), (ii) and (iii) of Definition 3.6, so that we defined a smooth linear action of GL ( p\|q ) on Rp\|q. The interesting aspect here is that we are able to compute the Zn2 -morphism (3.23). Indeed, in view of the proof of the full faithfulness of the standard Yoneda embedding c 7→ HomC(−, c) of an arbitrary locally small category C into the functor category Fun ( Cop, Set ) , the morphism σ ∈ HomC(c, c ′) that implements a natural transformation σ− : HomC(−, c)→ HomC(−, c′) is σ = σc(idc) ∈ HomC(c, c ′). In our case of interest C = Zn2Man, the previous Yoneda embedding is the functor (3.20) and the morphism σ ∈ HomZn2 Man ( GL ( p\|q ) × Rp\|q,Rp\|q ) is σ = σc(idc), with c = GL ( p\|q ) × Rp\|q. Since the pullback of the identity Zn2 -morphism idc is identity and the coordinate pullbacks (3.21) are ( Xa b ,x c ) ∈ O(c)×(p+\|q\|)(p+\|q\|+1). Equation (3.22) yields σ = σc(idc) ' σc ( Xa b ,x c ) = ( xb Xa b ) a , with sum and products in O(c). In other words: Proposition 3.7. The canonical action σ of the general linear Zn2 -group GL ( p\|q ) on the linear Zn2 -manifold or Zn2 -graded vector space Rp\|q, is the Zn2 -morphism that is defined by the coordinate pullbacks σ∗(xa) = xb Xa b , (3.24) where we denoted the coordinates of GL ( p\|q ) (resp., Rp\|q) by Xa b (resp., xc). 50 A.J. Bruce, E. Ibarguëngoytia and N. Poncin Example 3.8. We know that the general linear Z2 2-group GL(1\|1, 1, 1) can be identified with the open Z2 2-submanifold U4\|4,4,4 of R4\|4,4,4. We denote the global coordinates of this Cartesian Z2 2-manifold by ( xα, ξβ, θγ , zδ ) . The indices run over {1, 2, 3, 4} and the Z2 2-degrees of these coordinates are (0, 0), (0, 1), (1, 0) and (1, 1), respectively. We already mentioned that if we view U4\|4,4,4 as GL(1\|1, 1, 1), we must rearrange the coordinates: X= (Xa b )a,b =  x1 ξ1 θ1 z1 ξ2 x2 z2 θ2 θ3 z3 x3 ξ3 z4 θ4 ξ4 x4 . In view of (3.24), the action σ of GL(1\|1, 1, 1) on R1\|1,1,1 with global coordinates x = ( xa ) a = ( x0, ξ0, θ0, z0 ) , is given as σ∗ ( x0 ) = x0x1 + ξ0ξ1 + θ0θ1 + z0z1, σ∗ ( ξ0 ) = x0ξ2 + ξ0x2 + θ0z2 + z0θ2, σ∗ ( θ0 ) = x0θ3 + ξ0z3 + θ0x3 + z0ξ3, σ∗ ( z0 ) = x0z4 + ξ0θ4 + θ0ξ4 + z0x4. 3.5.2 Connection between the canonical action and the internal Hom Since GL ( p\|q ) = Ut\|u (see equation (3.10)) is an open Zn2 -submanifold (see equation (3.9)) of gl0 ( p\|q ) = Rt\|u = HomZn2 LinMan ( Rp\|q,Rp\|q ) (see Proposition 2.29 and equation (2.63)), we can expect a connection between the canonical action of GL ( p\|q ) on Rp\|q and HomZn2 LinMan ( Rp\|q,Rp\|q ) . It turns out that this link becomes apparent as soon as we understand the connection between the internal Hom of linear Zn2 - manifolds and the internal Hom of arbitrary Zn2 -manifolds. Indeed, for any Λ ' R0\|m, we have HomZn2 Man ( Rp\|q,Rp\|q ) (Λ) := HomZn2 Man ( Rp\|q,Rp\|q ) (see [13]). If we denote the coordinates of R0\|m by θ = (θd) and those of Rp\|q by x = (xa) =( xa, ξA ) , the RHS Hom-set can be identified with the set of (degree respecting) coordinate pullbacks: HomZn2 Man ( Rp\|q,Rp\|q ) (Λ) = { xa = xa(x, ξ, θ) = ∑ αβ faαβ(x)ξαθβ } . On the other hand, when denoting the coordinates of Rt\|u as above by X= (Xa b ), we get similarly HomZn2 LinMan ( Rp\|q,Rp\|q ) (Λ) = Rt\|u(Λ) = HomZn2 Man ( Λ,Rt\|u ) = { Xa b = Xa b (θ) = ∑ δ rab,δ θ δ } = gl0 ( p\|q,Λ ) . (3.25) Linear Zn2 -Manifolds and Linear Actions 51 An obvious identification leads now to HomZn2 LinMan ( Rp\|q,Rp\|q ) (Λ) = { xa = xa(x, ξ, θ) = ∑ b xb Xa b (θ) = ∑ b xbXa b (θ) + ∑ B ξB Xa B(θ) } . (3.26) When comparing (3.26) and (3.25), we see that the internal Hom of linear Zn2 -manifolds consists of the pullbacks of the internal Hom of arbitrary Zn2 -manifolds which are defined by the canonical action of gl0 ( p\|q ) (Λ) on Rp\|q, in the sense of (3.24). 3.5.3 Equivalent definitions of a smooth linear action Section 3.5 already implicitly contained the idea that a smooth linear action of a Zn2 -Lie group G on a linear Zn2 -manifold V in the sense of Definition 3.6, is equivalent to a Zn2 -morphism σ : G× V → V that satisfies the conditions (3.18) and (3.19) and additionally has a certain linearity property with respect to V . A natural idea is that σ∗ should send linear Zn2 -functions of V to Zn2 -functions of G × V that are linear along the fibers. The meaning of this concept becomes clear when we think of the classical differential geometric case in which the functions of a trivial vector bundle E = M × Rr are C∞(E) = Γ(∨E∗) = C∞(M)⊗ ∨ (Rr)∗ (∨ is the symmetric tensor product), i.e., are the functions that are smooth in the base and polynomial along the fiber. Hence, linear functions of E are the functions that are smooth in the base and linear along the fiber, i.e., C∞lin(E) = C∞(M)⊗ (Rr)∗ = C∞(M)⊗ C∞lin(Rr). We can choose the same definition in the case of the trivial Zn2 -vector bundle E = G× V : Olin E (\|G\| × \|V \|) := OG(\|G\|)⊗ Olin V (\|V \|). This definition is of course in particular valid for G = GL ( p\|q ) ∈ Zn2Lg. However, let us mention that the linear functions (“linear along the fibers”) of the trivial Zn2 -vector bundle E = GL ( p\|q ) × V that are defined on \|GL ( p\|q ) \| × \|V \| do not coincide with the linear functions (“globally linear”) of the linear Zn2 -manifold M = Rt\|u × V (see (2.57)) that are defined on the open subset \|GL ( p\|q ) \| × \|V \| of its base Rt × \|V \|: Olin E (∣∣GL ( p\|q )∣∣× \|V \|) 6= Olin M (∣∣GL ( p\|q )∣∣× \|V \|). Given what we have just said, we expect the following proposition to hold: Proposition 3.9. A smooth linear action σ− of the Zn2 -Lie group G = GL ( p\|q ) on a linear Zn2 -manifold V in the sense of Definition 3.6, is equivalent to a Zn2 -morphism σ : G × V → V that satisfies the conditions (3.18) and (3.19) and has the linearity property σ∗(Olin V (\|V \|)) ⊂ OG(\|G\|)⊗ Olin V (\|V \|). (3.27) Notice first that the pullback σ∗ is a morphism of Zn2 -algebras σ∗ : OV (\|V \|)→ OG×V (\|G\| × \|V \|). Since G and V have global coordinates, it follows from [15] that the target of σ∗ is given by OG×V (\|G\| × \|V \|) = OG(\|G\|)⊗̂OV (\|V \|), 52 A.J. Bruce, E. Ibarguëngoytia and N. Poncin which shows that it contains OG(\|G\|)⊗ Olin V (\|V \|) and that the requirement (3.27) actually makes sense. Another fact is also worth noting. We know from standard supergeometry that the classical Berezinian defines a super-Lie group morphism Ber: GL(p\|q)→ GL(1\|0), so that we get a linear action of GL(p\|q) on R1\|0. The point here is that linear actions of GL(p\|q) are not limited to actions on Rp\|q. Proof. In the light of the observations that follow Definition 3.6, it suffices to prove that the Λ0-linearity requirement (iii) in Definition 3.6 is equivalent to the linearity condition (3.27) in Proposition 3.9. Hence, let σ− be a smooth action of G on V and let σ be the corresponding Zn2 -morphism. If h : V → Rr\|s is a linear coordinate map of V , the Zn2 -morphism S := h ◦ σ ◦ ( idG×h−1 ) : G× Rr\|s → Rr\|s satisfies (3.27) if and only if the Zn2 -morphism σ does. Indeed, if σ has the property (3.27), then S∗ = ( idG×h−1 )∗ ◦ σ∗ ◦ h∗ = ( id∗G ⊗̂ ( h−1 )∗) ◦ σ∗ ◦ h∗ has obviously the same property. We similarly find that the converse implication holds. On the other hand, if we denote the coordinates of Rr\|s by y = (yc) = ( yc, ηC ) , the Λ-components of the natural transformations σ− and S− satisfy SΛ = hΛ ◦ σΛ ◦ ( idG(Λ)×h−1 Λ ) and SΛ ( gΛ, ∑ k λkyΛ,k ) = hΛ ( σΛ ( gΛ,h −1 Λ (∑ k λkyΛ,k ))) , for any gΛ ∈ G(Λ), any yΛ,k ∈ Rr\|s(Λ) and any λk ∈ Λ0 (where k runs through a finite set). Since hΛ : V (Λ)→ Rr\|s(Λ) is an isomorphism of Fréchet Λ0-modules, the Λ0-smooth map SΛ is Λ0-linear in yΛ if and only if the Λ0-smooth map σΛ is Λ0-linear in vΛ ∈ V (Λ). It is therefore sufficient to prove the equivalence “(iii) if and only if (3.27)” for V = Rr\|s. We refrain from writing down the proof of the implication “if (iii) then (3.27)”. It is technical and partially reminiscent of a part of the proof of Theorem 2.20 (for the super-case, see [17] and the references it contains). We now prove the converse implication from scratch. Assume that S∗ ( Olin Rr\|s(R r) ) ⊂ OG(\|G\|)⊗ Olin Rr\|s(R r). (3.28) In view of the universal property of the product of Zn2 -manifolds, we have G(Λ)× Rr\|s(Λ) 3 (gΛ, yΛ) ' uΛ ∈ ( G× Rr\|s ) (Λ). Linear Zn2 -Manifolds and Linear Actions 53 If we identify the Zn2 -morphisms gΛ, yΛ, uΛ with the corresponding continuous Zn2 -algebra morphisms g∗Λ ∈ HomZn2 Alg ( OG(\|G\|),Λ ) , y∗Λ ∈ HomZn2 Alg ( ORr\|s(R r),Λ ) , u∗Λ ∈ HomZn2 Alg ( OG(\|G\|)⊗̂ORr\|s(R r),Λ ) , we get u∗Λ = m̂Λ ◦ ( g∗Λ⊗̂ y∗Λ ) , where m̂Λ : Λ⊗̂Λ → Λ is continuous Zn2 -algebra morphism that extends the multiplication mΛ = ·Λ of Λ (see [15]). We denote the coordinates of G × Rr\|s as z = (zd) = (Xa b , y c). Since SΛ : ( G× Rr\|s ) (Λ) 3 uΛ ' (gΛ, yΛ) ' u∗Λ ' u∗Λ(zd) 7→ S ◦ uΛ ' u∗Λ ◦S∗ ' u∗Λ(S∗(yc)) ∈ Rr\|s(Λ), we find that SΛ ( gΛ, ∑ k λkyΛ,k ) ' m̂Λ (( g∗Λ⊗̂ (∑ k λkyΛ,k )∗)( S∗(yc) )) . In view of the definition of the Λ0-module structure on Rr\|s(Λ), we have(∑ k λkyΛ,k )∗ = ∑ k λky∗Λ,k and in view of the assumption (3.28), we get for any fixed c that S∗(yc) = M∑ n=1 snG ⊗R (∑ c′ rnc′y c′ ) , where M ∈ N, where snG ∈ OG(\|G\|) and where rnc′ ∈ R. Since λk ∈ Λ0, what we just said yields SΛ ( gΛ, ∑ k λkyΛ,k ) ' M∑ n=1 g∗Λ(snG) ·Λ ∑ k λk ·Λ (∑ c′ rnc′ y ∗ Λ,k(y c′) ) ' ∑ k λkSΛ(gΛ, yΛ,k). � 4 Future directions We view the current paper as the first steps towards understanding actions of Zn2 -Lie groups on Zn2 -manifolds and we claim that it will be vital in carefully constructing the total spaces of Zn2 -vector bundles, for example. In both these settings, the functor of points, and in particular Λ-points, are expected to be of fundamental importance. In particular, the typical fibres of Zn2 - vector bundles cannot be Zn2 -graded vector spaces, but rather they are linear Zn2 -manifolds. Moreover, the transition functions will correspond to an action of the general linear Zn2 -group and as such a careful understanding of linear actions is needed. This paper provides some of this technical background. We plan to explore the algebraic and geometric definitions of vector bundles in the category of Zn2 -manifolds in a future publication. 54 A.J. Bruce, E. Ibarguëngoytia and N. Poncin A The category of modules over a variable algebra We define the category AMod (resp., FAMod) of modules (resp., Fréchet modules) over any (unital) algebra (resp., any (unital) Fréchet algebra) A. The algebra A can vary from object to object. The objects are the modules over some A (resp., the Fréchet vector spaces that come equipped with a (compatible) continuous A-action). We denote such modules by MA. Morphisms consist of pairs (ϕ,Φ), where ϕ : A −→ B is an algebra morphism (resp., a continuous algebra morphism), and Φ: MA −→MB is a map (resp., a continuous map) that satisfies Φ(am+ a′m′) = ϕ(a)Φ(m) + ϕ(a′)Φ(m′), for all a, a′ ∈ A and m,m′ ∈MA. It is evident that we do indeed obtain a category in this way. The preceding categories AMod and FAMod are similar to the category AFMan that we used in [13]. They naturally appear when considering the zero degree rules functor or the functor of points. See for instance equations (2.1) and (2.22). B Basics of Zn 2 -geometry B.1 Zn 2 -manifolds and their morphisms The locally ringed space approach to Zn2 -manifolds was pioneered in [18]. We work over the field R of real numbers and set Zn2 := Z2×Z2×· · ·×Z2 (n-times). A Zn2 -graded algebra is an R- algebra Awith a decomposition into vector spaces A := ⊕γ∈Zn2 Aγ , such that the multiplication, say ·, respects the Zn2 -grading, i.e., Aα · Aβ ⊂ Aα+β. We will always assume the algebras to be associative and unital. If for any pair of homogeneous elements a ∈ Aα and b ∈ Aβ we have that a · b = (−1)〈α,β〉b · a, (B.1) where 〈−,−〉 is the standard scalar product on Zn2 , then A is a Zn2 -commutative algebra. Essentially, Zn2 -manifolds are “manifolds” equipped with both, standard commuting coordi- nates and formal coordinates of non-zero Zn2 -degree that Zn2 -commute according to the general sign rule (B.1). Note that in general we need to deal with formal coordinates that are not nilpotent. In order to keep track of the various formal coordinates, we need to introduce a convention on how we fix the order of elements in Zn2 and we choose the lexicographical order. For example, with this choice of ordering Z2 2 = {(0, 0), (0, 1), (1, 0), (1, 1)}. Note that other choices of ordering have appeared in the literature. A tuple q = (q1, q2, . . . , qN ) ∈ N×N (N = 2n − 1) provides the number of formal coordinates in each Zn2 -degree. We can now recall the definition of a Zn2 -manifold. Definition B.1. A (smooth) Zn2 -manifold of dimension p\|q is a locally Zn2 -ringed space M :=( \|M \|,OM ) , which is locally isomorphic to the Zn2 -ringed space Rp\|q := ( Rp, C∞Rp [[ξ]] ) . Local sections of M are formal power series in the Zn2 -graded variables ξ with smooth coefficients, OM (\|U \|) ' C∞(\|U \|)[[ξ]] := { ∞∑ α∈N×N fα ξ α : fα ∈ C∞(\|U \|) } , Linear Zn2 -Manifolds and Linear Actions 55 for “small enough” opens \|U \| ⊂ \|M \|. Morphisms between Zn2 -manifolds are morphisms of Zn2 - ringed spaces, that is, pairs Φ = (φ, φ∗) : ( \|M \|,OM ) → (\|N \|,ON ) consisting of a continuous map φ : \|M \| → \|N \| and a sheaf morphism φ∗ : ON (−) → OM ( φ−1(−) ) , i.e., a family of Zn2 -algebra morphisms φ∗\|V \| : ON (\|V \|) → OM ( φ−1(\|V \|) ) (\|V \| ⊂ \|N \| open) that commute with restrictions. We sometimes denote Zn2 -manifolds by M = (M,OM ) instead of M = ( \|M \|,OM ) and we some- times denote Zn2 -morphisms by φ = (\|φ\|, φ∗) instead of Φ = (φ, φ∗). Example B.2 (the local model). The locally Zn2 -ringed space Up\|q := ( Up, C∞Up [[ξ]] ) (Up ⊂ Rp open) is naturally a Zn2 -manifold – we refer to such Zn2 -manifolds as Zn2 -domains of dimension p\|q. We can employ (natural) coordinates ( xa, ξA ) on any Zn2 -domain, where the xa form a coordinate system on Up and the ξA are formal coordinates. Many of the standard results from the theory of supermanifolds pass over to Zn2 -manifolds. For example, the topological space \|M \| comes with the structure of a smooth manifold of dimension p, hence our suggestive notation. Moreover, there exists a canonical projection ε : OM → C∞\|M \|. What makes the category of Zn2 -manifolds a very tractable form of noncommutative geometry is the fact that we have local models. Much like in the theory of smooth manifolds, one can construct global geometric concepts via the gluing of local geometric concepts. That is, we can consider a Zn2 -manifold as being covered by Zn2 -domains together with specified gluing data. More precisely, a p\|q-chart (or p\|q-coordinate-system) over a (second-countable Hausdorff) smooth manifold \|M \| is a Zn2 -domain Up\|q = ( Up, C∞Up [[ξ]] ) , together with a diffeomorphism \|ψ\| : \|U \| → Up, where \|U \| is an open subset of \|M \|. Given two p\|q-charts( U p\|q α , \|ψα\| ) and ( U p\|q β , \|ψβ\| ) (B.2) over \|M \|, we set Vαβ := \|ψα\|(\|Uαβ\|) and Vβα := \|ψβ\|(\|Uαβ\|), where \|Uαβ\| := \|Uα\| ∩ \|Uβ\|. We then denote by \|ψβα\| the diffeomorphism \|ψβα\| := \|ψβ\| ◦ \|ψα\|−1 : Vαβ → Vβα. (B.3) Whereas in classical differential geometry the coordinate transformations are completely defined by the coordinate systems, in Zn2 -geometry, they have to be specified separately. A coordinate transformation between two charts, say the ones of (B.2), is an isomorphism of Zn2 -manifolds ψβα = ( \|ψβα\|, ψ∗βα ) : U p\|q α \|Vαβ → U p\|q β \|Vβα , (B.4) where the source and target are the open Zn2 -submanifolds U p\|q α \|Vαβ = ( Vαβ, C ∞ Vαβ [[ξ]] ) (note that the underlying diffeomorphism is (B.3)). A p\|q-atlas over \|M \| is a covering ( U p\|q α ,\|ψα\| ) α by charts together with a coordinate transformation (B.4) for each pair of charts, such that the usual cocycle condition ψβγψγα = ψβα holds (appropriate restrictions are understood). Moreover, we have the chart theorem [18, Theorem 7.10] that says that Zn2 -morphisms from a Zn2 -manifold (\|M \|,OM ) to a Zn2 -domain ( Up, C∞Up [[ξ]] ) are completely described by the pull- backs of the coordinates ( xa, ξA ) . In other words, to define a Zn2 -morphism valued in a Zn2 - domain, we only need to provide total sections ( sa, sA ) ∈ OM (\|M \|) of the source structure sheaf, whose degrees coincide with those of the target coordinates ( xa, ξA ) . Let us stress the condition (. . . , εsa, . . . )(\|M \|) ⊂ Up, where ε is the canonical projection, is often understood in the literature. 56 A.J. Bruce, E. Ibarguëngoytia and N. Poncin B.2 Zn 2 -Grassmann algebras, Zn 2 -points and the Schwarz–Voronov embedding It is clear that Zn2 -manifolds, as they are locally ringed spaces, are not fully determined by their topological points. To “claw back” a fully useful notion of a point, one can employ Grothendieck’s functor of points. This is, of course, an application of the Yoneda embedding (see [35, Chap- ter III, Section 2]). For the case of supermanifolds, it is well-known, via the seminal works of Schwarz and Voronov [39, 40, 46], that superpoints are sufficient to act as “probes” for the functor of points. That is, we only need to consider supermanifolds that have a single point as their underlying topological space. Dual to this, we may consider finite dimensional Grass- mann algebras Λ = Λ0 ⊕ Λ1 as parameterizing the “points” of a supermanifold. One can thus view supermanifolds as functors from the category of finite dimensional Grassmann algebras to sets. However, it turns out that the target category is not just sets, but (finite dimensional) Λ0-smooth manifolds. That is, the target category consists of smooth manifolds that have a Λ0-module structure on their tangent spaces. Morphisms in this category respect the module structure and are said to be Λ0-smooth (we will explain this further later on). In [13], it was shown how the above considerations generalize to the setting of Zn2 -manifolds. We will use the notations and results of [13] rather freely. We encourage the reader to consult this reference for the subtleties compared to the standard case of supermanifolds. A Zn2 -Grassmann algebra we define to be a formal power series algebra R[[θ]] in Zn2 -graded, Zn2 -commutative parameters θ`j . All the information about the number of generators is specified by the tuple q as before. We will denote a Zn2 -Grassmann algebra by Λ, as usually we do not need to specify the number of generators. A Zn2 -point is a Zn2 -manifold (that is isomor- phic to) R0\|q. It is clear, from Definition B.1, that the algebra of global sections of a Zn2 -point is precisely a Zn2 -Grassmann algebra. There is an equivalence between Zn2 -Grassmann algebras and Zn2 -points: Zn2GrAlg ∼= Zn2Ptsop. The Yoneda functor of points of the category Zn2Man of Zn2 -manifolds is the fully faithful embedding Y: Zn2Man 3M 7→ HomZn2 Man(−,M) ∈ Fun ( Zn2Manop, Set ) . 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A 44 (2001), 1167–1173, arXiv:math-ph/0212004. https://doi.org/10.1007/BFb0087788 https://doi.org/10.1070/RM1980v035n01ABEH001545 https://doi.org/10.7146/math.scand.a-12351 https://doi.org/10.1007/978-1-4757-4721-8 https://doi.org/10.1007/978-3-662-07386-5 https://doi.org/10.4064/bc110-0-14 https://doi.org/10.2307/2001189 https://doi.org/10.1007/BF01211055 https://doi.org/10.1007/BF01018248 https://doi.org/10.1016/j.aim.2004.05.004 https://arxiv.org/abs/math.AG/0207028 https://doi.org/10.1090/memo/0902 https://doi.org/10.1090/memo/0902 https://arxiv.org/abs/math.AG/0404373 https://doi.org/10.1007/978-4-431-55285-7_26 https://arxiv.org/abs/1610.01566 https://doi.org/10.1090/cln/011 https://doi.org/10.1007/BF01018249 https://doi.org/10.1007/BFb0061234 https://doi.org/10.1007/BF02877435 https://arxiv.org/abs/math-ph/0212004 1 Introduction 2 Z2n-graded vector spaces and Linear Z2n-manifolds 2.1 Z2n-graded vector spaces and the zero degree rules 2.2 Cartesian Z2n-graded vector spaces and Cartesian Z2n-manifolds 2.3 Finite dimensional Z2n-graded vector spaces and linear Z2n-manifolds 2.3.1 Finite dimensional Z2n-graded vector spaces 2.3.2 Linear Z2n-manifolds 2.3.3 Isomorphism between finite dimensional Z2n-graded vector spaces and linear Z2n-manifolds 3 Z2n-Lie groups and linear actions 3.1 Z2n-matrices 3.2 Invertibility of Z2n-matrices 3.3 Z2n-Lie groups and their functor of points 3.4 The general linear Z2n-group 3.5 Smooth linear actions 3.5.1 Canonical action of the general linear group 3.5.2 Connection between the canonical action and the internal Hom 3.5.3 Equivalent definitions of a smooth linear action 4 Future directions A The category of modules over a variable algebra B Basics of Z2n-geometry B.1 Z2n-manifolds and their morphisms B.2 Z2n-Grassmann algebras, Z2n-points and the Schwarz–Voronov embedding References
id	nasplib_isofts_kiev_ua-123456789-211363
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn	1815-0659
language	English
last_indexed	2026-03-14T22:43:45Z
publishDate	2021
publisher	Інститут математики НАН України
record_format	dspace
spelling	Bruce, Andrew James Ibarguëngoytia, Eduardo Poncin, Norbert 2025-12-30T15:56:40Z 2021 Linear ℤⁿ₂ -Manifolds and Linear Actions. Andrew James Bruce, Eduardo Ibarguëngoytia and Norbert Poncin. SIGMA 17 (2021), 060, 58 pages 1815-0659 2020 Mathematics Subject Classification: 58A50; 58C50; 14A22; 14L30; 13F25; 16L30; 17A70 arXiv:2011.01012 https://nasplib.isofts.kiev.ua/handle/123456789/211363 https://doi.org/10.3842/SIGMA.2021.060 We establish the representability of the general linear ℤⁿ₂-group and use the restricted functor of points – whose test category is the category of ℤⁿ₂-manifolds over a single topological point – to define its smooth linear actions on ℤⁿ₂-graded vector spaces and linear ℤⁿ₂-manifolds. Throughout the paper, particular emphasis is placed on the full faithfulness and target category of the restricted functor of points of a number of categories that we are using. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Linear ℤⁿ₂ -Manifolds and Linear Actions Article published earlier
spellingShingle	Linear ℤⁿ₂ -Manifolds and Linear Actions Bruce, Andrew James Ibarguëngoytia, Eduardo Poncin, Norbert
title	Linear ℤⁿ₂ -Manifolds and Linear Actions
title_full	Linear ℤⁿ₂ -Manifolds and Linear Actions
title_fullStr	Linear ℤⁿ₂ -Manifolds and Linear Actions
title_full_unstemmed	Linear ℤⁿ₂ -Manifolds and Linear Actions
title_short	Linear ℤⁿ₂ -Manifolds and Linear Actions
title_sort	linear ℤⁿ₂ -manifolds and linear actions
url	https://nasplib.isofts.kiev.ua/handle/123456789/211363
work_keys_str_mv	AT bruceandrewjames linearzn2manifoldsandlinearactions AT ibarguengoytiaeduardo linearzn2manifoldsandlinearactions AT poncinnorbert linearzn2manifoldsandlinearactions

Linear ℤⁿ₂ -Manifolds and Linear Actions

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