The Schwarz-Voronov Embedding of ℤⁿ₂-Manifolds
Informally, ℤⁿ₂-manifolds are 'manifolds' with ℤⁿ₂-graded coordinates and a sign rule determined by the standard scalar product of their ℤⁿ₂-degrees. Such manifolds can be understood in a sheaf-theoretic framework, as supermanifolds can, but with significant differences, in particular in i...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2020 |
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Інститут математики НАН України
2020
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | The Schwarz-Voronov Embedding of ℤⁿ₂-Manifolds. Andrew James Bruce, Eduardo Ibarguengoytia and Norbert Poncin. SIGMA 16 (2020), 002, 47 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860242684356067328 |
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| author | Bruce, Andrew James Ibarguengoytia, Eduardo Poncin, Norbert |
| author_facet | Bruce, Andrew James Ibarguengoytia, Eduardo Poncin, Norbert |
| citation_txt | The Schwarz-Voronov Embedding of ℤⁿ₂-Manifolds. Andrew James Bruce, Eduardo Ibarguengoytia and Norbert Poncin. SIGMA 16 (2020), 002, 47 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Informally, ℤⁿ₂-manifolds are 'manifolds' with ℤⁿ₂-graded coordinates and a sign rule determined by the standard scalar product of their ℤⁿ₂-degrees. Such manifolds can be understood in a sheaf-theoretic framework, as supermanifolds can, but with significant differences, in particular in integration theory. In this paper, we reformulate the notion of a ℤⁿ₂-manifold within a categorical framework via the functor of points. We show that it is sufficient to consider ℤⁿ₂-points, i.e., trivial ℤⁿ₂-manifolds for which the reduced manifold is just a single point, as 'probes' when employing the functor of points. This allows us to construct a fully faithful restricted Yoneda embedding of the category of ℤⁿ₂-manifolds into a subcategory of contravariant functors from the category of ℤⁿ₂-points to a category of Fréchet manifolds over algebras. We refer to this embedding as the Schwarz-Voronov embedding. We further prove that the category of ℤⁿ₂-manifolds is equivalent to the full subcategory of locally trivial functors in the preceding subcategory.
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| first_indexed | 2025-12-17T12:04:19Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 002, 47 pages
The Schwarz–Voronov Embedding of Zn
2 -Manifolds
Andrew James BRUCE, Eduardo IBARGUENGOYTIA and Norbert PONCIN
Mathematics Research Unit, University of Luxembourg, Maison du Nombre 6,
avenue de la Fonte, L-4364 Esch-sur-Alzette, Luxembourg
E-mail: andrew.bruce@uni.lu, eduardo.ibarguengoytia@uni.lu, norbert.poncin@uni.lu
Received July 10, 2019, in final form December 30, 2019; Published online January 08, 2020
https://doi.org/10.3842/SIGMA.2020.002
Abstract. Informally, Zn
2 -manifolds are ‘manifolds’ with Zn
2 -graded coordinates and a sign
rule determined by the standard scalar product of their Zn
2 -degrees. Such manifolds can
be understood in a sheaf-theoretic framework, as supermanifolds can, but with significant
differences, in particular in integration theory. In this paper, we reformulate the notion of
a Zn
2 -manifold within a categorical framework via the functor of points. We show that it
is sufficient to consider Zn
2 -points, i.e., trivial Zn
2 -manifolds for which the reduced manifold
is just a single point, as ‘probes’ when employing the functor of points. This allows us
to construct a fully faithful restricted Yoneda embedding of the category of Zn
2 -manifolds
into a subcategory of contravariant functors from the category of Zn
2 -points to a category
of Fréchet manifolds over algebras. We refer to this embedding as the Schwarz–Voronov
embedding. We further prove that the category of Zn
2 -manifolds is equivalent to the full
subcategory of locally trivial functors in the preceding subcategory.
Key words: supergeometry; superalgebra; ringed spaces; higher grading; functor of points
2010 Mathematics Subject Classification: 58C50; 58D1; 14A22
1 Introduction
Various notions of graded geometry play an important rôle in mathematical physics and can often
provide further insight into classical geometric constructions. For example, supermanifolds, as
pioneered by Berezin and collaborators, are essential in describing quasi-classical systems with
both bosonic and fermionic degrees of freedom. Very loosely, supermanifolds are ‘manifolds’
for which the structure sheaf is Z2-graded. Such geometries are of fundamental importance in
perturbative string theory, supergravity, and the BV-formalism, for example. While the theory
of supermanifolds is firmly rooted in theoretical physics, it has since become a respectable area
of mathematical research. Indeed, supermanifolds allow for an economical description of Lie
algebroids, Courant algebroids as well as various related structures, many of which are of direct
interest to physics. We will not elaborate any further and urge the reader to consult the ever-
expanding literature.
Interestingly, Zn2 -gradings (Zn2 = Z×n2 , n ≥ 2) can be found in the theory of parastatistics, see
for example [22, 24, 25, 58], behind an alternative approach to supersymmetry [51], in relation
to the symmetries of the Lévy-Lebond equation [2], and behind the theory of mixed symmetry
tensors [11]. Generalizations of the super Schrödinger algebra (see [3]) and the super Poincaré
algebra (see [10]) have also appeared in the literature. That said, it is unknown if these ‘higher
gradings’ are of the same importance in fundamental physics as Z2-gradings. It must also be
remarked that the quaternions and more general Clifford algebras can be understood as Zn2 -
graded Zn2 -commutative (see below) algebras [4, 5]. Thus, one may expect Zn2 -gradings to be
important in studying Clifford algebras and modules, though the implications for classical and
quantum field theory remain as of yet unexplored. It should be further mentioned that any ‘sign
mailto:andrew.bruce@uni.lu
mailto:eduardo.ibarguengoytia@uni.lu
mailto:norbert.poncin@uni.lu
https://doi.org/10.3842/SIGMA.2020.002
2 A.J. Bruce, E. Ibarguengoytia and N. Poncin
rule’ can be understood in terms of a Zn2 -grading (see [15]). A natural question here is to what
extent can Zn2 -graded geometry be developed.
A locally ringed space approach to Zn2 -manifolds has been constructed in a series of papers by
Bruce, Covolo, Grabowski, Kwok, Ovsienko & Poncin [11, 13, 15, 16, 17, 18, 19, 35]. It includes
the Zn2 -differential-calculus, the Zn2 -Berezinian, as well as a low-dimensional Zn2 -integration-
theory. Integration on Zn2 -manifolds turns out to be fundamentally different from integration
on Z1
2-manifolds (i.e., supermanifolds) and is currently being constructed in full generality by
authors of the present paper. The novel aspect of integration on Zn2 -manifolds is integration
with respect to the non-zero degree even parameters (for some preliminary results see [35]).
Loosely, Zn2 -manifolds are ‘manifolds’ for which the structure sheaf has a Zn2 -grading and
the commutation rule for the local coordinates comes from the standard scalar product of their
Zn2 -degrees. This is not just a trivial or straightforward generalization of the notion of a su-
permanifold 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 (for standard supermanifolds
all functions are polynomial in the Grassmann odd variables). The use of formal power series is
unavoidable in order to have a well-defined local theory (see [15]), and a well-defined differential
calculus (see [17]). Heuristically, one can view supermanifolds as ‘mild’ noncommutative geo-
metries: the noncommutativity is seen simply as anticommutativity of the odd coordinates. In
a similar vein, one can view Zn2 -manifolds (n > 1) as examples of ‘mild’ nonsupercommutative
geometries: the sign rule involved is not determined by the coordinates being even or odd, i.e.,
by their total degree, but by their Zn2 -degree.
The idea of understanding supermanifolds, i.e., Z1
2-manifolds, as ‘Grassmann algebra valued
manifolds’ can be traced back to the pioneering work of Berezin [9]. An informal understanding
along these lines has continuously been employed in physics, where one chooses a ‘large enough’
Grassmann algebra to capture the aspects to the theory needed. This informal understanding
leads to the DeWitt–Rogers approach to supermanifolds which seemed to avoid the theory of
locally ringed spaces altogether. However, arbitrariness in the choice of the underlying Grass-
mann algebra is somewhat displeasing. Furthermore, developing the mathematical consistency
of DeWitt–Rogers supermanifolds takes one back to the sheaf-theoretic approach of Berezin &
Leites: for a comparison of these approaches, the reader can consult Rogers [38] or Schmitt [42].
From a physics perspective, there seems no compelling reason to think that there is any physical
significance to the choice of underlying Grassmann algebra. To quote Schmitt [42]: “However,
no one has ever measured a Grassmann number, everyone measures real numbers”. The solution
here is, following Schwarz & Voronov [43, 44, 54], not to fix the underlying Grassmann alge-
bra, but rather understand supermanifolds as functors from the category of finite-dimensional
Grassmann algebras to, in the first instance, the category of sets. For a given, but arbitrary,
Grassmann algebra Λ, one speaks of the set of Λ-points of a supermanifold. It is well known that
the set of Λ-points of a given supermanifold comes with the further structure of a Λ0-smooth
manifold. That is we, in fact, do not only have a set, but also the structure of a finite-dimensional
manifold whose tangent spaces are Λ0-modules. Moreover, thinking of supermanifolds as func-
tors, not all natural transformations between the Λ-points correspond to genuine supermanifold
morphisms, only those that respect the Λ0-smooth structure do. A similar approach is used by
Molotkov [34], who defines Banach supermanifolds roughly speaking as specific functors from the
category of finite-dimensional Grassmann algebras to the category of smooth Banach manifolds
of a particular type. The classical roots of these ideas go back to Weil [56] who considered the
A-points of a manifold as the set of maps from the algebra of smooth functions on the mani-
fold to a specified finite-dimensional commutative local algebra A. Today one refers to Weil
functors and these have long been utilised in the theory of jet structures over manifolds, see for
example [28].
The Schwarz–Voronov Embedding of Zn2 -Manifolds 3
In this paper, we study Grothendieck’s functor of points [26] of a Zn2 -manifold M , which is
a contravariant functor M(−) from the category of Zn2 -manifolds to the category of sets, and
restrict it to the category of Zn2 -points, i.e., trivial Zn2 -manifolds R0|q that have no degree zero
coordinates. More precisely, we consider the restricted Yoneda functor M 7→ M(−) from the
category of Zn2 -manifolds to the category of contravariant functors from Zn2 -points to sets. Dual
to Zn2 -points R0|q are what we will call Zn2 -Grassmann algebras Λ (see Definition 2.3). The
aim of this paper is to carefully prove and generalise the main results of Schwarz & Voronov
[44, 54] to the ‘higher graded’ setting. In particular, we show that Zn2 -points R0|q ' Λ are
actually sufficient to act as ‘probes’ when employing the functor of points (see Theorem 3.8).
However, not all natural transformations ηΛ : M(Λ) → N(Λ) (where Λ is a variable) between
the sets M(Λ), N(Λ) of Λ-points correspond to morphisms φ : M → N of the underlying Zn2 -
manifolds. By carefully analysing the image of the functor of points, we prove that the set
M(Λ) of Λ-points of a Zn2 -manifold M comes with the extra structure of a Fréchet Λ0-manifold
(see Theorem 3.22; by Λ0 we mean the subalgebra of degree zero elements of the Zn2 -Grassmann
algebra Λ). Note that we are not trying to define infinite-dimensional Zn2 -manifolds, yet infinite-
dimensional manifolds, specifically Fréchet manifolds, are fundamental to our paper. Moreover,
we show that natural transformations ηΛ between sets of Λ-points arise from morphisms φ
of Zn2 -manifolds if and only if they respect the Fréchet Λ0-manifold structures (see Proposi-
tion 3.24). By restricting accordingly the natural transformations allowed, we get a full and
faithful embedding of the category of Zn2 -manifolds into the category of contravariant functors
from the category of Zn2 -points to the category of nuclear Fréchet manifolds over nuclear Fréchet
algebras. This embedding we refer to as the Schwarz–Voronov embedding (see Definition 3.28).
We finally study representability of such contravariant functors and prove that the category
of Zn2 -manifolds is equivalent to the full subcategory of locally trivial functors in the just de-
picted subcategory of contravariant functors from Zn2 -points to nuclear Fréchet manifolds (see
Theorem 3.34).
Methodology: As Zn2 -manifolds have well defined local models, we work with Zn2 -domains
and then ‘globalize’ the results to general Zn2 -manifolds. We modify the approach of Schwarz
& Voronov [44, 54] and draw on Balduzzi, Carmeli & Fioresi [7, 8] and Konechny & Schwarz
[29, 30], making all changes necessary to encompass Zn2 -manifolds. Let us mention that Balduzzi,
Carmeli & Fioresi study functors from the category of super Weil algebras and not that of
Grassmann algebras. However, if we truly want to build a restricted Yoneda embedding, the
source category of the functors of points must be a category of algebras that is opposite to
some category of supermanifolds – and super Weil algebras are not the algebras of functions of
some class of supermanifolds (unless they are Grassmann algebras). Moreover, the idea behind
our restriction of the Yoneda embedding is ‘the smaller the class of test algebras, the better’ –
which points again to Grassmann algebras as being the somewhat privileged objects. The most
striking difference between supermanifolds and Zn2 -manifolds (n > 1) is that we are forced,
due to the presence of non-zero degree even coordinates, to work with (infinite-dimensional)
Fréchet spaces, algebras and manifolds. Interestingly, nuclearity of the values M(Λ) of the
functor of points of a Zn2 -manifold M , i.e., nuclearity of the local models of the Fréchet Λ0-
manifolds M(Λ) or of their tangent spaces, does not play a rôle in the proofs of the statements
in this paper. More precisely, the functor of points M(−) has values M(Λ) that are nuclear
Fréchet Λ0-manifolds. Conversely, a functor F(−) whose values F(Λ) are Fréchet Λ0-manifolds
and which is representable, has nuclear values (nuclearity is encrypted in the representability
condition (see Theorem 3.34)). Although nuclearity of the tangent spaces of the manifolds M(Λ)
is not explicitly used throughout this work, we do not at all claim that nuclearity is not of
importance in the theory of Zn2 -manifolds. For instance, the function sheaf of a Zn2 -manifold is
a nuclear Fréchet sheaf of Zn2 -graded Zn2 -commutative algebras – a fact that is crucial for product
Zn2 -manifolds and Zn2 -Lie groups [13].
4 A.J. Bruce, E. Ibarguengoytia and N. Poncin
Applications: The functor of points has been used informally in physics as from the very
beginning. It is actually of importance in situations where there is no good notion of point (see
also Section 2.2), for instance in algebraic geometry and in super- and Zn2 -geometry. Construc-
ting a set-valued functor and showing that it is representable as a locally ringed space, e.g.,
a scheme or a Zn2 -manifold, is often easier than building that scheme or manifold directly.
Functors that are not representable can be interpreted as generalised schemes or generalised Zn2 -
manifolds. Further, the category of functors is better behaved than the corresponding category of
supermanifolds or of other types of spaces. Also homotopical algebraic geometry [49, 50], as well
as its generalisation that goes under the name of homotopical algebraic D-geometry (where D
refers to differential operators) [20, 21], are fully based on the functor of points approach. Finally,
the functor of points turns out to be an indispensable tool when it comes to the investigation
of Zn2 -Lie groups and their actions on Zn2 -manifolds, of geometric Zn2 -vector bundles . . . . These
concepts are explored in upcoming texts that are currently being written down.
Arrangement: In Section 2, we review the basic tenets of Zn2 -geometry and the theory of
Zn2 -manifolds. The bulk of this paper is to be found in Section 3. We rely on two appendices: in
Appendix A we recall the notion of a generating set of a category, and in Appendix B we review
indispensable concepts from the theory of Fréchet spaces, algebras and manifolds.
2 Rudiments of Zn
2 -graded geometry
2.1 The category of Zn
2 -manifolds
The locally ringed space approach to Zn2 -manifolds is presented in a series of papers [15, 16, 17,
18, 19, 35] by Covolo, Grabowski, Kwok, Ovsienko, and Poncin. We will draw upon these works
heavily and not present proofs of any formal statements.
Definition 2.1. A locally Zn2 -ringed space, n ∈ N, is a pair X := (|X|,OX), where |X| is
a second-countable Hausdorff space, and OX is a sheaf of Zn2 -graded Zn2 -commutative associative
unital R-algebras, such that the stalks Op, p ∈ |X|, are local rings.
In this context, Zn2 -commutative means that any two sections a, b ∈ OX(|U |), |U | ⊂ |X| open,
of homogeneous degrees deg(a) = a ∈ Zn2 and deg(b) = b ∈ Zn2 commute according to the sign
rule
ab = (−1)〈a,b〉ba,
where 〈−,−〉 is the standard scalar product on Zn2 . We will say that a section a is even or odd
if 〈a, a〉 ∈ Z2 is 0 or 1.
Just as in standard supergeometry, which we recover for n = 1, a locally Zn2 -ringed space
is a Zn2 -manifold if it is locally isomorphic to a specific local model. Given the central rôle
of (finite-dimensional) Grassmann algebras in the theory of supermanifolds, we consider here
Zn2 -Grassmann algebras.
Remark 2.2. In the following, we order the elements in Zn2 lexicographically, and refer to this
ordering as the standard ordering. For example, we thus get
Z2
2 = {(0, 0), (0, 1), (1, 0), (1, 1)}.
Definition 2.3. A Zn2 -Grassmann algebra Λq := R[[ξ]] is the Zn2 -graded Zn2 -commutative associa-
tive unital R-algebra of all formal power series with coefficients in R generated by homogeneous
parameters ξα subject to the commutation relation
ξαξβ = (−1)〈α,β〉ξβξα,
The Schwarz–Voronov Embedding of Zn2 -Manifolds 5
where α := deg(ξα) ∈ Zn2 \0, 0 = (0, . . . , 0). The tuple q = (q1, q2, . . . , qN ), N = 2n−1, provides
the number qi of generators ξα, which have the i-th degree in Zn2 \ 0 (endowed with its standard
order).
A morphism of Zn2 -Grassmann algebras, ψ∗ : Λq → Λp, is a map of R-algebras that preserves
the Zn2 -grading and the units.
We denote the category of Zn2 -Grassmann algebras and corresponding morphisms by Zn2GrAlg.
Example 2.4. For n = 0, we simply get R considered as an algebra over itself.
Example 2.5. If n = 1, we recover the classical concept of Grassmann algebra with the standard
supercommutation rule for generators. In this case, all formal power series truncate to polyno-
mials. In particular, the Grassmann algebra generated by a single odd generator is isomorphic
to the algebra of dual numbers.
Example 2.6. The Z2
2-Grassmann algebra Λ(1,1,1) is described by three generators(
ξ︸︷︷︸
(0,1)
, θ︸︷︷︸
(1,0)
, z︸︷︷︸
(1,1)
)
,
where we have indicated the Z2
2-degree. Note that ξθ = θξ, while ξ2 = 0 and θ2 = 0. Moreover,
ξz = −zξ and θz = −zθ, while z is not nilpotent. A general (inhomogeneous) element of Λ(1,1,1)
is then of the form
f(ξ, θ, z) = fz(z) + ξfξ(z) + θfθ(z) + ξθfξθ(z),
where fz(z), fξ(z), fθ(z) and fξθ(z) are formal power series in z. As a subalgebra we can con-
sider Λ(1,1,0), whose generators are ξ and θ. A general element of this subalgebra is a polynomial
in these generators.
Within any Zn2 -Grassmann algebra Λ := Λq, we have the ideal generated by the generators
of Λ, which we will denote as Λ̊. In particular we have the decomposition
Λ = R⊕ Λ̊,
which will be used later on. Moreover, the set of degree 0 elements, Λ0 ⊂ Λ, is a commutative
associative unital R-algebra.
Very informally, a Zn2 -manifold is a smooth manifold whose structure sheaf has been ‘de-
formed’ to now include the generators of a Zn2 -Grassmann algebra.
Definition 2.7. A (smooth) Zn2 -manifold of dimension p|q is a locally Zn2 -ringed space M :=
(|M |,OM ), which is locally isomorphic to the locally Zn2 -ringed space Rp|q := (Rp, C∞Rp [[ξ]]).
Local sections of OM are thus formal power series in the Zn2 -graded variables ξ with smooth
coefficients,
OM (|U |) ' C∞Rp(|U |)[[ξ]] :=
{ ∑
α∈N
∑
iqi
ξαfα : fα ∈ C∞Rp(|U |)
}
,
for ‘small enough’ open subsets |U | ⊂ |M |. A Zn2 -morphism, i.e., a morphism between two
Zn2 -manifolds, say M and N , is a morphism of Zn2 -ringed spaces, that is, a pair φ = (|φ|, φ∗) :
(|M |,OM ) → (|N |,ON ) consisting of a continuous map |φ| : |M | → |N | and a sheaf morphism
φ∗ : ON → |φ|∗OM , i.e., a family of Zn2 -graded unital R-algebra morphisms φ∗|V | : ON (|V |) →
OM (|φ|−1(|V |)) (|V | ⊂ |N | open), which commute with restrictions. We will refer to the global
sections of the structure sheaf OM as functions on M and denote them as C∞(M) := OM (|M |).
6 A.J. Bruce, E. Ibarguengoytia and N. Poncin
Example 2.8 (the local model). The locally Zn2 -ringed space Up|q :=
(
Up, C∞Up [[ξ]]
)
, where
Up ⊂ Rp is 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, ξα) on any Zn2 -domain, where the xa
form a coordinate system on Up and the ξα are formal coordinates.
Canonically associated to any Zn2 -graded algebra A is the homogeneous ideal J of A generated
by all homogeneous elements of A having nonzero degree. If f : A → A′ is a morphism of Zn2 -
graded algebras, then f(JA) ⊂ JA′ . The J-adic topology plays a fundamental rôle in the theory
of Zn2 -manifolds. In particular, these notions can be ‘sheafified’. That is, for any Zn2 -manifold M ,
there exists an ideal sheaf JM , defined by J (|U |) = 〈f ∈ OM (|U |) : deg(f) 6= 0〉. The JM -adic
topology on OM can then be defined in the obvious way.
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 and the continuous base map of any Zn2 -morphism is actually smooth. Further, for any
Zn2 -manifold M , there exists a short exact sequence of sheaves of Zn2 -graded Zn2 -commutative
associative R-algebras
0 −→ ker ε −→ OM
ε−→ C∞|M | −→ 0,
such that ker ε = JM .
The immediate problem with Zn2 -manifolds is that JM is not nilpotent – for supermanifolds
the ideal sheaf is nilpotent and this is a fundamental property that makes the theory of su-
permanifolds so well-behaved. However, this loss of nilpotency is compensated by Hausdorff
completeness of OM with respect to the JM -adic topology.
Proposition 2.9. Let M be a Zn2 -manifold. Then OM is JM -adically Hausdorff complete as
a sheaf of Zn2 -commutative associative unital R-algebras, i.e., the morphism
OM → lim
←k
OM/J kM ,
naturally induced by the filtration of OM by the powers of JM , is an isomorphism.
The presence of formal power series in the coordinate rings of Zn2 -manifolds forces one to rely
on the Hausdorff-completeness of the J -adic topology. This completeness replaces the standard
fact that supermanifold functions of Grassmann odd variables are always polynomials – a result
that is often used in extending results from smooth manifolds to supermanifolds.
What makes Zn2 -manifolds a very workable form of noncommutative geometry is the fact
that we have well-defined local models. Much like the theory of 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 information, i.e.,
coordinate transformations. Moreover, we have the chart theorem [15, 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 pullbacks of the coordinates (xa, ξα). In other words, to define a Zn2 -morphism
valued in a Zn2 -domain, we only need to provide total sections (sa, sα) ∈ OM (|M |) of the source
structure sheaf, whose degrees coincide with those of the target coordinates (xa, ξα). Let us
stress the condition (. . . , εsa, . . .)(|M |) ⊂ Up, which is often understood in the literature.
A few words about the atlas definition of a Zn2 -manifold are necessary. Let p|q be as above.
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]]
)
,
The Schwarz–Voronov Embedding of Zn2 -Manifolds 7
together with a diffeomorphism |ψ| : |U | → Up, where |U | is an open subset of |M |. Given two
p|q-charts(
Up|qα , |ψα|
)
and
(
Up|qβ , |ψβ|
)
(2.1)
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. (2.2)
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 (2.1), is an isomorphism of Zn2 -manifolds
ψβα = (|ψβα|, ψ∗βα) : Up|qα |Vαβ → U
p|q
β |Vβα , (2.3)
where the source and target are the open Zn2 -submanifolds
Up|qα |Vαβ =
(
Vαβ, C
∞
Vαβ
[[ξ]]
)
(note that the underlying diffeomorphism is (2.2)). A p|q-atlas over |M | is a covering
(
Up|qα ,|ψα|
)
α
by charts together with a coordinate transformation (2.3) for each pair of charts, such that the
usual cocycle condition ψβγψγα = ψβα holds (appropriate restrictions are understood).
Definition 2.10. A (smooth) Zn2 -manifold of dimension p|q is a (second-countable Hausdorff)
smooth manifold |M | together with a preferred p|q-atlas over it.
As in standard supergeometry, the Definitions 2.7 and 2.10 are equivalent [31]. For instance,
if M = (|M |,OM ) is a Zn2 -manifold of dimension p|q in the sense of Definition 2.7, there are
Zn2 -isomorphisms (isomorphisms of Zn2 -manifolds)
hα = (|hα|, h∗α) : Uα = (|Uα|,OM ||Uα|)→ U
p|q
α =
(
Upα, C∞Rp |Upα [[ξ]]
)
,
such that (|Uα|)α is an open cover of |M |. For any two indices α, β, the restriction hα|Uαβ of hα
to the open Zn2 -submanifold Uαβ = (|Uαβ|,OM ||Uαβ |), |Uαβ| = |Uα| ∩ |Uβ|, is a Zn2 -isomorphism
between Uαβ and
Up|qα |Vαβ =
(
Vαβ, C
∞
Rp |Vαβ [[ξ]]
)
, Vαβ = |hα|(|Uαβ|).
Therefore, the composite
ψβα = hβ|Uβαhα|
−1
Uαβ
is a Zn2 -isomorphism
ψβα : Up|qα |Vαβ → U
p|q
β |Vβα ,
such that the cocycle condition is satisfied.
As a matter of some formality, Zn2 -manifolds and their morphisms form a category. The
category of Zn2 -manifolds we will denote as Zn2Man. We remark this category is locally small.
Moreover, as shown in [13, Theorem 19], the category of Zn2 -manifolds admits (finite) products.
More precisely, let Mi, i ∈ {1, 2}, be Zn2 -manifolds. Then there exists a Zn2 -manifold M1 ×M2
and Zn2 -morphisms πi : M1 ×M2 →Mi (with underlying smooth manifold |M1 ×M2| = |M1| ×
|M2| and with underlying smooth morphisms |πi| : |M1| × |M2| → |Mi| given by the canonical
projections), such that for any Zn2 -manifold N and Zn2 -morphisms fi : N → Mi, there exists
a unique morphism h : N → M1 ×M2 making the obvious diagram commute. It follows that,
if φ ∈ HomZn2 Man(M,M ′) and ψ ∈ HomZn2 Man(N,N
′), there is a unique morphism φ × ψ ∈
HomZn2 Man(M ×N,M
′ ×N ′).
8 A.J. Bruce, E. Ibarguengoytia and N. Poncin
Remark 2.11. It is known that an analogue of the Batchelor–Gawȩdzki theorem holds in the
category of (real) Zn2 -manifolds, see [16, Theorem 3.2]. That is, any Zn2 -manifold is noncanoni-
cally isomorphic to a Zn2 \ {0}-graded vector bundle over a smooth manifold. While this result
is quite remarkable, we will not exploit it at all in this paper.
2.2 The functor of points
Similar to what happens in classical supergeometry, a Zn2 -manifold M is not fully described by
its topological points in |M |. To remedy this defect, we broaden the notion of ‘point’, as was
suggested by Grothendieck in the context of algebraic geometry.
More precisely, set V = {z ∈ Cn : P (z) = 0} ∈ Aff, where P denotes a polynomial
in n indeterminates with complex coefficients and Aff denotes the category of affine varieties.
Grothendieck insisted on solving the equation P (z) = 0 not only in Cn, but in An, for any
algebra A in the category CA of commutative (associative unital) algebras (over C). This leads
to an arrow
SolP : CA 3 A 7→ SolP (A) =
{
a ∈ An : P (a) = 0
}
∈ Set,
which turns out to be a functor
SolP ' HomCA(C[V ],−) ∈ [CA, Set],
where C[V ] is the algebra of polynomial functions of V . The dual of this functor, whose value
SolP (A) is the set of A-points of V , is the functor
HomAff(−, V ) ∈
[
Affop, Set
]
,
whose value HomAff(W,V ) is the set of W -points of V .
The latter functor can be considered not only in Aff, but in any locally small category, in
particular in Zn2Man. We thus obtain a covariant functor (functor in •)
•(−) = Hom(−, •) : Zn2Man 3M 7→M(−) = HomZn2 Man(−,M) ∈
[
Zn
2Man
op, Set
]
. (2.4)
As suggested above, the contravariant functor Hom(−,M) (we omit the subscript Zn2Man) (func-
tor in −) is referred to as the functor of points of M . If S ∈ Zn2Man, an S-point of M is just
a morphism πS ∈ Hom(S,M). One may regard an S-point of M as a ‘family of points of M
parameterised by the points of S’. The functor •(−) is known as the Yoneda embedding. For
any underlying locally small category C (here C = Zn2Man), the functor •(−) is fully faithful, what
means that, for any M,N ∈ Zn2Man, the map
•M,N (−) : Hom(M,N) 3 φ 7→ Hom(−, φ) ∈ Nat(Hom(−,M),Hom(−, N))
is bijective (here Nat denotes the set of natural transformations). It can be checked that the
correspondence •M,N (−) is natural in M and in N . Moreover, any fully faithful functor is
automatically injective up to isomorphism on objects: M(−) ' N(−) implies M ' N . Of
course, the functor •(−) is not surjective up to isomorphism on objects, i.e., not every functor
X ∈ [Zn2Manop, Set] is isomorphic to a functor of the type M(−). However, if such M does exist,
it is, due to the mentioned injectivity, unique up to isomorphism and it is called ‘the’ representing
Zn2 -manifold of X. Further, if X,Y ∈ [Zn2Manop, Set] are two representable functors, represented
by M , N respectively, a morphism or natural transformation between them, provides, due to
the mentioned bijectivity, a unique morphism between the representing Zn2 -manifolds M and N .
It follows that, instead of studying the category Zn2Man, we can just as well focus on the functor
category [Zn
2Man
op, Set] (which has better properties, in particular it has all limits and colimits).
The Schwarz–Voronov Embedding of Zn2 -Manifolds 9
A generalized Zn2 -manifold is an object in the functor category [Zn2Manop, Set] and morphisms
of such objects are natural transformations. The category [Zn2Manop, Set] of generalised Zn2 -
manifolds has finite products. Indeed, if F,G are two generalized manifolds, we define the functor
F ×G, given on objects S, by (F ×G)(S) = F (S)×G(S), and on morphisms Ψ: S → T , by
(F ×G)(Ψ) = F (Ψ)×G(Ψ): F (T )×G(T )→ F (S)×G(S).
It is easily seen that F × G respects compositions and identities. Further, there are canonical
natural transformations η1 : F × G → F and η2 : F × G → G. If now (H,α1, α2) is another
functor with natural transformations from it to F and G, respectively, it is straightforwardly
checked that there exists a unique natural transformation β : H → F ×G, such that αi = ηi ◦β.
One passes from the category of Zn2 -manifolds to the larger category of generalised Zn2 -manifolds
in order to understand, for example, the internal Hom objects. In particular, there always exists
a generalised Zn2 -manifold such that the so-called adjunction formula holds
HomZn2 Man(M,N)(−) := HomZn2 Man(−×M,N).
This internal Hom functor is defined on φ ∈ HomZn2 Man(P, S) by
HomZn2 Man(M,N)(φ) : HomZn2 Man(M,N)(S) −→ HomZn2 Man(M,N)(P ),
ΨS 7−→ ΨS ◦ (φ× 1M ).
In general, a mapping Zn2 -manifold HomZn2 Man(M,N) will not be representable. We will refer to
‘elements’ of a mapping Zn2 -manifold as maps reserving morphisms for the categorical morphisms
of Zn2 -manifolds.
Composition of maps between Zn2 -manifolds is naturally defined as a natural transformation
◦ : Hom(M,N)× Hom(N,L) −→ Hom(M,L),
defined, for any S ∈ Zn2Man, by
Hom(S ×M,N)× Hom(S ×N,L) −→ Hom(S ×M,L),
(ΨS ,ΦS) 7−→ (Φ◦Ψ)S := ΦS ◦ (1S ×ΨS) ◦ (∆× 1M ),
where ∆: S −→ S × S is the diagonal of S and 1S : S −→ S is its identity.
Similarly to the cases of smooth manifolds and supermanifolds, morphisms between Zn2 -
manifolds are completely determined by the corresponding maps between the global functions.
We remark that this is not, in general, true for complex (super)manifolds. More carefully, we
have the following proposition that was proved in [13, Theorem 3.7].
Proposition 2.12. Let M = (|M |,OM ) and N = (|N |,ON ) be Zn2 -manifolds. Then the natural
map
HomZn2 Man
(
M,N
)
−→ HomZn2 Alg
(
O(|N |),O(|M |)
)
,
where Zn
2Alg denotes the category of Zn2 -graded Zn2 -commutative associative unital R-algebras, is
a bijection.
It is worth recalling how a morphism ψ ∈ HomZn2 Alg
(
O(|N |),O(|M |)) defines a continuous
base map |φ| : |M | → |N |. We denote by εm ∈ HomZn2 Alg
(
O(|M |),R), m ∈ |M |, the morphism
εm : O(|M |) 3 f 7→ (ε|M |f)(m) ∈ R,
10 A.J. Bruce, E. Ibarguengoytia and N. Poncin
and by Spm(O(|M |)) the maximal spectrum of the algebra O(|M |). The map
[ : |M | 3 m 7→ ker εm ∈ Spm(O(|M |))
is a homeomorphism, both, when the target space is endowed with its Zariski topology and when
it is endowed with its Gel’fand topology. The continuous map |φ| : |M | → |N | that is induced
by the morphism ψ is now given by
|φ| : |M | ' Spm(O(|M |)) 3 m ' ker εm 7→ ker(εm ◦ ψ) ' n ∈ Spm(O(|N |)) ' |N |.
The fact that the functor HomZn2 Man(S,−) respects limits and in particular products directly
implies that(
M ×N
)
(S) 'M(S)×N(S). (2.5)
The latter result is essential in dealing with Zn2 -Lie groups. A (super) Lie group can be defined
as a group object in the category of smooth (super)manifolds. This leads us to the following
definition.
Definition 2.13. A Zn2 -Lie group is a group object in the category of Zn2 -manifolds.
A convenient fact here is that, if G is a Zn2 -Lie group, then the set G(S) is a group (see (2.5)).
In other words, G(−) is a functor from Zn2Manop → Grp.
Remark 2.14. We leave details and examples of Zn2 -Lie groups for future publications. However,
we will remark at this point that the idea of “colour supergroup manifolds” has already appeared
in the physics literature, albeit without a proper mathematical definition (see [1, 3, 36, 37], for
example). Another approach to Zn2 -Lie groups is via a generalisation of Harish-Chandra pairs
(see [33] for work in this direction).
3 Zn
2 -points and the functor of points
In view of (2.4), we need to ‘probe’ a given Zn2 -manifold M 'M(−) with all Zn2 -manifolds. We
will show that this is however not the case, since, much like for the category of supermanifolds,
we have a rather convenient generating set that we can employ, namely the set of Zn2 -points.
3.1 The category of Zn
2 -points
Definition 3.1. A Zn2 -point is a Zn2 -manifold R0|m with vanishing ordinary dimension. We
denote by Zn2Pts the full subcategory of Zn2Man, whose collection of objects is the (countable)
set of Zn2 -points.
Morphisms φ : R0|m → R0|n of Zn2 -points are exactly morphisms φ∗ : Λn → Λm of Zn2 -
Grassmann algebras:
Proposition 3.2. There is an isomorphism of categories
Zn2Pts ' Zn2GrAlgop.
We can think of Zn2 -points as formal thickenings of an ordinary point by the non-zero degree
generators. The simplest Zn2 -point is the one with trivial formal thickening, R0|0 :=
(
R0,R
)
:
Proposition 3.3. The Zn2 -point R0|0 = R0 is a terminal object in both, Zn2Man and Zn2Pts.
The Schwarz–Voronov Embedding of Zn2 -Manifolds 11
Proof. The unique morphism M −→ R0|0 corresponds to the morphism R 3 r · 1 7→ r · 1M ∈
OM (|M |), where 1M is the unit function. �
Proposition 3.4. The object set Ob(Zn2Pts) ' Ob(Zn2GrAlg) is a directed set.
Proof. Given any m = (m1,m2, . . . ,mN ) and n = (n1, n2, . . . , nN ), we write Λm ≤ Λn if and
only if mi ≤ ni, for all i. This preorder makes the non-empty set of Zn2 -Grassmann algebras into
a directed set, since, any Λm and Λn admit Λp, where pi = sup{mi, ni}, as upper bound. �
We will need the following functional analytic result in later sections of this paper. See
Definitions B.1 and B.5 for the notion of Fréchet space and Fréchet algebra, respectively.
Proposition 3.5. The algebra of functions of any Zn2 -point is a Zn2 -graded Zn2 -commutative
nuclear Fréchet algebra.
The proposition is a special case of the fact that the structure sheaf of any Zn2 -manifold is
a nuclear Fréchet sheaf of Zn2 -graded Zn2 -commutative algebras [12, Theorem 14].
Moreover, as a direct consequence of [13, Theorem 19, Definition 13], we observe that the
category of Zn2 -points admits all finite categorical products; in particular: R0|m×R0|n ' R0|m+n.
By restricting attention to elements of degree 0 ∈ Zn2 , we get the following corollary. See
Definition B.7 for the concept of Fréchet module.
Corollary 3.6. The set Λ0 of degree 0 elements of an arbitrary Zn2 -Grassmann algebra Λ is
a commutative nuclear Fréchet algebra. Moreover, the algebra Λ can canonically be considered
as a Fréchet Λ0-module.
Remark 3.7. Specialising to the n = 1 case, we recover the standard and well-known facts
about superpoints and their relation with Grassmann algebras.
3.2 A convenient generating set of Zn
2Man
It is clear that studying just the underlying topological points of a Zn2 -manifold is inadequate
to probe the graded structure. Much like the category of supermanifolds, where the set of
superpoints forms a generating set, the set of Zn2 -points forms a generating set for the cate-
gory of Zn2 -manifolds. For the classical case of standard supermanifolds, see for example [40,
Theorem 3.3.3]. For the general notion of a generating set, see Definition A.1.
Theorem 3.8. The set Ob
(
Zn2Pts
)
constitutes a generating set for Zn2Man.
Proof. Let φ = (|φ|, φ∗) and ψ = (|ψ|, ψ∗) be two distinct Zn2 -morphisms φ, ψ : M → N between
two Zn2 -manifolds M = (|M |,OM ) and N = (|N |,ON ). These morphisms have distinct smooth
base maps
|φ|, |ψ| : |M | → |N |,
or, if |φ| = |ψ|, they have distinct pullback morphisms of sheaves of algebras
φ∗, ψ∗ : ON → |φ|∗OM .
If |φ| 6= |ψ|, there is at least one point m ∈ |M |, such that |φ|(m) 6= |ψ|(m). Let now
s : R0|0 → M be the Zn2 -morphism, which corresponds to the Zn2Alg morphism s∗ : OM (|M |) 3
f 7→ (εf)(m) ∈ R, where ε is the sheaf morphism ε : OM → C∞|M |. It follows from the reconstruc-
tion theorem [13, Theorem 9] that the base morphism |s| : {?} → |M | maps ? to m. Hence, the
Zn2 -morphisms φ ◦ s and ψ ◦ s have distinct base maps.
12 A.J. Bruce, E. Ibarguengoytia and N. Poncin
Assume now that |φ| = |ψ|, so that there exists |V | ⊂ |N |, such that φ∗|V | 6= ψ∗|V |, i.e., such
that φ∗|V |f 6= ψ∗|V |f, for some function f ∈ ON (|V |). A cover of |V | by coordinate patches (Vi)i,
induces a cover |Ui| := |φ|−1(Vi) of |U | := |φ|−1(|V |). It follows that
(φ∗|V |f)||Ui| 6= (ψ∗|V |f)||Ui|,
for some fixed i, i.e., that
φ∗Vi(f |Vi) 6= ψ∗Vi(f |Vi),
so that φ∗Vi 6= ψ∗Vi .
Recall that, for any open subset |X| ⊂ |M |, there is a Zn2 -morphism
ιX : (|X|,OM ||X|)→ (|M |,OM ),
whose base map |ιX | is the inclusion and whose pullback ι∗X is the obvious restriction. Further,
any Zn2 -morphism φ : M → N , whose base map |φ| : |M | → |N | is valued in an open subset |Y |
of |N |, induces a Zn2 -morphism
φY : (|M |,OM )→ (|Y |,ON ||Y |),
whose base map |φY | is the map |φ| : |M | → |Y | and whose pullback φ∗Y is the pullback φ∗
restricted to ON ||Y |.
In view of the above, if (Uj)j is a cover of |Ui| by coordinate domains, we have
(φ∗Vi(f |Vi))|Uj 6= (ψ∗Vi(f |Vi))|Uj , (3.1)
for some fixed j. This implies that the Zn2 -morphisms (φ ◦ ιUj )Vi and (ψ ◦ ιUj )Vi from the Zn2 -
domain Uj = (Uj , C∞Uj [[ξ]]) to the Zn2 -domain Vi = (Vi, C∞Vi [[θ]]) are different. More precisely,
they have the same base map |φ| = |ψ| : Uj → Vi, but their pullbacks are distinct. Indeed, these
sheaf morphisms’ algebra maps at Vi are the maps ι∗Uj ,|Ui| ◦φ
∗
Vi and ι∗Uj ,|Ui| ◦ψ
∗
Vi from C∞Vi (y)[[θ]]
to C∞Uj (x)[[ξ]], where y runs through Vi and x through Uj , and the values of these algebra maps
at f |Vi are different (see equation (3.1)).
In view of Lemma 3.9, there is a Zn2 -morphism s : R0|m → Uj , such that
(φ ◦ ιUj )Vi ◦ s 6= (ψ ◦ ιUj )Vi ◦ s.
However, then the Zn2 -morphism ιUj ◦ s : R0|m → M separates φ and ψ, since the algebra maps
at Vi of the pullbacks (s∗ ◦ ι∗Uj ) ◦ φ
∗ and (s∗ ◦ ι∗Uj ) ◦ ψ
∗ differ. Indeed, as the Zn2 -morphisms
(φ ◦ ιUj )Vi and (ψ ◦ ιUj )Vi are fully determined by the pullbacks of the target coordinates, their
pullbacks at Vi differ for at least one coordinate yb, θB. It follows from the proof of Lemma 3.9
that the pullback s∗Uj ◦ (ι∗Uj ,|Ui| ◦ φ
∗
Vi) at Vi of (φ ◦ ιUj )Vi ◦ s and the similar pullback for ψ differ
for the same coordinate. However, the pullback at Vi considered is also the algebra map at Vi
of the pullback (s∗ ◦ ι∗Uj ) ◦ φ
∗, so that the pullbacks (s∗ ◦ ι∗Uj ) ◦ φ
∗ and (s∗ ◦ ι∗Uj ) ◦ψ
∗ are actually
distinct. �
It remains to prove the following
Lemma 3.9. The statement of Theorem 3.8 holds for any two distinct Zn2 -morphisms between
Zn2 -domains.
The Schwarz–Voronov Embedding of Zn2 -Manifolds 13
Proof. We consider two Zn2 -domains Up|q and Vr|s together with two distinct Zn2 -morphisms
Up|q
φ
−→−→
ψ
Vr|s.
As in the general case above, there are two cases to consider: either |φ| 6= |ψ|, or |φ| = |ψ| and
φ∗ 6= ψ∗. In the proof of Theorem 3.8, we showed that in the first case, the maps φ and ψ can
be separated. In the second case, since a Zn2 -morphism valued in a Zn2 -domain is fully defined by
the pullbacks of the coordinates, these global Zn2 -functions φ∗Vr(Y
i), ψ∗Vr(Y
i) ∈ C∞Up(x)[[ξ]] differ
for at least one coordinate Y i = yb or Y i = θB. Let B be an index, such that
φ∗Vr
(
θB
)
=
∞∑
|α|=1
φBα (x)ξα, ψ∗Vr
(
θB
)
=
∞∑
|α|=1
ψBα (x)ξα,
where we denoted the coordinates of Up|q by
(
xa, ξA
)
and used the standard multi-index notation,
differ. This means that the functions φBα (x) and ψBα (x) differ for at least one α and at least
one x ∈ Up, say for α = a and x = x ∈ Up ⊂ Rp. From this, we can construct the separating
Zn2 -morphism
R0|q s−→ Up|q
φ
−→−→
ψ
Vr|s.
Let us denote the coordinates of R0|q by χA. We then define the Zn2 -morphism s by setting
s∗Upx
a = xa ∈ R[[χ]], deg
(
xa
)
= deg
(
xa
)
,
s∗Upξ
A = χA ∈ R[[χ]], deg
(
χA
)
= deg
(
ξA
)
.
It is clear that φ ◦ s 6= ψ ◦ s, since
∞∑
|α|=1
φBα (x)χα = s∗Up
(
φ∗Vr
(
θB
))
6= s∗Up
(
ψ∗Vr
(
θB
))
=
∞∑
|α|=1
ψBα (x)χα.
The case where φ∗Vr(Y
i) 6= ψ∗Vr(Y
i) for Y i = yb is almost identical. In particular, we then have
φ∗Vr
(
yb
)
= |φ|b(x) +
∞∑
|α|=2
φbα(x)ξα,
ψ∗Vr
(
yb
)
= |ψ|b(x) +
∞∑
|α|=2
ψbα(x)ξα.
Since we know that |φ| = |ψ|, we can proceed as for Y i = θB. �
In view of Proposition A.3, we get the
Corollary 3.10. The restricted Yoneda functor
YZn2 Pts : Zn2Man 3M 7→ HomZn2 Man
(
−,M
)
∈
[
Zn2Ptsop, Set
]
is faithful.
Above, we wrote M(−) ∈ [Zn2Manop, Set] for the image of M ∈ Zn2Man by the non-restricted
Yoneda functor. If no confusion arises, we will use the same notation M(−) for the image
YZn2 Pts(M) ∈ [Zn2Ptsop, Set] of M by the restricted Yoneda functor.
14 A.J. Bruce, E. Ibarguengoytia and N. Poncin
Definition 3.11. Let M be an object of Zn2Man and Λ ' R0|m an object of Zn2GrAlg ' Zn2Ptsop.
We refer to the set
M(Λ) := HomZn2 Man
(
R0|m,M
)
' HomZn2 Alg
(
O(|M |),Λ
)
as the set of Λ-points of M .
Proposition 3.12. Let
m∗ ∈ HomZn2 Alg
(
O(|M |),Λ
)
be a Λ-point of M and let s ∈ O(|M |). The Λ-point m∗ can equivalently be viewed as a Zn2 -
morphism
m = (|m|,m∗) ∈ HomZn2 Man
(
R0|m,M
)
and therefore it defines a unique topological point x := |m|(?) ∈ |M |. If |U | ⊂ |M | is an open
neighbourhood of x, such that s||U | = 0, then m∗(s) = 0.
Proof. Since m∗ : OM → OR0|m is a sheaf morphism, it commutes with restrictions, i.e., for any
open subsets |V | ⊂ |U | ⊂ |M | and any s ∈ OM (|U |), we have m∗|U |(s) ∈ OR0|m
(
|m|−1(|U |)
)
and
(m∗|U |(s))||m|−1(|V |) = m∗|V |(s||V |) ∈ OR0|m
(
|m|−1(|V |)
)
.
It follows that m∗(s) = m∗|M |(s) ∈ Λ = OR0|m({?}) reads
m∗(s) = (m∗|M |(s))|{?} = (m∗|M |(s))||m|−1(|U |) = m∗|U |(s||U |) = 0. �
Lemma 3.13. There is a 1 : 1 correspondence
M(Λ) '
⋃
x∈|M |
HomZn2 Alg
(
OM,x,Λ
)
between the set of Λ-points of M and the set of morphisms from the stalks of OM to Λ. The set
Mx(Λ) := HomZn2 Alg
(
OM,x,Λ
)
is referred to as the set of Λ-points near x.
Proof. Any Λ-point m∗ or m = (|m|,m∗) defines a topological point x = |m|(?) ∈ |M |, as well
as a Zn2Alg-morphism φx ∈ HomZn2 Alg(OM,x,Λ) between stalks. This morphism is given, for any
tU ∈ O(|U |) defined in some neighbourhood |U | of x in |M |, by
φx[tU ]x = m∗?[tU ]x = [m∗|U |tU ]? = m∗|U |tU .
Conversely, any morphism ψy ∈ HomZn2 Alg(OM,y,Λ) (y ∈ |M |) between stalks defines a Λ-
point µ∗ ∈ HomZn2 Alg
(
O(|M |), Λ
)
. It suffices to set
µ∗t = ψy[t]y ∈ Λ,
for all t ∈ O(|M |).
It remains to check that the composites m∗ 7→ φx 7→ µ∗ and ψy 7→ µ∗ 7→ φx are identities. In
the first case, for any t ∈ O(|M |), we get µ∗t = φx[t]x = m∗t, so that µ∗ = m∗. In the second
case, we need the following reconstruction results. Let |U | ⊂ |M | be an open subset and set
SU =
{
s ∈ O0(|M |) : (εs)||U | is invertible in C∞(|U |)
}
.
The Schwarz–Voronov Embedding of Zn2 -Manifolds 15
Then the localization map λU : O(|M |) · S−1
U → O(|U |) is an isomorphism in Zn2Alg. More
precisely, for any tU ∈ O(|U |), there is a unique Fs−1 ∈ O(|M |) · S−1
U , such that tU = F ||U |s|−1
|U |
(if s ∈ SU , then s||U | is invertible in O(|U |)), and we identify Fs−1 with tU . For the proof of
these statements or more details on them, see [13, Proposition 3.5.]. It is further clear from the
results of [13, Proposition 3.1.] that x = |µ|(?) is the topological point y.
We now compute the second composite above. For any tU defined in a neighborhood |U | of x,
we get
φx[tU ]x = µ∗|U |
(
Fs−1
)
= µ∗(F )µ∗(s)−1
= ψx[F ]x(ψx[s]x)−1 = ψx[F ]xψx
(
[s]−1
x
)
= ψx
(
[F ||U |]x
[
s|−1
|U |
]
x
)
= ψx[tU ]x,
where the second equality is part of the reconstruction theorem of Zn2 -morphisms [13]. �
Let us consider an open cover (|UI |)I∈A of the smooth manifold |M |, as well as the open
Zn2 -submanifolds UI :=
(
|UI |,OM ||UI |
)
of the Zn2 -manifold M (which need not be coordinate
charts).
Proposition 3.14. For any Zn2 -Grassmann algebra Λ and Zn2 -manifold M =
(
|M |,OM
)
, we
have a natural 1 : 1 correspondence
M(Λ) '
⋃
I∈A
UI(Λ),
so that the family of sets (UI(Λ))I∈A is a cover of the set M(Λ).
Proof. Since it is clear from the definition of a stalk that OUI ,x = OM,x, for any x ∈ |UI |, it
follows from Lemma 3.13 that⋃
I∈A
UI(Λ) '
⋃
I∈A
⋃
x∈|UI |
HomZn2 Alg
(
OM,x,Λ
)
=
⋃
x∈|M |
HomZn2 Alg
(
OM,x,Λ
)
'M(Λ). �
Recall that
HomZn2 Man(−,−) ∈
[
Zn2Man,
[
Zn2Ptsop, Set
]]
,
so that,
(i) any Zn2 -morphism φ = (|φ|, φ∗) : M → N is mapped (injectively) to a natural transforma-
tion
φ ' HomZn2 Man(−, φ) : HomZn2 Man(−,M)→ HomZn2 Man(−, N),
whose Λ-component (Λ ' R0|m) is the Set-map given by
φΛ := HomZn2 Man(Λ, φ) : M(Λ) = HomZn2 Man
(
R0|m,M
)
' HomZn2 Alg(O(|M |),Λ) 3 m∗
7→ m∗ ◦ φ∗ ∈ HomZn2 Alg(O(|N |),Λ) ' HomZn2 Man
(
R0|m, N
)
= N(Λ), and,
(ii) for any fixed M ∈ Zn2Man, given a morphism ψ = (|ψ|, ψ∗) : R0|m′ → R0|m of Zn2 -points,
or, equivalently, a morphism ψ∗ : Λ → Λ′ of Zn2 -Grassmann algebras, we get the induced
Set-map
M(ψ∗) := HomZn2 Man(ψ,M) : M(Λ) = HomZn2 Man
(
R0|m,M
)
' HomZn2 Alg(O(|M |),Λ) 3 m∗ 7→ ψ∗ ◦m∗ ∈ HomZn2 Alg(O(|M |),Λ′)
' HomZn2 Man
(
R0|m′ ,M
)
= M(Λ′). (3.2)
16 A.J. Bruce, E. Ibarguengoytia and N. Poncin
When reading the maps φΛ and M(ψ∗) through the 1 : 1 correspondence
M(Λ) 3 m∗ 7→ (x,m∗?) ∈
⋃
y∈|M |
HomZn2 Alg
(
OM,y,Λ
)
,
where x = |m|(?), we obtain
φΛ : M(Λ) −→ N(Λ),
(x,m∗?) 7→ (|φ|(x),m∗? ◦ φ∗x), and
M(ψ∗) : M(Λ) −→M(Λ′),
(x,m∗?) 7→ (x, ψ∗ ◦m∗?).
3.3 Restricted Yoneda functor and fullness
The Yoneda functor from any locally small category C into the category of Set-valued contravari-
ant functors on C, is fully faithful. This holds in particular for C = Zn2Man. When we restrict
the contravariant functors to the generating set Zn2Pts, the resulting restricted Yoneda functor
is automatically faithful. In the following, we show that it is not full, i.e., that not all natural
transformations are induced by a Zn2 -morphism.
Naturality of any transformation φ : M(−)→ N(−) between Set-valued contravariant (resp.,
covariant) functors on Zn2Pts (resp., Zn2GrAlg), means that the diagram
M(Λ) N(Λ)
M(Λ′) N(Λ′)
//
φΛ
��
M(ψ∗)
��
N(ψ∗)
//
φΛ′
commutes, for any morphism ψ∗ : Λ→ Λ′ of Zn2 -Grassmann algebras.
A Λ-point of a Zn2 -manifold M is denoted by m∗ or m = (|m|,m∗). If the manifold is a Zn2 -
domain Up|q, we use the notation x∗ or x = (|x|, x∗). If
(
xa, ξA
)
are the coordinates of Up|q,
a Λ-point x∗ in Up|q is completely determined by the degree-respecting pullbacks(
xaΛ, ξ
A
Λ
)
:=
(
x∗
(
xa
)
, x∗
(
ξA
))
.
Since xaΛ ∈ Λ0 = R⊕ Λ̊0, we write xaΛ =
(
xa||, x̊
a
Λ
)
. Hence, any Λ-point x∗ in Up|q can be identified
with
x∗ '
(
xaΛ, ξ
A
Λ
)
=
(
xa||, x̊
a
Λ, ξ
A
Λ
)
∈ Rp × Λ̊p0 × Λ̊q1γ1
× · · · × Λ̊qNγN , (3.3)
where
x|| =
(
xa||
)
=
(
. . . , xa||, . . .
)
∈ Up,
and where γ1, . . . , γN denote the non-zero Zn2 -degrees in standard order. Here the x̊aΛ (resp.,
the ξAΛ ) are formal power series containing at least 2 (resp., at least 1) of the generators
(
θC
)
of
the Zn2 -Grassmann algebra Λ.
As mentioned above, any Zn2 -morphism, in particular any morphism φ : Up|q → Vr|s between
Zn2 -domains, naturally induces a natural transformation, with Λ-component
φΛ : Up|q(Λ) 3 x∗ 7→ x∗ ◦ φ∗ ∈ Vr|s(Λ).
The Schwarz–Voronov Embedding of Zn2 -Manifolds 17
If
(
yb, ηB
)
are the coordinates of Vr|s, the morphism φ reads
φ∗
(
yb
)
=
∑
|α|≥0
φbα(x)ξα, φ∗
(
ηB
)
=
∑
|α|>0
φBα (x)ξα,
where the right-hand sides have the appropriate degrees and where φ0(Up) ⊂ Vr. Further, the
image Λ-point x∗ ◦ φ∗ in Vr|s by φΛ of the Λ-point x∗ '
(
x∗(xa); x∗
(
ξA
))
=
(
xa||, x̊
a
Λ; ξAΛ
)
in Up|q,
is given by
ybΛ =
∑
|α|≥0
∑
|β|≥0
1
β!
(
∂βxφ
b
α
)
(x||)̊x
β
Λξ
α
Λ, (3.4a)
ηBΛ =
∑
|α|>0
∑
|β|≥0
1
β!
(
∂βxφ
B
α )(x||
)
x̊βΛξ
α
Λ. (3.4b)
Let us recall that there is no convergence issue with terms in x|| [15]. Thus the components
of a natural transformation implemented by a Zn2 -morphism between Zn2 -domains, are very
particular formal power series in the formal variables x̊aΛ and ξAΛ , which are themselves formal
power series in the generators
(
θC
)
of Λ.
We are now able to prove that not all natural transformations between the restricted functors
M(−), N(−) ∈ [Zn2Pts, Set] associated with M,N ∈ Zn2Man, arise from a Zn2 -morphism M → N .
Since it suffices to give one counter-example, we choose M = N = Rp|0 = Rp.
Example 3.15. Consider an arbitrary diffeomorphism φ : Rp −→ Rp. The Λ-component of the
associated natural transformation is
φΛ : Rp|0(Λ) −→ Rp|0(Λ),
(xbΛ, 0) 7→
(
φb(x||) +
∑
|β|>0
1
β!
(∂βxφ
b)(x||)̊x
β
Λ, 0
)
.
From this data we obtain another natural transformation
αΛ : Rp|0(Λ) −→ Rp|0(Λ),(
xbΛ, 0
)
7→
(
φb(x||), 0
)
.
The natural transformation α is not implemented by a morphism ψ : Rp → Rp. Indeed, otherwise
αΛ = ψΛ, for all Λ. This means that
(
φb(x||), 0
)
=
(
ψb(x||) +
∑
|β|>0
1
β!
(∂βxψ
b)(x||)̊x
β
Λ, 0
)
,
for all Λ and all Λ-points. Since φb(x) ≡ ψb(x), we have ∂βxφb ≡ ∂βxψb. Take now any β : |β| = 1,
so that βa = 1, for some fixed a ∈ {1, . . . , p}. As we can choose Λ and xbΛ, for all b ∈ {1, . . . , p},
arbitrarily, we can choose x̊bΛ = 0, for all b 6= a, and x̊aΛ = θDθE , where θD and θE are two
different generators of Λ that have the same degree. The coefficient of θDθE in the sum over
all β is then
(
∂xaψ
b
)
(x||), hence ∂xaφ
b ≡ ∂xaψ
b ≡ 0. The latter observation is a contradiction,
since the Jacobian determinant of φ does not vanish anywhere in Rp.
We now generalise a technical result [54, Theorem 1] to Zn2 -domains Up|q. Let
Bp|q
(
Up
)
:= F
(
Up,R
)
[[X,Ξ]],
18 A.J. Bruce, E. Ibarguengoytia and N. Poncin
be the Zn2 -graded Zn2 -commutative associative unital R-algebra of formal power series in p pa-
rameters Xa of Zn2 -degree 0 and q1, . . . , qN parameters ΞA of non-zero Zn2 -degree γ1, . . . , γN , and
with coefficients in arbitrary R-valued functions on Up, i.e., we do not ask that these functions be
continuous let alone smooth. Following [43, 44, 54], we will refer to this algebra as a Zn2 -Berezin
algebra. Any element of this algebra is of the form
F =
∑
|α|≥0
∑
|β|≥0
Fαβ(x)XβΞα, (3.5)
where the xa are coordinates in Up.
Theorem 3.16. For any Zn2 -domains Up|q and Vr|s, there is a 1 : 1 correspondence
Nat
(
Up,q,Vr,s
)
→
(
Bp|q
(
Up
))r|s
between
– the set of natural transformations in [Zn2Ptsop, Set] between Up|q(−) and Vr|s(−), and
– the set of ‘vectors’ F with r (resp., with s1, . . . , sN ) components F b of degree 0 (resp.,
components FB of degrees γ1, . . . , γN ) of the type (3.5), such that the r-tuple
(
F b00
)
made
of the coefficients F b00(x) of the r series F b satisfies(
F b00
)(
Up
)
⊂ Vr.
Proof. Let F be such a ‘vector’. For any Λ, we define the map
βΛ : Up|q(Λ) 3
(
xa||, x̊
a
Λ, ξ
A
Λ
)
7→
(
ybΛ, η
B
Λ
)
∈ Vr|s(Λ),
where
ybΛ :=
∑
|α|≥0
∑
|β|≥0
F bαβ(x||)̊x
β
Λξ
α
Λ and ηBΛ :=
∑
|α|≥0
∑
|β|≥0
FBαβ(x||)̊x
β
Λξ
α
Λ.
Since x̊aΛ, ξAΛ have the same degrees as Xa, ΞA, the right-hand sides of (3.3) have the same
degrees as F b, FB, hence, ybΛ, ηBΛ have the degrees required to be a Λ-point in Vr|s. Moreover,
we have
yb|| = F b00(x||),
so that y|| ∈ Vr. The target of the map βΛ is thus actually Vr|s(Λ). The naturality of β under
morphisms of Zn2 -Grassmann algebras is obvious: β is a natural transformation in [Zn2Ptsop, Set]
between Up|q(−) and Vr|s(−). Finally, we defined a map
I :
(
Bp|q
(
Up
))r|s → Nat
(
Up,q,Vr,s
)
.
We will explain now that any natural transformation β : Up|q(−) −→ Vr|s(−) is the image
by I of a unique ‘vector’ F. We first show that, for any Λ ' R0|m, the image βΛ(x∗) ∈ Vr|s(Λ)
of any Λ-point
x∗ '
(
xa||, x̊
a
Λ, ξ
A
Λ
)
∈ Up × Λ̊p0 × Λ̊q1γ1
× · · · × Λ̊qNγN
in Up|q, has components ybΛ and ηBΛ of the type (3.3).
The Schwarz–Voronov Embedding of Zn2 -Manifolds 19
Step 1. We prove that any Λ-point in Up|q is the image by a Zn2 -Grassmann algebra map
ϕ∗ : Λ′ → Λ of a Λ′-point in Up|q, some of whose defining series are series in formal pairings.
Let
(
θC
)
be the generators of Λ. The Λ-point x∗ then reads
x∗ '
(
xa||,
∑
λκ θ
λθκKa
κλ, ξ
A
Λ
)
,
where the degree of Ka
κλ ∈ Λ is the sum of the degrees of θλ and θκ. Recall that a (resp., A)
runs through {1, . . . , p} (resp., through {1, . . . , |q|}), and that λ, κ run through {1, . . . , |m|}.
Consider now the set S of generators
θ′ =
(
ηaλ, ζbκ, ψ
A
)
,
where b has the same range as a, and define their (non-zero) Zn2 -degrees by
deg
(
ηaλ
)
= deg
(
θλ
)
, deg
(
ζbκ
)
= deg
(
θκ
)
, deg
(
ψA
)
= deg
(
ξAΛ
)
= deg
(
ξA
)
.
Let Λ′ be the Zn2 -Grassmann algebra defined by S, and set
x′∗ '
(
xa||,
∑
λ η
aλζaλ, ψ
A
)
∈ Up × Λ̊′p0 × Λ̊′q1γ1 × · · · × Λ̊′qNγN
(no sum over a in the formal pairings
∑
λ η
aλζaλ). The degree-respecting equalities
ϕ∗
(
ηaλ
)
= θλ, ϕ∗
(
ζbκ
)
=
∑
λ
θλKb
λκ, ϕ∗
(
ψA
)
= ξAΛ
define a morphism of Zn2 -Grassmann algebras ϕ∗ : Λ′ −→ Λ. It suffices to set
ϕ∗
(∑
ε
rεθ
′ε
)
:=
∑
ε
rε(ϕ
∗θ′)ε.
Indeed, any term of the right-hand side is a series in θ whose terms contain at least |ε| genera-
tors. Hence, for any ε, only the terms |ε| ≤ |ε| can contribute to θε, and therefore there is no
convergence issue with the coefficient of θε. Since the Λ-point ϕ∗ ◦ x′∗ in Up|q reads
ϕ∗ ◦ x′∗ ' ϕ∗
(
xa||,
∑
λ η
aλζaλ, ψ
A
)
=
(
xa||,
∑
λκ θ
λθκKa
κλ, ξ
A
Λ
)
' x∗,
naturality of the transformation β : Up|q(−) −→ Vr|s(−) implies that(
ybΛ, η
B
Λ
)
:' βΛ(x∗) = βΛ(ϕ∗ ◦ x′∗) = βΛ
(
Up|q(ϕ∗)(x′∗)
)
= Vr|s(ϕ∗)(βΛ′(x
′∗)) = ϕ∗ ◦ (βΛ′(x
′∗)) ' ϕ∗
(
ybΛ′ , η
B
Λ′
)
, (3.6)
where ybΛ′ and ηBΛ′ are series in the generators of Λ′.
Step 2. We define formal rotations under which the formal pairings are invariant. Moreover,
we show that any formal series that is invariant under the formal rotations is a series in the
formal pairings.
The formal part of each degree 0 component of x′∗ can be viewed as a formal pairing ηa ·ζa =∑
λ η
aλζaλ, which is stable under formal rotations R∗. More precisely, we set
R∗
(
ηaλ
)
=
∑
κ
ηaκ(Oa)λκ, R
∗(ζbκ) =
∑
λ
(
Obt
)λ
κ
ζbλ, R
∗(ψA) = ψA,
where Oa and Obt are any (m1 + · · · + mN ) × (m1 + · · · + mN ) block-diagonal matrices with
entries in R that satisfy∑
λ
(Oa)λρ
(
Oat
)ω
λ
= δωρ . (3.7)
20 A.J. Bruce, E. Ibarguengoytia and N. Poncin
Since, for any fixed a (resp., b), the components ηaλ (resp., ζbκ) are ordered such that the m1
first components have degree γ1, the next m2 degree γ2, and so on, these equalities are degree-
preserving. Hence, they define a Zn2 -Grassmann algebra morphism R∗ : Λ′ → Λ′ via
R∗
(∑
ε
rεθ
′ε
)
= R∗
(∑
αβγ
rαβγη
αζβψγ
)
:=
∑
αβγ
rαβγ(R∗η)α(R∗ζ)βψγ .
Since the images R∗(ηaλ) (resp., R∗(ζbκ)) are linear in the ηaκ (resp., ζbλ) (of the same degree),
the term indexed by αβγ is a homogeneous polynomial of order |α|+|β|+|γ| in the generators θ′.
Hence, for any ε, only the terms |α|+ |β|+ |γ| = |ε| can contribute to θ′ε, so that no convergence
problems arise. In view of (3.7), it is clear that, as mentioned above, the formal pairing ηa ·ζa =∑
λ η
aλζaλ is invariant underR∗. As any Zn2 -Grassmann algebra morphism, the formal rotationR∗
induces maps Up|q(R∗) and Vr|s(R∗), and due to naturality of β, we find
Vr|s(R∗)(βΛ′x
′∗) = βΛ′
(
Up|q(R∗)(x′∗)
)
= βΛ′(R
∗ ◦ x′∗)
' βΛ′
(
R∗
(
xa||,
∑
λ
ηaλζaλ, ψ
A
))
' βΛ′x
′∗,
so that βΛ′x
′∗ is invariant under rotations.
We are now prepared to continue the computation (3.6). Since
βΛ′(x
′∗) '
(
ybΛ′ , η
B
Λ′
)
=
(
yb||, ẙ
b
Λ′ , η
B
Λ′
)
(3.8)
is invariant under the rotations R∗, the series ẙbΛ′ , η
B
Λ′ in the generators θ′ are invariant. More
explicitly, for each series, we have an equality of the type∑
γ
(∑
k,`
∑
|α|=k, |β|=`
Fαβγη
αζβ
)
ψγ =
∑
γ
(∑
k,`
∑
|α|=k, |β|=`
Fαβγ(R∗η)α(R∗ζ)β
)
ψγ ,
which is equivalent to∑
|α|=k, |β|=`
Fαβγ · · · ηaληbµζcν · · · =
∑
|α|=k, |β|=`
Fαβγη
αζβ =
∑
|α|=k, |β|=`
Fαβγ(R∗η)α(R∗ζ)β
=
∑
|α|=k, |β|=`
Fαβγ · · · ηaδ(Oa)λδ ηbδ
′(
Ob
)µ
δ′
(
Oct
)δ′′
ν
ζcδ′′ · · · ,
and holds for all (!) formal rotations. This is only possible, if the power series considered, i.e.,
the series ẙbΛ′ and ηBΛ′ , are series in pairings ηa ·ζa =
∑
λ η
aλζaλ. In the classical setting, the result
is known under the name of first fundamental theorem of invariant theory for the orthogonal
group [23, 57]. It has been extended to the graded situation in [7, Proposition 4.13]. In view
of (3.6), we thus get(
yb||, ẙ
b
Λ, η
B
Λ
)
= βΛ(x∗) = βΛ
(
xa||, x̊
a
Λ, ξ
A
Λ
)
=
(
yb||, ϕ
∗(ẙbΛ′), ϕ
∗(ηBΛ′)),
where any image by ϕ∗ is of the type∑
(α,β)6=(0,0)
Fαβϕ
∗((η · ζ)β
)
ϕ∗
(
ψα
)
=
∑
(α,β) 6=(0,0)
Fαβx̊
β
Λξ
α
Λ.
It is clear from (3.8) and (3.3) that the coefficients
F bαβ, F
B
αβ((α, β) 6= (0, 0)), and F b00 := yb||
The Schwarz–Voronov Embedding of Zn2 -Manifolds 21
depend (only) on x|| ∈ Up. Hence, the image(
ybΛ, η
B
Λ
)
= βΛ(x∗) =
(
F b(x||, x̊Λ, ξΛ), FB(x||, x̊Λ, ξΛ)
)
is actually of the type (3.3). Since βΛ(x∗) is a Λ-point in Vr|s, the r series F b(x||, x̊Λ, ξΛ) and
the si series FB(x||, x̊Λ, ξΛ) are of degree 0 and degree γi, respectively, i.e., the r series F b(x,X,Ξ)
and the si series FB(x,X,Ξ) are of degree 0 and degree γi, respectively. For the same reason,
we have F00(x||) ∈ Vr, for all x|| ∈ Up, so that we constructed a ‘vector’ F ∈ (Bp|q(Up))r|s, whose
image by I is obviously β.
Step 3. We show that F is unique (which concludes the proof). If there is another ‘vector’ F′,
such that I(F′) = β, we have∑
|α|≥0,|β|≥0
F b
αβ(x||)̊x
β
Λξ
α
Λ =
∑
|α|≥0,|β|≥0
F ′bαβ(x||)̊x
β
Λξ
α
Λ, (3.9)
for all b ∈ {b, B}, all Λ, and all x∗. Notice first that any x̊aΛ (resp., any ξAΛ ) is a series of degree 0
(resp., of degree deg
(
ξA
)
= γA) in the θ-s that contains at least two parameters θCθC
′
(resp.,
at least one parameter θC
′′
). Hence, both sides are series in θ, and the left-hand side and right-
hand side coefficients of any monomial θε coincide. A term (α, β) 6= (0, 0) cannot contribute to
the independent term θ0. Hence F b
00(x||) = F ′b00(x||). We now show that F b
αβ(x||) = F ′bαβ(x||),
for an arbitrarily fixed (α, β) 6= (0, 0). Since Λ is arbitrary, we can choose as many different
generators θ in each non-zero degree as necessary, and, since x∗ is arbitrary, we can choose x||
arbitrarily in Up and we can choose the coefficients of the series x̊aΛ and ξAΛ arbitrarily (except
that we have to observe that the coefficient of a monomial θε, which does not have the required
degree, must be zero). Let now α1, . . . , αµ and β1, . . . , βν be the non-zero components in the
fixed α and β. For each factor ξAiΛ of
ξαΛ =
(
ξA1
Λ
)α1 · · ·
(
ξ
Aµ
Λ
)αµ ,
we choose a monomial in one generator θCi of degree γAi , set its coefficient rCi to 1, and all
the other coefficients in the series ξAiΛ to zero. Further, for different ξAiΛ , we choose different
generators θCi . Similarly, for each factor x̊
aj
Λ of
x̊βΛ =
(
x̊a1
Λ
)β1 · · ·
(
x̊aνΛ
)βν ,
we choose monomials θDjkθEjk (k ∈ {1, . . . , βj}) in two generators of the same odd degree (for
all Zn2 -manifolds with n ≥ 1, there is at least one odd degree), set their coefficient rDjkEjk to 1,
and all the other coefficients in the series x̊
aj
Λ to zero. Further, we choose the generators so that
all generators θCi , θDjk , and θEjk are different. When setting
θω =
ν∏
j=1
θDj1θEj1 · · · θDjβj θEjβj
µ∏
i=1
(
θCi
)αi 6= 0,
the terms indexed by (the fixed) (α, β) in both sides of (3.9), read
β!F b
αβ(x||)θ
ω and β!F ′bαβ(x||)θ
ω.
For any term (α′, β′) 6= (α, β), we either get a new series ξAΛ or x̊aΛ (i.e., a series that is not
present in ξαΛ or x̊βΛ), or we get an old series a different number of times. In the second case, the
term (α′, β′) does not contribute to the coefficient of θω; in the first, we set all the coefficients
of the new series to 0, so that the term (α′, β′) vanishes. Finally, we obtain F b
αβ(x||) = F ′bαβ(x||),
for any x|| ∈ Up. �
22 A.J. Bruce, E. Ibarguengoytia and N. Poncin
We now show that Rp|q(Λ) is a Fréchet space and that Up|q(Λ) is an open subset of Rp|q(Λ).
This means that we have a notion of directional derivative, as well as a notion of smoothness of
continuous maps between the Λ-points of Zn2 -domains. For more details on Fréchet objects, we
refer the reader to Appendix B.
Proposition 3.17. For any Λ ∈ Zn2GrAlg, the set Rp|q(Λ) is a nuclear Fréchet space and
a Fréchet Λ0-module. Moreover, the set Up|q(Λ) is an open subset of Rp|q(Λ).
Proof. Let Λ ∈ Zn2GrAlg. As explained above, there is a 1 : 1 correspondence between the
Λ-points x∗ of Rp|q (resp., of Up|q) and the (p+ |q|)-tuples
x∗ '
(
xaΛ, ξ
A
Λ
)
∈ Λp0 × Λq1γ1
× · · · × ΛqNγN
(resp., the same (p+ |q|)-tuples, but with the additional requirement that the p-tuple (xa||) made
of the independent terms of (xaΛ) be a point in Up ⊂ Rp). Note now that Λ is the Zn2 -graded
Zn2 -commutative nuclear Fréchet R-algebra of global Zn2 -functions of some R0|m. 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 nuclear Fréchet. The latter statements can be
found in [12].
As for the second claim in Proposition 3.17, recall that Λ0 is a (commutative) Fréchet algebra,
see Corollary 3.6. The Fréchet Λ0-module structure on Rp|q(Λ) is then defined by
m : Λ0 × Rp|q(Λ) 3 (a, x∗) 7→
(
a · xaΛ, a · ξAΛ
)
∈ Rp|q(Λ). (3.10)
Since this action is defined using the continuous associative multiplication · : Λγi×Λγj → Λγi+γj
of the Fréchet algebra Λ, it is (jointly) continuous.
As any closed subspace of a Fréchet space is itself a Fréchet space, the space
Λ̊0 ' {0} × Λ̊0 ⊂ R× Λ̊0 = Λ0
is Fréchet. We thus see that
Up|q(Λ) ' Up × Λ̊p0 ×
N∏
i=1
Λqiγi ⊂ Rp × Λ̊p0 ×
N∏
i=1
Λqiγi ' Rp|q(Λ) (3.11)
is open. �
Remark 3.18. In the following, we will use the isomorphisms (3.11) (and similar ones) without
further reference.
The just described Λ0-module structure is vital in understanding the structure of the Λ-points
of any Zn2 -manifold. In particular, morphisms between Zn2 -domains induce natural transforma-
tions between the associated functors that respect this module structure. The converse is also
true, that is, any natural transformation between the associated functors that respects the Λ0-
module structure comes from a morphism between the underlying Zn2 -domains. More carefully,
we have the following proposition.
Theorem 3.19. Let Up|q and Vr|s be Zn2 -domains. A natural transformation β : Up|q(−) −→
Vr|s(−) comes from a Zn2 -manifold morphism Up|q → Vr|s if and only if βΛ : Up|q(Λ) −→ Vr|s(Λ)
is Λ0-smooth, for all Λ ∈ Zn2GrAlg. That is, for all Λ, the map βΛ must be a smooth map
(from the open subset Up|q(Λ) of the Fréchet space Rp|q(Λ) to the Fréchet space Rr|s(Λ), see
Appendix B) and its Gâteaux derivative (see Appendix B) must be Λ0-linear, i.e.,
dx∗βΛ(a · v) = a · dx∗βΛ(v),
for all x∗ ∈ Up|q(Λ), a ∈ Λ0, and v ∈ Rp|q(Λ).
The Schwarz–Voronov Embedding of Zn2 -Manifolds 23
Proof. Part I. Let β : Up|q(−) −→ Vr|s(−) be a natural transformation with Λ0-smooth com-
ponents βΛ, Λ ∈ Zn2GrAlg. From Theorem 3.16, we know that βΛ is completely specified by the
systems
ybΛ =
∑
|α|≥0,|β|≥0
F bαβ(x||)̊x
β
Λξ
α
Λ and ηBΛ =
∑
|α|>0,|β|≥0
FBαβ(x||)̊x
β
Λξ
α
Λ, (3.12)
where the coefficients F b
αβ (b ∈ {b, B}) are set-theoretical maps from Up to R.
Part Ia. Smoothness of βΛ implies that these coefficients are smooth. Indeed, we will show
that F b
αβ ∈ C0(Up) and that, if F b
αβ ∈ Ck(Up) (k ≥ 0), then F b
αβ ∈ Ck+1(Up).
Step 1. Since
βΛ : Up|q(Λ)→ Λr0 ×
N∏
i=1
Λsiγi
is continuous, any of its components
ybΛ : Up|q(Λ)→ Λγi(b)
= R[[θ]]γi(b)
'
∏
γi(b)
R
is continuous. For simplicity, we wrote yBΛ instead of ηBΛ , and we will continue doing so. Moreover,
the target space are the formal power series in θ with coefficients in R, all whose terms have
the degree γi(b) of yb, and this space is identified with the corresponding space of families of
reals. For any ω such that θω has the degree γi(b), the corresponding real coefficient gives rise
to a continuous map
yb,ωΛ : Up|q(Λ)→ R.
Since this joint continuity implies separate continuity with respect to x|| ∈ Up, for any fixed
(̊xΛ, ξΛ) and any Λ, we can proceed as at the end of the proof of Theorem 3.16. More precisely,
select any (α, β) and select (for an appropriate Λ) the pair (̊xΛ, ξΛ) such that x̊βΛξ
α
Λ = β!θω,
where θω is now the degree γi(b) monomial defined in the proof just mentioned. The real
coefficient of this monomial is β!F b
αβ(x||), which, as said, is an R-valued continuous map on Up,
so that F b
αβ ∈ C0(Up), for all b and all (α, β).
Step 2. Since
Up|q(Λ) ⊂ R×
(
Rp−1 × Λ̊p0 ×
N∏
i=1
Λqiγi
)
is an open subset of a product of two Fréchet spaces, smoothness of βΛ implies (via an iterated
application of Proposition B.4) that, for any b ∈ {b, B}, any ` ∈ N and any γ ∈ Np (|γ| = `),
the partial derivative
dγx|| y
b
Λ : Up|q(Λ)× R×` →
∏
γi(b)
R
is continuous.
Assume now that F b
αβ ∈ Ck(Up) (k ≥ 0), for any b and any (α, β), as well as that, for any
γ ∈ Np (|γ| = k) and any b, the continuous partial Gâteaux derivative
dγx|| y
b
Λ(1, . . . , 1) : Up|q(Λ)→
∏
γi(b)
R
24 A.J. Bruce, E. Ibarguengoytia and N. Poncin
is given by
dγx||,x∗ y
b
Λ(1, . . . , 1) =
∑
αβ
(
∂γxF
b
αβ
)
(x||)̊x
β
Λξ
α
Λ. (3.13)
Observe that for k = 0, this condition is automatically satisfied. We will now show that, under
these assumptions, the same statements hold at order k + 1. In view of (3.13), any order k + 1
continuous partial Gâteaux derivative
dxa|| d
γ
x||
ybΛ(1, . . . , 1) : Up|q(Λ)→
∏
γi(b)
R
(a ∈ {1, . . . , p}, |γ| = k) is given, at any x∗ ' (x||, x̊Λ, ξΛ) ∈ Up|q(Λ), by∑
αβ
lim
t→0
1
t
((
∂γxF
b
αβ
)
(x1
||, . . . , x
a
|| + t, . . . , xp||)−
(
∂γxF
b
αβ
)(
x1
||, . . . , x
a
||, . . . , x
p
||
))
x̊βΛξ
α
Λ. (3.14)
When proceeding as in Step 1, we get that the limit is an R-valued continuous function in Up. In
other words, the partial derivative ∂xa∂
γ
xF b
αβ exists and is continuous in Up, i.e., F b
αβ ∈ Ck+1(Up).
Moreover, formula (3.13) pertaining to order k derivatives, extends to the order k+1 derivatives,
see (3.14).
Part Ib. We examine the further consequences of Λ0-smoothness, in particular those of Λ0-
linearity. Since βΛ is of class C1, its components ybΛ : Up|q(Λ)→
∏
γi(b) R are of class C1. Further,
as
Up|q(Λ) ⊂
(
R× Λ̊0
)
×
(
Rp−1 × Λ̊p−1
0 ×
N∏
i=1
Λqiγi
)
is an open subset of a product of two Fréchet spaces, the partial Gâteaux derivative
d(xa|| ,̊x
a
Λ) y
b
Λ : Up|q(Λ)×
(
R× Λ̊0
)
→
∏
γi(b)
R
is continuous. It is given by
d(xa|| ,̊x
a
Λ),x∗ y
b
Λ(v||, v̊Λ) = dxa||,x∗ y
b
Λ(v||) + dx̊aΛ,x∗ y
b
Λ(̊vΛ)
= v||
∑
αβ
(
∂xaF
b
αβ
)
(x||)̊x
β
Λξ
α
Λ +
∑
αβ
F b
αβ(x||) lim
t→0
1
t
(
(̊xaΛ + t̊vΛ)βa − (̊xaΛ)βa
)∏
b 6=a
(
x̊bΛ
)βbξαΛ
=: v||T1 + T2.
As Λ̊0 is a commutative algebra, it follows from the binomial formula that
T2 = v̊Λ
∑
αβ
βaF
b
αβ(x||)̊x
β−ea
Λ ξαΛ =: v̊ΛT2,
where (ea)a is the canonical basis of Rp. Observe now that, in view of (3.10), the Λ0-linearity of
the total Gâteaux derivative of ybΛ with respect to x∗ is equivalent to the Λ0-linearity of all its
partial Gâteaux derivatives with respect to the xaΛ =
(
xa||, x̊
a
Λ
)
and the ξAΛ . For a = 0 + v̊Λ ∈ Λ0
and v = 1 + 0 ∈ R + Λ̊0 = Λ0, this implies that
v̊ΛT2 = d(xa|| ,̊x
a
Λ),x∗ y
b
Λ(̊vΛ · 1) = v̊Λ · d(xa|| ,̊x
a
Λ),x∗ y
b
Λ(1) = v̊ΛT1,
The Schwarz–Voronov Embedding of Zn2 -Manifolds 25
i.e., that
v̊Λ
∑
αβ
(βa + 1)F b
α,β+ea(x||)̊x
β
Λξ
α
Λ
= v̊Λ
∑
α,γ : γa 6=0
γaF
b
αγ(x||)̊x
γ−ea
Λ ξαΛ = v̊Λ
∑
αβ
(
∂xaF
b
αβ
)
(x||)̊x
β
Λξ
α
Λ.
Since Λ ∈ Zn2GrAlg, v̊Λ ∈ Λ̊0, and x∗ ∈ Up|q(Λ) are arbitrary, we can repeat the θω-argument
used above. More precisely, we select (α, β), select (̊xΛ, ξΛ) such that x̊βΛξ
α
Λ = β!θω, and select
v̊Λ = θDθE ∈ Λ̊0 such that θDθEθω 6= 0. The coefficients of the latter monomial in the left and
right hand sides do coincide, which means that
(βa + 1)F b
α,β+ea(x||) =
(
∂xaF
b
αβ
)
(x||),
or, equivalently,
F b
αγ(x||) =
1
γa
(
∂xaF
b
α,γ−ea
)
(x||), (3.15)
for all b, α, a, all γ : γa 6= 0, and all x|| ∈ Up. For any b, α, and x||, we now set
φbα(x||) := F b
α0(x||) ∈ C∞
(
Up
)
.
An iterated application of (3.15) shows that
F b
αγ(x||) =
1
γ!
(
∂γxφ
b
α
)
(x||).
Hence, the ybΛ have the form (3.4a) and (3.4b). This means that the natural transformation β
is implemented by the φbα, which define actually a Zn2 -morphism from Up|q to Vr|s. Indeed, the
property
(
φb0
)
(Up) ⊂ Vr follows from the similar property of
(
F b00
)
. On the other hand, the
pullback
φ∗
(
yb
)
:=
∑
α
φbα(x)ξα
must have the same degree as yb. However, if deg(ξα) 6= deg
(
yb
)
, then deg(ξαΛ) 6= deg
(
ybΛ
)
,
whatever ξΛ. It follows therefore from (3.12) that φbα = F b
α0 = 0.
Part II. The proof of the converse implication is less demanding. Let β : Up|q(−) → Vr|s(−)
be a natural transformation that is induced by a Zn2 -morphism φ : Up|q → Vr|s, i.e., that is of
the form (3.4a) and (3.4b). For any Λ ∈ Zn2GrAlg, the map βΛ is smooth and its derivative
is Λ0-linear if and only if its components ybΛ have these properties. The total derivative of ybΛ
with respect to x∗ exists, is continuous, and is Λ0-linear if and only if its partial derivatives with
respect to the xaΛ and the ξAΛ exist, are continuous, and are Λ0-linear. When computing the
derivative ybΛ with respect to ξAiΛ ∈ Λγi at x∗ ∈ Up|q(Λ) in the direction of wΛ ∈ Λγi , we get∑
αβ
1
β!
(
∂βxφ
b
α
)
(x||)̊x
β
Λ
(
ξA1
Λ
)α1 · · · lim
t→0
1
t
((
ξAiΛ + twΛ
)αi − (ξAiΛ
)αi) · · · (ξA|q|Λ
)α|q| .
If γi is odd, the exponent αi is 0 or 1. In the first (resp., the second) case, the limit vanishes
(resp., is wΛ). If γi is even, the multiplication of vectors in Λγi is commutative and the binomial
formula shows that the limit is wΛαi
(
ξAiΛ
)αi−1
. The derivative thus exists, is continuous, and is
Λ0-linear. Similarly, the derivative of ybΛ with respect to xaΛ exists if and only if its derivatives
26 A.J. Bruce, E. Ibarguengoytia and N. Poncin
with respect to xa|| and with respect to x̊aΛ exist. The (standard) computation of the derivative
with respect to xaΛ at x∗ in the direction of
vΛ = (v||, v̊Λ) ∈ R× Λ̊0
thus leads to the sum of the terms
v||
∑
αβ
1
β!
(
∂β+ea
x φbα
)
(x||)̊x
β
Λξ
α
Λ
and
v̊Λ
∑
α,γ : γa 6=0
1
γ!
(
∂γxφ
b
α
)
(x||)γax̊
γ−ea
Λ ξαΛ = v̊Λ
∑
αβ
1
β!
(
∂β+ea
x φbα
)
(x||)̊x
β
Λξ
α
Λ.
The derivative considered does therefore exist, is continuous, and is Λ0-linear (note that it is
essential that the derivative is the series over αβ multiplied by vΛ – as a ∈ Λ0 does not act
on v||). �
Remark 3.20. The Λ0-linearity is a strong constraint that takes us from the category of ge-
neralized Zn2 -manifolds to the one of Zn2 -manifolds. A similar phenomenon exists in complex
analysis. Indeed, for any real differentiable function f = u + iv : Ω ⊂ C ' R2 → C ' R2, the
Jacobian is an R-linear map Jf : R2 → R2. However, if we further insist that the Jacobian be
C-linear, then we see that f must be holomorphic, that is, it must satisfy the Cauchy–Riemann
equations on Ω. Imposing C-linearity thus greatly restricts class of functions and takes us from
real analysis to complex analysis.
It will also be important to understand what happens to the Λ-points of a given Zn2 -domain
under morphisms of Zn2 -Grassmann algebras.
Proposition 3.21. Let Up|q be a Zn2 -domain and let ψ∗ : Λ → Λ′ be a morphism of Zn2 -
Grassmann algebras. The induced map (see (3.2))
Ψ := Up|q(ψ∗) : Up|q(Λ) 3 x∗ ' (xΛ, ξΛ) 7→ ψ∗ ◦ x∗ ' ψ∗(xΛ, ξΛ) ∈ Up|q(Λ′)
is a smooth map from the open subset Up|q(Λ) of the Fréchet space and Fréchet Λ0-module
Rp|q(Λ) to the open subset Up|q(Λ′) of the Fréchet space and Fréchet Λ′0-module Rp|q(Λ′), such
that
dx∗Ψ(a · v) = ψ∗(a) · dx∗Ψ(v),
for all x∗ ∈ Up|q(Λ), v ∈ Rp|q(Λ) and a ∈ Λ0.
Proof. Since Λ = OR0|m({?}), so that
ψ∗ ∈ HomZn2 Alg(OR0|m({?}),OR0|m′ ({?})),
there is a unique morphism
Φ = (|φ|, φ∗) ∈ HomZn2 Man
(
R0|m′ ,R0|m),
such that ψ∗ = φ∗{?}. Hence, the morphism ψ∗ is continuous from Λ = R[[θ]] to Λ′ = R[[θ′]]
endowed with their standard locally convex topologies [12], and so are its restrictions ψ∗|Λγi
from Λγi to Λ′γi . We thus see that the induced map
Ψ = (ψ∗|Λ0)×p ×
N∏
i=1
(ψ∗|Λγi )
×qi
is continuous.
The Schwarz–Voronov Embedding of Zn2 -Manifolds 27
At x∗ ' (xΛ, ξΛ) =: uΛ ∈ Up|q(Λ) and v ' vΛ ∈ Rp|q(Λ), the derivative
dΨ: Up|q(Λ)× Rp|q(Λ) −→ Rp|q(Λ′)
is defined as
dx∗Ψ(v) = lim
t→0
Ψ(x∗ + tv)−Ψ(x∗)
t
= lim
t→0
(
. . . ,
ψ∗(uaΛ + tvaΛ)−ψ∗(uaΛ)
t
, . . .
)
=
(
. . . , ψ∗(vaΛ), . . .
)
=: (ψ∗(vaΛ)),
where a is the label a ∈ {1, . . . , p} or A ∈ {1, . . . , |q|} of any coordinate in Rp|q(Λ), and where
we used the R-linearity of the Zn2 -algebra morphism ψ∗ : Λ→ Λ′. Hence, for any a ∈ Λ0, we get
dx∗Ψ(a · v) = (ψ∗(a · vaλ)) = (ψ∗(a) · ψ∗(vaλ)) = ψ∗(a) · dx∗ψ(v).
Since the higher order derivatives of Ψ vanish, all its derivatives exist and are continuous, hence,
the map Ψ is actually smooth. �
3.4 The manifold structure on the set of Λ-points
The next theorem generalizes Propositions 3.17 and 3.21. For information about Fréchet man-
ifolds, we refer to Appendix B. We recall that the Λ-points M(Λ) of a Zn2 -manifold M can be
equivalently viewed as the maps m = (|m|,m∗) ∈ HomZn2 Man
(
R0|m,M
)
, as the global pullbacks
m∗ = m∗|M | ∈ HomZn2 Alg(OM (|M |),Λ), or as the induced morphisms
m∗? ∈ HomZn2 Alg(OM,x,Λ),
where x = |m|(?) ∈ |M |. If M = Up|q is a Zn2 -domain, we often write x instead of m and we can
identify x ' x∗ ' x∗? with the pullbacks
(u||, ůΛ, ρΛ) ∈ Up × Λ̊p0 ×
∏
i
Λqiγi
by x∗ of the coordinate functions (u, ρ) in Up|q. Recall as well that Zn2 -morphisms φ : M → N
are mapped injectively to natural transformations φ : M(−)→ N(−) with Λ-component
φΛ : M(Λ) 3 (x,m∗?) 7→ (|φ|(x),m∗? ◦ φ∗x) ∈ N(Λ), (3.16)
and that, for any fixed M , a Zn2 -Grassmann algebra morphism ψ∗ : Λ→ Λ′ induces a map
M(ψ∗) : M(Λ) 3 (x,m∗?) 7→ (x, ψ∗ ◦m∗?) ∈M(Λ′).
Theorem 3.22. Let M be a Zn2 -manifold, and let Λ and Λ′ be Zn2 -Grassmann algebras. Then
(i) M(Λ) has the structure of a nuclear Fréchet Λ0-manifold, and,
(ii) given a morphism of Zn2 -Grassmann algebras ψ∗ : Λ −→ Λ′, the induced mapping M(ψ∗)
is ψ∗-smooth.
Proof. (i) Let p|q be the dimension of the Zn2 -manifold M . The local Zn2 -isomorphisms
hα = (|hα|, h∗α) : Uα = (|Uα|,OM ||Uα|)→ U
p|q
α =
(
Upα, C∞Rp |Upα [[ρ]]
)
,
where α varies in some A and where |Uα| ⊂ |M | is open, provide an atlas on M (see paragraph
below Definition 2.10). As recalled above, the Zn2 -isomorphisms
hα : Uα → U
p|q
α
28 A.J. Bruce, E. Ibarguengoytia and N. Poncin
implement natural isomorphisms hα with Λ-components
hα,Λ : Uα(Λ) 3 (x,m∗?) 7→ (|hα|(x),m∗? ◦ (hα)∗x) ∈ Up|qα (Λ), (3.17)
whose inverses are the similar maps defined using∣∣h−1
α
∣∣ = |hα|−1 and
(
h−1
α
)∗
y
= ((hα)∗|hα|−1(y))
−1(y ∈ Upα).
The family (Uα(Λ), hα,Λ) (α ∈ A) is an atlas that endows M(Λ) with a nuclear Fréchet Λ0-
manifold structure. Indeed:
(a) Any hα,Λ : Uα(Λ)→ Up|qα (Λ) is a bijection valued in the open subset Up|qα (Λ) of the nuclear
Fréchet vector space Rp|q(Λ), which is also a Fréchet module over the nuclear Fréchet
algebra Λ0. Moreover, as the |Uα| are an open cover of |M |, we have
M(Λ) =
⋃
α∈A
Uα(Λ),
in view of Proposition 3.14.
(b) The image hα,Λ(Uα(Λ) ∩ Uβ(Λ)) is open in Rp|q(Λ). To see this, set |Uαβ| = |Uα| ∩ |Uβ| ⊂
|Uα| and consider the open Zn2 -submanifold Uαβ = (|Uαβ|,OM ||Uαβ |) of Uα. The Zn2 -
isomorphism hα restricts to a Zn2 -isomorphism
hα : Uαβ → U
p|q
αβ ,
where the target is the open Zn2 -subdomain Up|qαβ of Up|qα defined over the open subset
Upαβ := |hα|(|Uαβ|) ⊂ Upα,
obtained as the image of the open subset |Uαβ| ⊂ |Uα| by the diffeomorphism |hα|. The
restricted Zn2 -isomorphism hα induces a natural isomorphism hα, whose Λ-component is
a bijection
hα,Λ : Uαβ(Λ)→ Up|qαβ (Λ).
Further, we have
Uαβ(Λ) =
⋃
x∈|Uαβ |
HomZn2 Alg(OM,x,Λ)
=
⋃
x∈|Uα|
HomZn2 Alg(OM,x,Λ)
⋂ ⋃
x∈|Uβ |
HomZn2 Alg(OM,x,Λ) = Uα(Λ) ∩ Uβ(Λ).
Hence, the image hα,Λ(Uα(Λ) ∩ Uβ(Λ)) = Up|qαβ (Λ) ⊂ Rp|q(Λ) is open.
(c) We have still to prove that the transition bijections
hβ,Λ(hα,Λ)−1 : Up|qαβ (Λ)→ Up|qβα (Λ)
are Λ0-smooth. In view of Theorem 3.19, the Zn2 -isomorphism
hβh
−1
α : Up|qαβ → U
p|q
βα
The Schwarz–Voronov Embedding of Zn2 -Manifolds 29
induces a natural isomorphism hβh
−1
α with a Λ0-smooth Λ-component(
hβh
−1
α
)
Λ
: Up|qαβ (Λ)→ Up|qβα (Λ).
In view of equations (3.16) and (3.17), we get(
hβh
−1
α
)
Λ
(u, x∗?) =
(∣∣hβh−1
α
∣∣(u), x∗? ◦
(
hβ ◦ h−1
α
)∗
u
)
=
(
|hβ|
(
|hα|−1(u)
)
, x∗? ◦
(
(hα)∗|hα|−1(u)
)−1 ◦ (hβ)∗|hα|−1(u)
)
= hβ,Λ
(
(hα,Λ)−1(u, x∗?)
)
,
for any (u, x∗?) ∈ U
p|q
αβ (Λ). It follows that hβ,Λ(hα,Λ)−1 =
(
hβh
−1
α
)
Λ
is Λ0-smooth.
(ii) The statement of part (ii) is purely local, see Appendix B. Let (x,m∗?) ∈ M(Λ), let
(Uα(Λ), hα,Λ) be a chart of M(Λ) around (x,m∗?), and let (Uβ(Λ′), hβ,Λ′) be a chart of M(Λ′),
such that M(ψ∗)(Uα(Λ)) ⊂ Uβ(Λ′). We must show that the local form
hβ,Λ′ ◦M(ψ∗) ◦ (hα,Λ)−1
of M(ψ∗) is ψ∗-smooth. Actually, we can choose (Uα(Λ′), hα,Λ′) as second chart, since the image
by M(ψ∗) of a point (y, n∗?) in Uα(Λ), i.e., a point
(y, n∗?) ∈ HomZn2 Alg(OM,y,Λ)
with y ∈ |Uα|, is the point
(y, ψ∗ ◦ n∗?) ∈ HomZn2 Alg(OM,y,Λ
′),
i.e., in Uα(Λ′). From here, we omit subscript α. Since h : U(−)→ Up|q(−) is a natural transfor-
mation, the diagram
U(Λ) U(Λ′)
Up|q(Λ) Up|q(Λ′)
//
M(ψ∗)
��
hΛ
��
hΛ′
//
Up|q(ψ∗)
commutes. Since h is in fact a natural isomorphism, we get that
hΛ′ ◦M(ψ∗) ◦ (hΛ)−1 = Up|q(ψ∗).
From Proposition 3.21 we conclude that this local form is indeed ψ∗-smooth. �
In view of (3.3), in general, the local model Rp|q(Λ) of M(Λ) is infinite-dimensional, due to the
non-zero degree even coordinates of Λ. If the particular Zn2 -Grassmann algebra has no non-zero
degree even coordinates, then it is a polynomial algebra and the resulting local model Rp|q(Λ)
will, of course, be finite-dimensional. Further, we have the
Corollary 3.23. For any Zn2 -manifold M , the associated functor
M(−) ∈
[
Zn2Ptsop, Set
]
can be considered as a functor
M(−) ∈
[
Zn2Ptsop, A(N)FMan
]
,
30 A.J. Bruce, E. Ibarguengoytia and N. Poncin
where the target category is either the category AFMan of Fréchet manifolds over a Fréchet al-
gebra or the category ANFMan of nuclear Fréchet manifolds over a nuclear Fréchet algebra, see
Appendix B. Therefore, the faithful restricted Yoneda functor YZn2 Pts, see Corollary 3.10, can be
viewed as a faithful functor
YZn2 Pts : Zn2Man→
[
Zn2Ptsop, A(N)FMan
]
.
The latter statement requires that the natural transformation φ : M(−) → N(−) induced
by a Zn2 -morphism φ : M → N have components φΛ : M(Λ) → N(Λ) that are morphisms in
A(N)FMan between the Fréchet Λ0-manifolds M(Λ) and N(Λ), i.e., that the φΛ be ρ-smooth for
some morphism ρ : Λ0 → Λ0 of Fréchet algebras. We will show in the next subsection that this
condition is satisfied for ρ = idΛ0 , i.e., we will show that:
Proposition 3.24. Any natural transformation φ : M(−)→ N(−) that is implemented by a Zn2 -
morphism φ : M → N has Λ0-smooth components φΛ : M(Λ)→ N(Λ).
Theorem 3.25. Let M ∈ Zn2Man be of dimension p|q and let Λ ∈ Zn2GrAlg.
(i) The nuclear Fréchet Λ0-manifold M(Λ) is a fiber bundle in the category ANFMan. Its base
is the nuclear Fréchet R-manifold M(R), i.e., the smooth manifold |M |, and its typical
fiber is the nuclear Fréchet Λ0-manifold
Λp|q := Λ̊p0 ×
N∏
i=1
Λqiγi .
(ii) The topology of M(Λ), which is defined, as in the case of smooth manifolds, by the at-
las providing the nuclear Fréchet Λ0-structure, is a Hausdorff topology, so that M(Λ) is
a genuine Fréchet manifold.
Proof. (i) We think of fiber bundles in ANFMan exactly as of fiber bundles in the category of
smooth manifolds. Of course, in such a fiber bundle, all objects and arrows are ANFMan-objects
and ANFMan-morphisms.
Let p∗ : Λ→ R be, as above, the canonical Zn2GrAlg-morphism. The induced map
π := M(p∗) : M(Λ) 3 (x,m∗?) 7→ (x, p∗ ◦m∗?) ' x ∈M(R) ' |M |
is p∗-smooth, i.e., is a morphism in the category ANFMan.
We will show that π is surjective and that the local triviality condition is satisfied.
Let z ∈ |M |. There is a Zn2 -chart (U, h) of M , such that |U | ⊂ |M | is a neighborhood of z.
The Zn2 -isomorphism h : U → Up|q induces a natural isomorphism h, whose Λ-components are
Λ0-diffeomorphisms, i.e., Λ0-smooth maps that have a Λ0-smooth inverse. We have the following
commutative diagram:
U(Λ) Up|q(Λ) ' Up × Λp|q
U(R) ' |U | Up|q(R) ' Up,
oo //
hΛ
��
U(p∗) = π|U(Λ)
��
Up|q(p∗) ' prj1
oo //
hR = |h|
where prj1 is the canonical projection. Let us explain that Up|q(p∗) ' prj1, when read through
[ : Up × Λp|q ↔ Up|q(Λ). We need a more explicit description of the equivalent views on Λ-
points of a Zn2 -domain, see beginning of Section 3.4. As elsewhere in this text, we denote
The Schwarz–Voronov Embedding of Zn2 -Manifolds 31
a Zn2 -morphism R0|m → Up|q by x = (|x|, x∗) and we denote the morphism it induces between
the stalks OUp|q ,|x|(?) → Λ by x∗?. The morphism [ is the succession of identifications
Up × Λp|q 3 (x||, x̊Λ, ξΛ) ' x = (|x|, x∗) ' (|x|(?), x∗?) ∈ Up|q(Λ), (3.18)
where the components of the base morphism |x| are obtained (see [15]) by applying the base pro-
jection ε? : Λ→R of R0|m, i.e., the canonical morphism p∗, to the components xaΛ = (xa||, x̊
a
Λ)∈Λ0.
Hence, we get
|x|(?) = |x| =
(
. . . , p∗(xaΛ), . . .
)
= x||. (3.19)
Therefore, we actually obtain that
Up|q(p∗)([(x||, x̊Λ, ξΛ)) = (|x|(?), p∗ ◦ x∗?) ' |x|(?) = x|| = prj1(x||, x̊Λ, ξΛ).
Since π|U(Λ) = |h|−1 ◦ prj1 ◦hΛ, the local projection π|U(Λ) is surjective, so that z is in the
image of π, which is thus surjective as well.
As just mentioned, we started from z ∈ |M | and found a neighborhood |U | of z and a Λ0-
diffeomorphism hΛ. When identifying |U | with Up via |h| (which then becomes id), we get the
Λ0-diffeomorphism
hΛ : π−1(|U |) ' U(Λ) 3 (y,m∗?) 7→ (y,m∗? ◦ h∗y) ∈ |U | × Λp|q. (3.20)
Observe that in equation (3.20) we used [−1 defined in equations (3.18) and (3.19), thus identi-
fying
(y,m∗? ◦ h∗y) ∈ HomZn2 Alg(OUp|q ,y,Λ) ⊂ Up|q(Λ)
with h ◦m ∈ HomZn2 Man
(
R0|m,Up|q
)
, and then with(
y,pr2(m∗(h∗(x))),m∗(h∗(ξ))
)
∈ |U | × Λp|q,
where we denoted the projection of Λ0 onto Λ̊0 by pr2. Notice also that the conclusion that
Λp|q is a nuclear Fréchet Λ0-manifold comes from the facts that any subspace (resp., any closed
subspace) of a nuclear (resp., a Fréchet) space is a nuclear (resp., a Fréchet) space.
Hence, the trivialization condition is satisfied as well, and M(Λ) is a fiber bundle in ANFMan,
as announced.
(ii) Now consider two different Λ-points m∗ = (x,m∗?) and n∗ = (y, n∗?) in M(Λ). If x 6= y,
then, as |M | is Hausdorff, there exist open neighborhoods |U | of x and |V | of y, such that
|U |∩|V | = ∅. When denoting the corresponding open Zn2 -submanifolds by U and V , respectively,
we get open neighborhoods U(Λ) and V (Λ) of m∗ and n∗, such that U(Λ)∩V (Λ) = ∅. We have
of course to check that, for any Zn2 -chart (Uα, hα), the image
hα,Λ(Uα(Λ) ∩ U(Λ))
is open in Rp|q(Λ), and similarly for V (Λ). To see this, it suffices to proceed as in the proof of
Theorem 3.22.
Next, consider the situation where x = y =: z ∈ |M |, use the trivialization constructed in (i),
and denote the canonical projection from Up × Λp|q onto Λp|q by prj2. As m∗ 6= n∗, we have
hΛ(m∗) 6= hΛ(n∗), i.e.,
(|h|(z), prj2(hΛ(m∗))) 6= (|h|(z), prj2(hΛ(n∗))).
32 A.J. Bruce, E. Ibarguengoytia and N. Poncin
Since prj2(hΛ(m∗)) 6= prj2(hΛ(n∗)) are points in the Hausdorff space Λp|q, there are open neigh-
borhoods Vm∗ and Vn∗ of these projections that do not intersect. The preimages Um∗ and Un∗
of Vm∗ and Vn∗ by the continuous map
prj2 ◦hΛ : U(Λ)→ Λp|q
are then open neighborhoods of m∗ and n∗ that do not intersect.
Finally, the space M(Λ) is indeed a Hausdorff topological space. �
3.5 The Schwarz–Voronov embedding
In order to get a fully faithful functor, hence, to embed the category Zn2Man as full subca-
tegory into a functor category, we need to replace the target category [Zn2Ptsop, A(N)FMan] by
a subcategory that we denote by [[Zn2Ptsop, A(N)FMan]] and that we define as follows:
Definition 3.26. The category [[Zn2Ptsop, A(N)FMan]] is the subcategory of the category [Zn2Ptsop,
A(N)FMan],
(i) whose objects are the functors F , such that, for any Λ ∈ Zn2Ptsop, the value F(Λ) is
a (nuclear) Fréchet Λ0-manifold, and
(ii) whose morphisms are natural transformations η : F → G, such that, for any Λ, the com-
ponent ηΛ : F(Λ)→ G(Λ) is Λ0-smooth.
Proposition 3.27. The restricted Yoneda functor YZn2 Pts can be considered as a faithful functor
S : Zn2Man→ [[Zn2Ptsop, A(N)FMan]].
Proof. The image YZn2 Pts(M) of an object M ∈ Zn2Man is a functor M(−)∈ [Zn2Ptsop, A(N)FMan],
such that, for any Λ, the value M(Λ) is a (nuclear) Fréchet Λ0-manifold. Further, the image
YZn2 Pts(φ) of a Zn2 -morphism φ : M → N is a natural transformation φ : M(−) → N(−), such
that, for any Λ, the component φΛ : M(Λ)→ N(Λ) is Λ0-smooth.
The proof of Λ0-smoothness uses the following construction, which we will also need later on.
Let M,N ∈ Zn2Man be manifolds of dimension p|q and r|s, respectively, let |φ| ∈ C∞(|M |, |N |),
and let (|Vβ|)β be an open cover of |N | by Zn2 -charts
gβ : Vβ → V
r|s
β , where Vβ = (|Vβ|,ON ||Vβ |).
The open subsets |Uβ| := |φ|−1(|Vβ|) ⊂ |M | cover |M |, and each |Uβ| can be covered by Zn2 -charts
hβα : Uβα → U
p|q
βα , where Uβα = (|Uβα|,OM ||Uβα|).
The Zn2 -morphism φ : M → N restricts to a Zn2 -morphism φ|Uβα : Uβα → Vβ. In particular,
the composite
gβ ◦ φ|Uβα ◦ (hβα)−1 : Up|qβα → V
r|s
β
is a Zn2 -morphism.
We now show that φΛ is Λ0-smooth. Therefore, let (x,m∗?) ∈ M(Λ). There is a Zn2 -chart
(Vβ, gβ) of N such that |φ|(x) ∈ |Vβ|, and there is a Zn2 -chart (Uβα, hβα) of M such that x ∈ |Uβα|.
These charts (we omit in the following the subscripts β and α) induce charts (U(Λ), hΛ) of M(Λ)
around (x,m∗?), and (V (Λ), gΛ) of N(Λ) such that φΛ(U(Λ)) ⊂ V (Λ). It suffices to show (see
Appendix B) that the local form
gΛ ◦ φΛ ◦ (hΛ)−1 =
(
g ◦ φ|U ◦ h−1
)
Λ
is Λ0-smooth. This is the case in view of Theorem 3.19. Finally, the faithfulness is established
in Corollary 3.10. This completes the proof. �
The Schwarz–Voronov Embedding of Zn2 -Manifolds 33
We will prove that the functor S is fully faithful, hence, injective (up to isomorphism) on
objects. Therefore, it embeds the category Zn2Man of Zn2 -manifolds as full subcategory into the
larger functor category [[Zn2Ptsop, A(N)FMan]].
Definition 3.28. We refer to the faithful functor
S : Zn2Man −→ [[Zn2Ptsop, A(N)FMan]]
as the Schwarz–Voronov embedding.
Theorem 3.29. The Schwarz–Voronov embedding S is a fully faithful functor. That is, given
two Zn2 -manifolds M and N , the injective map
SM,N : HomZn2 Man
(
M,N
)
→ Hom[[Zn2 Ptsop,A(N)FMan]]
(
M(−), N(−)
)
is bijective.
Proof. Notice first that it follows from the results of [13] and Lemma 3.13 that there is a 1 : 1
correspondence
|M | ' HomZn2 Alg(OM (|M |),R) '
⋃
x∈|M |
HomZn2 Alg(OM,x,R) = M(R),
which is given by
x 7→ εx 7→ (x, εx),
where εx is the evaluation map εx(f) = (εf)(x) (f ∈ OM (|M |)) and where ε is the base map
ε : OM → C∞|M |. Hence, any (x,m∗?) ∈M(R) is equal to (x, εx) and can be identified with x. In
view of (3.17), this 1 : 1 correspondence identifies the nuclear Fréchet R-manifold structure on
M(R) with the smooth manifold structure on |M |.
Let now
η : M(−)→ N(−)
be a natural transformation in the target set of SM,N , i.e., a natural transformation such that,
for any Λ, the Λ-component ηΛ is Λ0-smooth. In particular, the map
|φ| := ηR : |M | → |N |,
is a smooth map between the reduced manifolds. As in the proof of Proposition 3.27, let
(Vβ, gβ)β be an open cover of |N | by Zn2 -charts, and, for any β, let (Uβα, hβα)α be an open
cover of |Uβ| := |φ|−1(|Vβ|) by Zn2 -charts. When denoting the canonical Zn2 -Grassmann algebra
morphism Λ→ R by p∗, we get the commutative diagram⋃
βα Uβα(Λ)
⋃
β Vβ(Λ)
⋃
βα |Uβα|
⋃
β |Vβ|,
//
ηΛ
��
M(p∗)
��
N(p∗)
//
|φ|
which shows that, for any β, α, we get the Λ0-smooth map
(ηΛ)|Uβα(Λ) : Uβα(Λ)→ Vβ(Λ).
34 A.J. Bruce, E. Ibarguengoytia and N. Poncin
Indeed, if, for (x,m∗?) ∈ Uβα(Λ), we set ηΛ(x,m∗?) = (y, n∗?), the commutativity of the diagram
implies that
y ' (y, p∗ ◦ n∗?) = (N(p∗) ◦ ηΛ)(x,m∗?) = (ηR ◦M(p∗))(x,m∗?) = ηR(x, p∗ ◦m∗?)
' |φ|(x) ∈ |Vβ|.
Therefore, the restriction
η|Uβα(−) : Uβα(−)→ Vβ(−)
is a natural transformation with Λ0-smooth components.
Note that
hβα : Uβα → U
p|q
βα and gβ : Vβ → V
r|s
β
are Zn2 -isomorphisms and induce natural isomorphisms, also denoted by hβα and gβ, whose
components are chart diffeomorphisms
hβα,Λ : Uβα(Λ)→ Up|qβα (Λ) and gβ,Λ : Vβ(Λ)→ Vr|sβ (Λ)
of nuclear Fréchet Λ0-manifolds. The local form
gβ,Λ ◦ (ηΛ)|Uβα(Λ) ◦ (hβα,Λ)−1 : Up|qβα (Λ)→ Vr|sβ (Λ)
of ηΛ is thus Λ0-smooth. In other words, any Λ-component of the natural transformation
ϕβα := gβ ◦ η|Uβα(−) ◦ h−1
βα : Up|qβα (−)→ Vr|sβ (−) (3.21)
between functors associated to Zn2 -domains, is Λ0-smooth. It therefore follows from Theorem 3.19
that ϕβα is implemented by a Zn2 -morphism
ϕβα : Up|qβα → V
r|s
β ,
so that the composite
φβα := g−1
β ◦ ϕβα ◦ hβα : Uβα → N (3.22)
is a Zn2 -morphism that is defined on an open Zn2 -submanifold of M . The question is whether
we can patch together these locally defined Zn2 -morphisms, which are inherited from η, and get
a globally defined Zn2 -morphism φ : M → N that induces η.
Let φβα|Uβα,νµ and φνµ|Uβα,νµ be the Zn2 -morphisms obtained by restriction to the open Zn2 -
submanifold Uβα,νµ with base manifold |Uβα,νµ| := |Uβα|∩|Uνµ|. They coincide as Zn2 -morphisms,
if they do as associated natural transformations, i.e., if all Λ-components of those transformations
coincide. This is the case since both Λ-components are equal to ηΛ|Uβα,νµ(Λ). It follows that the
Zn2 -algebra morphisms
φβα|∗Uβα,νµ , φνµ|
∗
Uβα,νµ
: ON (|N |)→ OM (|Uβα,νµ|)
coincide. This implies that we can glue the Zn2 -algebra morphisms φ∗βα : ON (|N |)→ OM (|Uβα|)
and get a Zn2 -algebra morphism
φ∗ : ON (|N |)→ OM (|M |).
The Schwarz–Voronov Embedding of Zn2 -Manifolds 35
Indeed, for any f ∈ ON (|N |), the φ∗βα(f) ∈ OM (|Uβα|) are a family of Zn2 -functions on an open
cover of |M |, which do coincide on the intersections. To see this, note that
(φ∗βα(f))||Uβα,νµ| = φβα|∗Uβα,νµ(f) = φνµ|∗Uβα,νµ(f) = (φ∗νµ(f))||Uβα,νµ|.
Hence, there is a unique global section F ∈ OM (|M |) of the sheaf OM , such that F ||Uβα| =
φ∗βα(f). The Set-morphism, which is defined by
φ∗|N | : ON (|N |) 3 f 7→ F ∈ OM (|M |),
is actually a morphism of Zn2 -algebras. Indeed, note that
ρ
|M |
|Uβα| ◦ φ
∗
|N | = φ∗βα
(ρ is the restriction) and observe that, for any element |Uβα| of the open cover of |M | considered,
we have
(φ∗|N |(f · g))||Uβα| = φ∗βα(f) · φ∗βα(g) = (φ∗|N |(f) · φ∗|N |(g))||Uβα|.
The Zn2 -algebra morphism φ∗|N | fully characterizes a Zn2 -morphism φ = (||φ||, φ∗) : M → N . We
will show that φ induces the natural transformation η, which then completes the proof.
Since φ is glued from the Zn2 -morphisms φβα, we get, in view of equations (3.21) and (3.22),
in particular that
||φ||||Uβα| = |φβα| = ηR|Uβα(R) = |φ|||Uβα|, (3.23)
so that ||φ|| = |φ|. Further, for any |Vβ|,
ρ
|Uβ |
|Uβα| ◦ φ
∗
|Vβ | = φ∗βα,|Vβ | : ON (|Vβ|)→ OM (|Uβα|). (3.24)
Let now Λ be any Zn2 -Grassmann algebra and let (x,m∗?) ∈ Uβα(Λ). As x ∈ |Uβα| and |φ|(x) ∈
|Vβ|, it follows from equations (3.23), (3.24), (3.21), and (3.22), that the image of (x,m∗?) by the
Λ-component of the natural transformation induced by φ is
φΛ(x,m∗?) = (|φ|(x),m∗? ◦ φ∗x) = (|φβα|(x),m∗? ◦ φ∗βα,x) = (φβα)Λ(x,m∗?) = ηΛ(x,m∗?). �
The following theorem is of importance in the study of Zn2 -Lie groups.
Theorem 3.30. The Schwarz–Voronov embedding S sends Zn2 -Lie groups G to functors S(G) =
G(−) from the category Zn2Ptsop of Zn2 -Grassmann algebras to the category ANFLg of nuclear
Fréchet Lie groups over nuclear Fréchet algebras.
The proof is not entirely straightforward and will be given in a paper on Zn2 -Lie groups, which
is currently being written down.
3.6 Representability and equivalence of categories
As the Schwarz–Voronov embedding is fully faithful, the category Zn2Man can be viewed as a full
subcategory of the category [[Zn2Ptsop, A(N)FMan]]. Functor categories are known to be well-suited
for geometric constructions. Hence, when trying to build a Zn2 -manifold M (possibly from other
Zn2 -manifolds Mι), it is often easier to build a functor F in [[Zn2Ptsop, A(N)FMan]] (from the given
Zn2 -manifolds interpreted as functors Mι(−)). However, one has then to show that F can be
represented by a Zn2 -manifold M , i.e., that there is a Zn2 -manifold M , such that M(−) ' F .
36 A.J. Bruce, E. Ibarguengoytia and N. Poncin
Definition 3.31. A functor
F ∈ [[Zn2Ptsop, A(N)FMan]]
is said to be representable, if there exists a Zn2 -manifold M ∈ Zn2Man (which is then unique up
to unique isomorphism), such that
M(−) ' F in [[Zn2Ptsop, A(N)FMan]].
We define the restriction F||U | of a functor F ∈ [[Zn2Ptsop, A(N)FMan]] to an open subset
|U | ⊂ F(R) ∈ (N)FMan.
For any Λ ∈ Zn2GrAlg, let
p∗Λ : Λ −→ R
be the canonical projection, let
F(p∗Λ) : F(Λ) −→ F(R)
be the corresponding smooth map. The preimage
F||U |(Λ) := (F(p∗Λ))−1(|U |) (3.25)
is an open (nuclear) Fréchet Λ0-submanifold of F(Λ).
Consider now a morphism ϕ∗ : Λ −→ Λ′ in Zn2GrAlg. As p∗Λ′ ◦ϕ∗ = p∗Λ, we get the restriction
F||U |(ϕ∗) := F(ϕ∗)
∣∣
F||U|(Λ)
: F||U |(Λ) −→ F||U |(Λ′), (3.26)
which is a morphism in A(N)FMan.
Definition 3.32. For any functor
F ∈ [[Zn2Ptsop, A(N)FMan]]
and any open subset |U | ⊂ F(R), the restriction of F to |U | is the functor
F||U | ∈ [[Zn2Ptsop, A(N)FMan]]
that is defined by equations (3.25) and (3.26).
Example 3.33. Let M ∈ Zn2Man, let M(−) be the corresponding functor, and let |U | ⊂ |M | '
M(R) be an open subset. The restriction M(−)||U | is given:
(i) on objects Λ, by
M(−)||U |(Λ) := {(x,m∗?) ∈M(Λ): (x, p∗Λ ◦m∗?) ' x ∈ |U |} = U(Λ), (3.27)
where U = (|U |,OM ||U |) is the open Zn2 -submanifold of M over |U |, and
(ii) on morphisms ϕ∗ : Λ→ Λ′, by
M(−)||U |(ϕ∗) := M(ϕ∗)|U(Λ) = U(ϕ∗), (3.28)
since both maps are given by
U(Λ) 3 (x,m∗?) 7→ (x, ϕ∗ ◦m∗?) ∈ U(Λ′).
The Schwarz–Voronov Embedding of Zn2 -Manifolds 37
Let F be representable, let M be ‘its’ representing Zn2 -manifold, and let
η : F →M(−) (3.29)
be the corresponding natural isomorphism in [[Zn2Ptsop, A(N)FMan]]. The maps ηΛ and η−1
Λ are
then Λ0-smooth, i.e., ηΛ is a Λ0-diffeomorphism, for any Λ. In particular, the map ηR : F(R)→
M(R) is a diffeomorphism of (nuclear) Fréchet manifolds. This means that the (nuclear) Fréchet
structures on F(R) ' M(R) coincide. Further, if one identifies F(R) ' M(R) with |M |, the
(nuclear) Fréchet structure on F(R) ' M(R) coincides with the smooth structure on |M |. We
can therefore view F(R) as being the smooth manifold |M |. Consider now a Zn2 -atlas (Uα, hα)α
of M . If we denote the dimension of M by p|q, the Zn2 -chart map hα is a Zn2 -isomorphism
hα : Uα → U
p|q
α
valued in a Zn2 -domain of dimension p|q, which implies that
hα : Uα(−)→ Up|qα (−) (3.30)
is a natural isomorphism in [[Zn2Ptsop, A(N)FMan]]. In view equations (3.29), (3.27), (3.28),
and (3.30), the family (|Uα|)α is an open cover of |M | ' F(R), such that, for any α, we have
F||Uα| 'M(−)||Uα| = Uα(−) ' Up|qα (−)
in [[Zn2Ptsop, A(N)FMan]].
Theorem 3.34. A functor F ∈ [[Zn2Ptsop, A(N)FMan]] is representable if and only if there exists
an open cover (|Uα|)α of F(R), such that, for each α, we have
F||Uα| ' U
p|q
α (−) (3.31)
in [[Zn2Ptsop, A(N)FMan]], where Up|qα is a Zn2 -domain in a fixed Rp|q.
Proof. We showed already that the condition is necessary. Assume now that condition (3.31)
is satisfied, i.e., that we have natural isomorphisms
kα : F||Uα| → U
p|q
α (−)
in [[Zn2Ptsop, A(N)FMan]]. This means that the Λ-components
kα,Λ : F||Uα|(Λ)→ Up|qα (Λ)
are Λ0-diffeomorphisms.
In particular, we have a diffeomorphism
|hα| := kα,R : F||Uα|(R) = (F(p∗R))−1(|Uα|) = |Uα| → U
p|q
α (R) ' Upα.
Notice that (|Uα|, |hα|)α can be interpreted as a smooth atlas on |M | := F(R). The direct
image of the structure sheaf O
Up|qα
over Upα by the continuous map |hα|−1 : Upα → |Uα| is a sheaf
over |Uα|, which we denote by OUα :
OUα :=
(
|hα|−1)
∗OUp|qα .
38 A.J. Bruce, E. Ibarguengoytia and N. Poncin
The Zn2 -ringed space
Uα := (|Uα|,OUα)
is isomorphic to the Zn2 -domain Up|qα . The isomorphism is hα := (|hα|, h∗α), where h∗α is the
identity map (a composite of direct images is the direct image by the composite). In other
words, we have an isomorphism of Zn2 -manifolds
hα : Uα → U
p|q
α .
Consider now an overlap |Uαβ| := |Uα| ∩ |Uβ| 6= ∅. Omitting restrictions, we get that kβk
−1
α
is a natural isomorphism (in [[Zn2Ptsop, A(N)FMan]])
kβα := kβk
−1
α : Up|qαβ (−)→ Up|qβα (−)
between functors corresponding to Zn2 -domains (defined as usual). In view of Theorem 3.19, the
natural isomorphism kβα is implemented by a Zn2 -isomorphism
kβα : Up|qαβ → U
p|q
βα .
It follows that
ψβα := h−1
β kβαhα : Uαβ → Uβα
is an isomorphism of Zn2 -manifolds, where Uαβ := (|Uαβ|,OUα ||Uαβ |). The Zn2 -manifolds Uα
can thus be glued and provide then a Zn2 -manifold M over |M | = F(R), such that there are
Zn2 -isomorphisms (|Uα|,OM ||Uα|)→ Uα, if the ψβα satisfy the cocycle condition.
Since the Schwarz–Voronov embedding is fully faithful, we have that ψγβψβα = ψγα as Zn2 -
morphisms if and only if the induced natural transformations coincide. However, for any Λ, we
get
(ψγβψβα)Λ = (hγ,Λ)−1kγ,Λ(kβ,Λ)−1hβ,Λ(hβ,Λ)−1kβ,Λ(kα,Λ)−1hα,Λ = ψγα,Λ.
It remains to show that M actually represents F , i.e., that we can find a natural isomorphism
η : M(−) → F in the category [[Zn2Ptsop, A(N)FMan]], i.e., that, for any Λ ∈ Zn2GrAlg, there is
a Λ0-diffeomorphism ηΛ : M(Λ)→ F(Λ) that is natural in Λ. As (|Uα|)α is an open cover of |M |,
the source decomposes as
M(Λ) =
⋃
α
Uα(Λ),
the Uα(Λ) being open (nuclear) Fréchet Λ0-submanifolds. On any Uα(Λ), we define ηΛ by setting
ηΛ|Uα(Λ) := (kα,Λ)−1hα,Λ : Uα(Λ)→ F||Uα|(Λ) ⊂ F(Λ).
These restrictions provide a well-defined map
ηΛ : M(Λ)→ F(Λ).
Indeed, if (x,m∗?) ∈ Uα(Λ) ∩ Uβ(Λ), we have
(kα,Λ)−1(hα,Λ(x,m∗?)) = (kβ,Λ)−1(hβ,Λ(x,m∗?)) if and only if ψβα,Λ(x,m∗?) = (x,m∗?).
The Schwarz–Voronov Embedding of Zn2 -Manifolds 39
However, since we glued M from the Uα, the gluing Zn2 -isomorphisms ψβα became identities and
so did the induced natural isomorphisms. The definition of η−1
Λ is similar. The source F(Λ)
decomposes as
F(Λ) =
⋃
α
F||Uα|(Λ),
the F||Uα|(Λ) being open (nuclear) Fréchet Λ0-submanifolds. On any F||Uα|(Λ), we define η−1
Λ
by setting
η−1
Λ |F||Uα|(Λ) := (hα,Λ)−1kα,Λ : F||Uα|(Λ)→ Uα(Λ) ⊂M(Λ).
The condition for these restrictions to give a well-defined map
η−1
Λ : F(Λ)→M(Λ)
is equivalent to the condition for ηΛ. Clearly, the maps ηΛ and η−1
Λ are inverses. Naturality
and Λ0-smoothness are local questions and are therefore consequences of the naturality and the
Λ0-smoothness of (kα,Λ)−1hα,Λ and of (hα,Λ)−1kα,Λ. �
We are now prepared to show that the category Zn2Man is equivalent to a functor category.
Theorem 3.35. The category Zn2Man of Zn2 -manifolds (defined as Zn2 -ringed spaces that are lo-
cally isomorphic to Zn2 -domains) and Zn2 -morphisms (defined as morphisms of Zn2 -ringed spaces)
is equivalent to the full subcategory [[Zn2Ptsop, A(N)FMan]] rep of representable functors in [[Zn2Ptsop,
A(N)FMan]].
In other words, the category Zn2Man is equivalent to the category of locally trivial functors in
the subcategory of the functor category [Zn2Ptsop, A(N)FMan], whose objects F have values F(Λ)
in (nuclear) Fréchet Λ0-manifolds and whose morphisms are the natural transformations with
Λ0-smooth components.
Remark 3.36. This result is reminiscent of the identification of schemes with those contrava-
riant functors from affine schemes to sets that are sheaves (for the Zariski topology on affine
schemes) and have a cover by open immersions of affine schemes.
Proof. The Schwarz–Voronov embedding viewed as functor valued in [[Zn2Ptsop, A(N)FMan]]rep is
obviously fully faithful and essentially surjective. It thus induces an equivalence of categories. �
A Generating sets of categories
We will freely use Mac Lane’s book [32] as our source of categorical notions and proofs of general
statements. For completeness, we recall the concept of generating set of a category.
Definition A.1 ([32, p. 127]). Let C be a category. A set S = {Si ∈ Ob(C) : i ∈ I}, where I is
any index set, is said to be a generating set of C, if, for any pair of distinct C-morphisms
φ, ψ : A −→ B,
i.e., φ 6= ψ, there exists some i ∈ I and a C-morphism
s : Si −→ A,
such that the compositions
Si
s−→ A
φ
⇒
ψ
B
are distinct, i.e., φ ◦ s 6= ψ ◦ s. In this case, we say that the object Si separates the morphisms φ
and ψ, and that the set S generates the category C.
40 A.J. Bruce, E. Ibarguengoytia and N. Poncin
Example A.2. The set {R} is a generating set of the category of finite-dimensional real vector
spaces. This is easily seen, as, if we have two distinct linear maps φ, ψ : V → W , then there
exists a vector v ∈ V (v 6= 0), such that φ(v) 6= ψ(v). Thus, the two linear maps differ on the
one-dimensional subspace generated by v. Now let z be a basis of R. Then, the linear map
s : R→ V given by s(z) = v, keeps φ and ψ separate.
Proposition A.3. For any locally small category C, a set S ⊂ Ob(C) generates C if and only if
the restricted Yoneda embedding
YS : C→
[
S op, Set
]
,
where S is viewed as full subcategory of C, is faithful.
Proof. The restricted embedding is defined on objects by
YS(A) = HomC(−, A) ∈
[
S op, Set
]
and on morphisms by
YS(φ) = HomC(−, φ) : YS(A)→ YS(B),
where
(YS(φ))Si : HomC(Si, A) 3 s 7→ φ ◦ s ∈ HomC(Si, B).
The embedding YS is faithful if and only if, for any different φ, ψ : A → B, the corresponding
natural transformations are distinct, i.e., there is at least one i ∈ I and one s ∈ HomC(Si, A),
such that φ ◦ s 6= ψ ◦ s. �
B Fréchet spaces, modules and manifolds
Manifolds over algebras A, also known as A-manifolds, are manifolds for which the tangent
spaces are endowed with a module structure over a given finite-dimensional commutative algebra.
For details, the reader may consult Shurygin [46, 47, 48], and for a discussion of the specific
case of (the even part of) Grassmann algebras one may consult Azarmi [6]. A comprehensive
introduction to the subject can be found in the book (in Russian) by Vishnevskĭı, Shirokov,
and Shurygin [53]. The concept needed in this paper is a infinite-dimensional generalisation of
an A-manifold to the category of Fréchet algebras and Fréchet manifolds. For an introduction
to locally convex spaces, including Fréchet vector spaces, we refer the reader to Conway [14,
Chapter IV], Trèves [52, Part I], or Rudin [39, Chapter 1]. A brief introduction to Fréchet
algebras can be found in Waelbroeck [55, Chapter VII]. For Fréchet manifolds, the reader can
consult Saunders [41, Chapter 7] and Hamilton [27, Part I.4].
Definition B.1. A Fréchet (vector) space is a complete Hausdorff metrizable locally convex
topological vector space.
There exist a few other, equivalent, definitions of Fréchet spaces. The topology on a locally
convex space is metrizable if and only if it can be derived from a countable family of semi-
norms || − ||k, k ∈ N. The topology is Hausdorff if and only if the family of semi-norms is
separating, i.e., if ||x||k = 0, for all k, implies x = 0. Given such a family of semi-norms, one
defines a translationally invariant metric that induces the topology by setting
d(x, y) =
∞∑
k=0
2−k
||x− y||k
1 + ||x− y||k
,
for all x and y.
The Schwarz–Voronov Embedding of Zn2 -Manifolds 41
Example B.2. Let M = (|M |,O) be a Zn2 -manifold. For any open subset U ⊂ |M |, the
spaceO(U) of Zn2 -functions on U is a Fréchet space. An inducing family of semi-norms is given by
||f ||C,D = sup
x∈C
|ε(D(f))(x)|,
where ε is the projection ε : O(U)→ C∞(U) of Zn2 -functions to base functions, where C is any
compact subset of U , and where D is any Zn2 -differential operator over U . Details on the con-
struction of a countable family of semi-norms that is equivalent to (|| − ||C,D)C,D, can be found
in the proof of the last lemma in [12].
Given two Fréchet spaces
(
F,
(
|| − ||Fk
)
k∈N
)
and
(
G,
(
|| − ||Gk
)
k∈N
)
, a linear map
φ : F −→ G
is continuous if and only if, for every semi-norm || − ||Gk , there exists a semi-norm || − ||Fl and
a positive real number C > 0, such that
||φ(x)||Gk ≤ C||x||Fl ,
for every x ∈ F . A similar result holds for continuous bilinear maps
φ : F ×G→ H.
The morphisms of Fréchet spaces are the continuous linear maps, so that the category of Fréchet
spaces is a full subcategory of the category of topological vector spaces.
What makes Fréchet spaces interesting, is the fact that they have just enough structure to
define a derivative of a mapping between such spaces. This leads to a meaningful notion of
a smooth map between Fréchet spaces, and so much of finite-dimensional differential geometry
can be transferred to the infinite-dimensional setting, using Fréchet spaces as local models. The
well known Gâteaux (directional) derivative is defined as follows.
Definition B.3. Let F and G be Fréchet spaces and U ⊂ F be open, and let φ : U → G be
a (nonlinear) continuous map. Then the derivative of φ in the direction of v ∈ F at x ∈ U is
defined as
dxφ(v) := lim
t→0
φ(x + tv)− φ(x)
t
provided the limit exists. We say that φ is continuously differentiable, if the limit exists for all
x ∈ U and v ∈ F , and if the mapping
dφ : U × F −→ G
is (jointly) continuous.
Higher order derivatives are defined inductively, i.e.,
dk+1
x φ(v1, v2, . . . , vk+1) := lim
t→0
dkx+tvk+1
φ(v1, v2, . . . , vk) − dkxφ(v1, v2, . . . , vk)
t
.
A continuous map φ : U → G is then said to be k times continuously differentiable or to be of
class Ck, if
dkφ : U × F×k −→ G
is continuous (or, more explicitly, if all its derivatives of order ≤ k exist everywhere and are
continuous). If φ is of class Ck, its derivative dkxφ(v1, v2, . . . , vk) is multilinear and symmetric
in F×k [45]. Furthermore, we say that φ is smooth, if it is of class Ck, for all k.
42 A.J. Bruce, E. Ibarguengoytia and N. Poncin
Proposition B.4. Let F1, F2 be Fréchet spaces and let U ⊂ F1×F2 be an open subset. A con-
tinuous map φ : U → G valued in a Fréchet space G is of class C1 if and only if its (total)
derivative
dφ : U × (F1 × F2) 3 ((f1, f2), (v1, v2)) 7→ d(f1,f2) φ(v1, v2) ∈ G
is continuous, which is the case if and only if the naturally defined partial derivatives
df1 φ : U × F1 3 ((f1, f2), v1) 7→ df1,(f1,f2) φ(v1) ∈ G
and
df2 φ : U × F2 3 ((f1, f2), v2) 7→ df2,(f1,f2) φ(v2) ∈ G
are continuous. In this case, we have
d(f1,f2) φ(v1, v2) = df1,(f1,f2) φ(v1) + df2,(f1,f2) φ(v2).
The Gâteaux or Fréchet–Gâteaux derivative gives a rather weak notion of differentiation,
however, most of the standard results from calculus in the finite-dimensional setting remain
true. Specifically, the fundamental theorem of calculus and the chain rule still hold. However,
the inverse function theorem is in general lost. For a special class of Fréchet spaces, known
as ‘tame’ Fréchet spaces, there is an analogue of the inverse function theorem known as the
Nash–Moser inverse function theorem, see Hamilton [27] for details.
A nuclear space is a locally convex topological vector space F , such that, for any locally
convex topological vector space G, the natural map F ⊗̂πG −→ F ⊗̂ιG from the projective
to the injective tensor product of F and G is an isomorphism of locally convex topological
vector spaces. In particular, a nuclear Fréchet space is a locally convex topological vector space
that is a nuclear space and a Fréchet space. Loosely, if a space F is nuclear, then, for any
locally convex space G, the complete topological vector space F ⊗̂G is independent of the locally
convex topology considered on F ⊗ G. Because of this, and their nice dual properties, nuclear
spaces provide a reasonable setting for infinite-dimensional analysis. All the Fréchet spaces we
encounter in this paper are in fact nuclear.
The following definition is standard.
Definition B.5. A Fréchet algebra is a Fréchet vector space A, which is equipped with an
associative bilinear and (jointly) continuous multiplication · : A× A→ A. If (pi)i∈I is a family
of semi-norms that induces the topology on A, (joint) continuity is equivalent to the existence,
for any i ∈ I, of j ∈ I, k ∈ I, and C > 0, such that
pi(x · y) ≤ Cpj(x)pk(y), ∀x, y ∈ A.
We can always consider an equivalent increasing countable family of semi-norms (|| − ||n)n∈N.
The preceding condition then requires that, for any n ∈ N, there is r ∈ N (r ≥ n) and C > 0,
such that
||x · y||n ≤ C||x||r||y||r, ∀x, y ∈ A.
In particular, the topology can be induced by a countable family of submultiplicative semi-norms,
i.e., by a family (qn)n∈N that satisfies
qn(x · y) ≤ qn(x)qn(y), ∀n ∈ N, ∀x, y ∈ A.
The Schwarz–Voronov Embedding of Zn2 -Manifolds 43
Note that many authors define a Fréchet algebra simply as a Fréchet vector space, which
carries an associative bilinear multiplication, and whose topology can be induced by a countable
family of submultiplicative semi-norms. This latter definition is equivalent to the former.
In general, a Fréchet algebra need not be unital, and, if it is, one does not require pi(1A) = 1,
in contrast to what is usually required for Banach algebras.
Example B.6 (formal power series). Consider the space
R[[z1, z2, . . . , zq]]
of formal power series in q parameters and coefficients in reals. We set j := (j1, j2, . . . , jq) ∈ N×q
and |j| := j1 + j2 + · · ·+ jq. A general series x now reads
x =
∑
j
zjaj =
∑
j
zj11 z
j2
2 · · · z
jq
q ajq ...j2j1 ,
with no question on the convergence. The algebra structure is the standard multiplication of
formal power series. The topology of coordinate-wise convergence is metrizable and given by
the family of semi-norms
||x||k :=
∑
|j|≤k
|aj |, ∀ k ∈ N.
This algebra is unital with the obvious unit, and it is submultiplicative.
Let us denote the category of Fréchet algebras (resp., commutative Fréchet algebras) as FAlg
(resp., CFAlg). Morphisms in this category are defined to be continuous algebra morphisms. If
we restrict attention to nuclear Fréchet algebras (resp., commutative nuclear Fréchet algebras),
then we work in the full subcategory NFAlg (resp., CNFAlg).
Definition B.7. Fix A ∈ FAlg. A Fréchet A-module is a Fréchet vector space F , together with
a continuous action
A× F µ−→ F,
(a, v) 7→ µ(a, v),
which we will write as µ(a, v) := a ·v (and which is of course compatible with the multiplication
in A).
We give a short survey on Fréchet manifolds.
Definition B.8. Let M be a set. An F -chart of M is a bijective map φ : U → φ(U) ⊂ F ,
where U ⊂M and φ(U) is an open subset of a Fréchet space F .
A Fréchet atlas can be defined using charts valued in various Fréchet spaces. For our purposes,
it is sufficient to consider a fixed Fréchet model.
Definition B.9. A smooth F -atlas on a set M is a collection of F -charts ((Uα, φα))α∈A, such
that
(i) the subsets Uα cover the set M,
(ii) the subsets φα(Uα ∩ Uβ) are open in F ,
(iii) the transition maps
φβα := φβ ◦ φ−1
α : φα(Uα ∩ Uβ) ⊂ F −→ φβ(Uβ ∩ Uα) ⊂ F
are smooth.
44 A.J. Bruce, E. Ibarguengoytia and N. Poncin
A new F -chart (U, φ) on M is compatible with a given smooth F -atlas, if and only if their
union is again a smooth F -atlas, i.e., the subsets φ(U ∩Uα) ⊂ F and φα(Uα ∩U) ⊂ F are open,
and the transition maps
φα ◦ φ−1 : φ(U ∩ Uα) −→ φα(Uα ∩ U) and φ ◦ φ−1
α : φα(Uα ∩ U) −→ φ(U ∩ Uα)
are smooth (for every α ∈ A). Similarly, two smooth F -atlases are compatible provided their
union is also a smooth F -atlas. Compatibility is an equivalence relation on all possible smooth
F -atlases on M.
Definition B.10. A smooth F -structure on a setM is a choice of an equivalence class of smooth
F -atlases on M. We say that M is a Fréchet manifold modelled on the Fréchet space F , if M
comes equipped with a smooth F -structure. If the model vector space F is nuclear, we speak of
a nuclear Fréchet manifold.
A smooth F -atlas on a Fréchet manifold M allows us to define in the obvious way a topo-
logy on M, which is independent of the atlas considered in the chosen equivalence class. The
domain U of an F -chart (U, φ) is open in this topology and the bijective map φ : U ⊂ M →
φ(U) ⊂ F is a homeomorphism for the induced topologies. Most authors confine themselves to
Fréchet manifolds, whose topology is Hausdorff.
Morphisms between two Fréchet manifolds are the smooth maps between them, where smooth-
ness is defined, just as in the finite-dimensional case of smooth manifolds, in terms of charts and
smoothness of local representatives of the maps. We denote the category of Fréchet manifolds
and the morphisms between them by FMan.
Further, the tangent space TfM to a Fréchet manifoldM at a point f ∈M can be defined as
usual, using the tangency equivalence relation for the smooth curves of M that pass through f
at time 0. One can easily see that TfM is a Fréchet space. The concept of Fréchet vector bundle
is the natural generalization of the notion of smooth vector bundle to the category of Fréchet
manifolds. The tangent bundle TM of a Fréchet manifold M is an example of a Fréchet vector
bundle.
In general, we must make a distinction between the (kinematic) tangent bundle as defined here
and the operational tangent bundle defined in terms of derivations of the algebra of functions of
a Fréchet manifold. Indeed, the two notions do not, in general, coincide, there are derivations
that do not correspond to tangent vectors. However, it is known that for nuclear Fréchet
manifolds the two concepts do coincide.
Let F : M→ N be a smooth map between Fréchet manifolds modelled on Fréchet spaces F
and G, respectively. There is a tangent map TF of F, which is a smooth map
TF : TM→ TN ,
and restricts, for any f ∈M, to a linear map
TfF : TfM→ TF(f)N .
As in the finite-dimensional case, the local representative of TfF is the derivative dφ(f)
(
ψFφ−1
)
of the corresponding local representative
ψFφ−1 : φ(U) ⊂ F → G
of F at the point φ(f).
Fundamental to the work in this paper are Fréchet manifolds with a further module structure
on their tangent bundle.
The Schwarz–Voronov Embedding of Zn2 -Manifolds 45
Definition B.11. LetM be a Fréchet manifold, whose model Fréchet space F is a module over
a Fréchet algebra A. We say that M is a Fréchet A-manifold, if and only if all transition maps
are A-linear, i.e.,
dφα(f)φβα(a · v) = a · dφα(f)φβα(v),
for all f ∈ Uα ∩ Uβ, a ∈ A, and v ∈ F .
Morphisms between Fréchet A-manifolds M and N are the A-smooth maps between them,
i.e., are the smooth maps F : M → N that are A-linear at every point. This means that, for
any point f ∈M, there is anM-chart (U, φ) around f and an N -chart (V, ψ) around F(f) that
contains F(U), such that the local representative
dφ(f)
(
ψFφ−1
)
of the derivative TfF is an A-linear endomorphism of the A-module F . The requirement actually
means that the derivative TfF must be A-linear at any point f ∈ M. In this way, we obtain
the category of Fréchet A-manifolds, which we denote as AFMan.
In this paper, we will use the category AFMan, whose objects are the Fréchet A-manifolds,
where A is not a fixed Fréchet algebra, but any Fréchet algebra. The definition of AFMan-
morphisms generalizes the definition of AFMan-morphisms. Suppose that M is a Fréchet A-
manifold modelled on an A-module F andN is a Fréchet B-manifold modelled on a B-module G.
The AFMan-morphisms from M to N are the A-smooth maps between them, i.e., those smooth
maps F : M → N that are at any point compatible with the module structures of F and G.
This means that there is a Fréchet algebra morphism ρ : A → B, and, for any f ∈ M, there
exist charts (U, φ) and (V, ψ) as above, such that
dφ(f)
(
ψFφ−1
)
(a · v) = ρ(a) · dφ(f)
(
ψFφ−1
)
(v),
for any a ∈ A and v ∈ F . This requirement actually means that, for any f , the derivative TfF
is compatible with the induced actions on the tangent spaces. We will refer to an A-smooth map
with associated Fréchet algebra morphism ρ, as a ρ-smooth map. If we restrict our attention to
nuclear objects, i.e., the model Fréchet vector space and the Fréchet algebra are both nuclear,
then we denote the corresponding category as ANFMan.
Acknowledgements
The authors cordially thank the anonymous referees for their valuable remarks and comments,
which have served to improve this article, as well as for their suggestions for future research.
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1 Introduction
2 Rudiments of Z2n-graded geometry
2.1 The category of Z2n-manifolds
2.2 The functor of points
3 Z2n-points and the functor of points
3.1 The category of Z2n-points
3.2 A convenient generating set of Z2nMan
3.3 Restricted Yoneda functor and fullness
3.4 The manifold structure on the set of -points
3.5 The Schwarz–Voronov embedding
3.6 Representability and equivalence of categories
A Generating sets of categories
B Fréchet spaces, modules and manifolds
References
|
| id | nasplib_isofts_kiev_ua-123456789-210608 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-17T12:04:19Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Bruce, Andrew James Ibarguengoytia, Eduardo Poncin, Norbert 2025-12-12T10:41:28Z 2020 The Schwarz-Voronov Embedding of ℤⁿ₂-Manifolds. Andrew James Bruce, Eduardo Ibarguengoytia and Norbert Poncin. SIGMA 16 (2020), 002, 47 pages 1815-0659 2010 Mathematics Subject Classification: 58C50; 58D1; 14A22 arXiv:1906.09834 https://nasplib.isofts.kiev.ua/handle/123456789/210608 https://doi.org/10.3842/SIGMA.2020.002 Informally, ℤⁿ₂-manifolds are 'manifolds' with ℤⁿ₂-graded coordinates and a sign rule determined by the standard scalar product of their ℤⁿ₂-degrees. Such manifolds can be understood in a sheaf-theoretic framework, as supermanifolds can, but with significant differences, in particular in integration theory. In this paper, we reformulate the notion of a ℤⁿ₂-manifold within a categorical framework via the functor of points. We show that it is sufficient to consider ℤⁿ₂-points, i.e., trivial ℤⁿ₂-manifolds for which the reduced manifold is just a single point, as 'probes' when employing the functor of points. This allows us to construct a fully faithful restricted Yoneda embedding of the category of ℤⁿ₂-manifolds into a subcategory of contravariant functors from the category of ℤⁿ₂-points to a category of Fréchet manifolds over algebras. We refer to this embedding as the Schwarz-Voronov embedding. We further prove that the category of ℤⁿ₂-manifolds is equivalent to the full subcategory of locally trivial functors in the preceding subcategory. The authors cordially thank the anonymous referees for their valuable remarks and comments, which have served to improve this article, as well as for their suggestions for future research. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Schwarz-Voronov Embedding of ℤⁿ₂-Manifolds Article published earlier |
| spellingShingle | The Schwarz-Voronov Embedding of ℤⁿ₂-Manifolds Bruce, Andrew James Ibarguengoytia, Eduardo Poncin, Norbert |
| title | The Schwarz-Voronov Embedding of ℤⁿ₂-Manifolds |
| title_full | The Schwarz-Voronov Embedding of ℤⁿ₂-Manifolds |
| title_fullStr | The Schwarz-Voronov Embedding of ℤⁿ₂-Manifolds |
| title_full_unstemmed | The Schwarz-Voronov Embedding of ℤⁿ₂-Manifolds |
| title_short | The Schwarz-Voronov Embedding of ℤⁿ₂-Manifolds |
| title_sort | schwarz-voronov embedding of ℤⁿ₂-manifolds |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210608 |
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