Weak Frobenius monads and Frobenius bimodules
As observed by Eilenberg and Moore (1965), for a monad F with right adjoint comonad G on any category A, the category of unital F-modules AF is isomorphic to the category of counital G-comodules AG. The monad F is Frobenius provided we have F=G and then AF≃AF. Here we investigate which kind of is...
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| Опубліковано в: : | Algebra and Discrete Mathematics |
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Інститут прикладної математики і механіки НАН України
2016
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| Цитувати: | Weak Frobenius monads and Frobenius bimodules / R. Wisbauer // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 2. — С. 287–308. — Бібліогр.: 8 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859851612409823232 |
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| author | Wisbauer, R. |
| author_facet | Wisbauer, R. |
| citation_txt | Weak Frobenius monads and Frobenius bimodules / R. Wisbauer // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 2. — С. 287–308. — Бібліогр.: 8 назв. — англ. |
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| description | As observed by Eilenberg and Moore (1965), for a monad F with right adjoint comonad G on any category A, the category of unital F-modules AF is isomorphic to the category of counital G-comodules AG. The monad F is Frobenius provided we have F=G and then AF≃AF. Here we investigate which kind of isomorphisms can be obtained for non-unital monads and non-counital comonads. For this we observe that the mentioned isomorphism is in fact an isomorphisms between AF and the category of bimodules AFF subject to certain compatibility conditions (Frobenius bimodules). Eventually we obtain that for a weak monad (F,m,η) and a weak comonad (F,δ,ε) satisfying Fm⋅δF=δ⋅m=mF⋅Fδ and m⋅Fη=Fε⋅δ, the category of compatible F-modules is isomorphic to the category of compatible Frobenius bimodules and the category of compatible F-comodules.
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| first_indexed | 2025-12-07T15:42:22Z |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 21 (2016). Number 2, pp. 287–308
© Journal “Algebra and Discrete Mathematics”
Weak Frobenius monads
and Frobenius bimodules
Robert Wisbauer∗
Abstract. As observed by Eilenberg and Moore (1965),
for a monad F with right adjoint comonad G on any category A,
the category of unital F -modules AF is isomorphic to the category
of counital G-comodules A
G. The monad F is Frobenius provided
we have F = G and then AF ≃ A
F . Here we investigate which
kind of isomorphisms can be obtained for non-unital monads and
non-counital comonads. For this we observe that the mentioned iso-
morphism is in fact an isomorphisms between AF and the category of
bimodules AF
F
subject to certain compatibility conditions (Frobenius
bimodules). Eventually we obtain that for a weak monad (F, m, η)
and a weak comonad (F, δ, ε) satisfying Fm · δF = δ · m = mF · Fδ
and m · Fη = Fε · δ, the category of compatible F -modules is iso-
morphic to the category of compatible Frobenius bimodules and
the category of compatible F -comodules.
Introduction
A monad (F, m, η) on a category A is called a Frobenius monad pro-
vided the functor F is (right) adjoint to itself (e.g. Street [6]). Then F also
allows for a comonad structure (F, δ, ε) and the (Eilenberg-Moore) cate-
gory AF of F -modules is isomorphic to the category A
F of F -comodules.
As shown in [5, Theorem 3.13], this isomorphism characterises a functor
with monad and comonad structure as Frobenius monad. It is not diffi-
cult to see that the categories AF and A
F are in fact isomorphic to the
∗The author wants to thank Bachuki Mesablishvili for proofreading.
2010 MSC: 18A40, 18C20, 16T1.
Key words and phrases: pairing of functors; adjoint functors; weak (co)monads;
Frobenius monads; firm modules; cofirm comodules; separability.
288 Weak Frobenius monads
category A
F
F of (unital and counital) Frobenius bimodules. In this setting
units and counits play a crucial role.
Here we are concerned with the question what is left from these
correspondences when the conditions on units and counits are weakened.
An elementary approach to this setting is offered in [7] and [8] where
adjunctions between functors are replaced by regular pairings (L, R) of
functors L : A → B, R : B → A (see 1.4). The composition LR (resp. RL)
yields endofunctors on A (resp. B) and these are closely related to weak
(co)monads as considered by Böhm et al. in [1, 3] (see Remark 1.12). In
Section 1 we recall the definitions and collect basic results needed for our
investigations.
Given a non-unital monad (F, m) on any category A, a non-unital
module ̺ : F (A) → A is called firm (see [2]) if the defining fork
FF (A)
mA //
F (̺)
// F (A)
̺ // A
is a coequaliser in the category of non-unital F -modules. This notion
is generalised in Section 2 by restricting the coequaliser requirement to
certain classes K of morphisms of F -modules. It turns out that compatible
modules of a weak monad (F, m, η) satisfy the resulting conditions for a
suitable class K (Proposition 2.10). Similar results hold for weak comonads.
In Section 3, we return to parings of the functors L and R. Given
natural transformations η : IA → RL and ε̃ : RL → IB, one obtains a
non-unital monad (LR, Lε̃R) and a non-counital comonad (LR, LηR) on
B for which the Frobenius condition is satisfied and this motivates the
definition of Frobenius bimodules (see 3.1). Given a non-counital LR-
comodule ω : B → LR(B), the question arises when it can be extended
to a Frobenius bimodule by some ̺ : LR(B) → B. As sufficient condition
it turns out that the defining cofork for ̺ is a coequaliser in the category
of non-counital comodules (see Proposition 3.6). Further situations are
investigated, in particular for regular pairings (Theorems 3.9, 3.10).
In Section 4, the results about the pairings (L, R) from Section 3 are
reformulated for the (co)monad LR, that is, we consider an endofunctor
F on B endowed with a weak monad structure (F, m, η), a weak comonad
structure (F, δ, ε), and the compatibility between m and δ is postulated as
the Frobenius property (see 4.1). (For LηR and Lε̃R the latter follows by
naturality, see (3.1)). The constructions lead to various functors between
(compatible) module, comodule and bimodule categories (see 4.2, 4.3,
4.6). For proper (co)monads we get some results obtained by Böhm and
Gómez-Torrecillas in [2] as Corollaries 4.7, 4.8.
R. Wisbauer 289
1. Regular pairings
Throughout A and B will denote any categories. The symbols IA, A,
or just I will stand for the identity morphism on an object A, IF or F
denote the identity transformation on the functor F , and IA means the
identity functor on A.
Given an endofunctor T on A, an idempotent natural transformation
e : T → T is said to split if there are an endofunctor T on A and natural
transformations p : T → T and i : T → T such that e = i · p and p · i = IT .
We recall some notions from [7], [8], [3].
1.1. Non-counital comodules. Let (G, δ) be a pair with an endofunctor
G : A → A and a coassociative natural transformation (coproduct)
δ : G → GG. Then (non-counital) G-comodules are defined as objects
A ∈ A with a morphism υ : A → G(A) satisfying G(υ) · υ = δA · υ and
the category of these G-comodules is denoted by A
−→
G.
Consider a triple (G, δ, ε), with (G, δ) a pair as above and ε : G → IA
any natural transformation (quasi-counit). Then a G-comodule (A, υ) is
said to be compatible provided υ = GεA · δA · υ. The full subcategory of
A
−→
G consisting of compatible comodules is denoted by A
G.
(G, δ, ε) is called a weak comonad if
ε = ε · Gε · δ, δ = GεG · Gδ · δ, and Gε · δ = εG · δ.
Then a G-comodule (A, υ) is compatible if εGA · δA · υ = υ = υ · εA · υ.
Furthermore, Gε · δ : G → G is idempotent and in case this is split by
G
p
−→ G
i
−→ G, one obtains a comonad (G, δ, ε) by putting
δ : G
i
−→ G
δ
−→ GG
pp
−→ GG, ε : G
i
−→ G
ε
−→ IA.
1.2. Non-unital modules. Let (F, m) be a pair with an endofunctor
F : A → A and an associative natural transformation (product) m :
FF → F . Then (non-unital) F -modules are defined as objects A ∈ A
with a morphism ̺ : F (A) → A satisfying ̺ ·F̺ = ̺ ·mA and the category
of these F -modules is denoted by A
−→
F .
Consider a triple (F, m, η), with (F, m) a pair as above and any natural
transformation η : IA → F (quasi-unit). An F -module (A, ̺) is said to
be compatible provided ̺ = ̺ · mA · FηA and the full subcategory of A
−→
F
consisting of compatible modules is denoted by AF .
(F, m, η) is called a weak monad if
η = m · Fη · η, m = m · mF · FηF , and m · Fη = m · ηF .
290 Weak Frobenius monads
Then an F -module (A, ̺) is compatible if ̺ · mA · ηFA = ̺ = ̺ · ηA · ̺.
Furthermore, m · Fη : F → F is idempotent and in case this is split by
F
p
−→ F
i
−→ F , one obtains a monad (F , m, η) by putting
m : FF
ii
−→ FF
m
−→ F
p
−→ F , η : IA
η
−→ F
p
−→ F .
1.3. Pairings of functors. For functors L : A → B and R : B → A,
pairings are defined as maps, natural in A ∈ A and B ∈ B,
MorB(L(A), B)
α // MorA(A, R(B)),
β
oo
MorA(R(B), A)
α̃ // MorA(B, L(A))
β̃
oo .
These - and their compositions - are determined by natural transformations
obtained as images of the corresponding identity morphisms,
map natural transformation map natural transformation
α η : IA → RL, α̃ η̃ : IB → LR,
β ε : LR → IB, β̃ ε̃ : RL → IA,
β · α ℓ : L
Lη
−→ LRL
εL
−→ L β̃ · α̃ r̃ : R
Rη̃
−−→ RLR
ε̃R
−→ R
α · β r : R
ηR
−−→ RLR
Rε
−→ R α̃ · β̃ ℓ̃ : L
η̃L
−→ LRL
Lε̃
−→ L .
β (resp. α) is said to be symmetric if Lr = ℓR (resp. Rℓ = rL) (see
[8, §3]). Under the given conditions (see [8]),
(i) (LR, LηR, ε) is a non-counital comonad on B with quasi-counit ε;
(ii) (RL, RεL, η) is a non-unital monad on A with quasi-unit η;
(iii) (LR, Lε̃R, η̃) is a non-unital monad on B with quasi-unit η̃;
(iv) (RL, Rη̃L, ε̃) is a non-counital comonad on A with quasi-counit ε̃.
Clearly, if α is a bijection, then (L, R) is an adjoint pair, if α̃ is a
bijection, then (R, L) is an adjoint pair, and if α and α̃ are bijections,
then LR and RL are Frobenius functors.
1.4. Regular pairings. A pairing (L, R, α, β) is said to be regular if
α · β · α = α and β · α · β = β.
In this case, ℓ : L → L and r : R → R (see 1.3) are idempotents and
ε = ε · ℓr = ε · ℓR = ε · Lr ,
η = rℓ · η = Rℓ · η = rL · η.
R. Wisbauer 291
If β is symmetric, ℓr = Lr = ℓR; if α is symmetric, rℓ = Rℓ = rL.
Assume the idempotents ℓ, r to be splitting, that is,
L
ℓ
−→ L = L
p
−→ L
i
−→ L, R
r
−→ R = R
p′
−→ R
i′
−→ R.
Then, for the natural morphisms
η : IA
η // RL
p′p // RL, ε : LR
ii′
// LR
ε // IB,
one gets εL · Lη = IL and Rε · ηR = IR, hence yielding an adjunction
(L, R, α, β).
1.5. Proposition. For functors A
L //
B
R
oo , there are equivalent:
(a) (L, R) allows for a regular pairing (L, R, α, β) with splitting idem-
potents ℓ, r;
(b) there are retractions L
i
−→ L
p
−→ L and R
i′
−→ R
p′
−→ R such that (L, R)
allows for an adjunction.
Proof. (a)⇒(b) The data from 1.4 yield an adjunction (L, R, α, β) and
the commutative diagram
MorB(L(A), B)
α //
Mor(iA,B)
��
MorA(A, R(B))
β //
Mor(A,p′
B
)
��
MorB(L(A), B)
Mor(iA,B)
��
MorB(L(A), B)
α // MorA(A, R(B))
β
// MorB(L(A), B).
(b)⇒(a) Given an adjunction (L, R, α, β) and retracts L
i
−→ L
p
−→ L
and R
i′
−→ R
p′
−→ R, the above diagram tells us how to define (new) α and
β to get commutativity. Then it is routine to check that (L, R, α, β) is
a regular pairing and the resulting idempotents are split by (p, i) and
(p′, i′), respectively.
Now assume that (L, R, α, β) and (R, L, α̃, β̃) are regular pairings.
Then ℓ, ℓ̃ are two natural transformations on L and r , r̃ are two natural
transformations on R. We are interested in the case when they coincide.
Applying 1.5 and its dual yields:
1.6. Proposition. For functors A
L //
B
R
oo , there are equivalent:
292 Weak Frobenius monads
(a) (L, R) allows for regular pairings (L, R, α, β) and (R, L, α̃, β̃) with
splitting idempotents ℓ = ℓ̃, r = r̃ ;
(b) there are retractions L
i
−→ L
p
−→ L and R
i′
−→ R
p′
−→ R such that (L, R)
and (R, L) allow for adjunctions, that is, (L, R) is a Frobenius pair
of functors.
1.7. Remark. Let (G, δ, ε) be a non-counital comonad on the category A
with quasi-unit ε. For the Eilenberg-Moore category A
−→
G of non-counital
G-comodules there are the free and the forgetful functors
φG : A → A
−→
G, A 7→ (G(A), δA), UG : A
−→
G
→ A, (A, ω) 7→ A.
There is a pairing (φG, UG, αG, βG) with the maps, for X ∈A, (A, ω) ∈ A
−→
G,
αG : MorA(UG(A), X) → MorG(A, φG(X)), f 7→ G(f) · ω,
βG : MorG(A, φG(X)) → MorA(UG(A), X), g 7→ εX · g.
Compatible G-comodules υ : A → G(A) are those with αGβG(υ) = υ.
(G, δ, ε) is a weak comonad if and only if (φG, UG, αG, βG) is a regular
pairing with βG symmetric (see [8, Proposition 4.4]).
Similar characterisations hold for weak monads ([8, Proposition 3.4]).
1.8. Related comonads. Let (L, R, α, β) be a regular pairing (see 1.4).
(1) For the coproduct
δ : LR
LηR
−−−→ LRLR
ℓRLr
−−−→ LRLR,
(LR, δ, ε) is a weak comonad on B. If β is symmetric, δ = LηR.
(2) ℓr : LR → LR induces morphisms of non-counital comonads re-
specting the quasi-counits,
(LR, LηR, ε) → (LR, LηR, ε) and (LR, LηR, ε) → (LR, δ, ε),
and an endomorphism of weak comonads (LR, δ, ε) → (LR, δ, ε).
Proof. Direct verification shows εLR · δ = ℓr = LRε · δ, the conditions for
a weak comonad. For the next claims, consider the commutative diagram
LR
ℓr //
LηR
��
LR
LηR
��
δ
&&
LRLR
ℓRLr //
ℓrℓr %%
LRLR
ℓRLr //
LrℓR
��
LRLR
LrℓR
��
LRLR
ℓRLr
// LRLR ;
R. Wisbauer 293
the left hand part proves the assertion about the first morphism and the
outer paths show the properties of the second and third morphisms.
1.9. Related monads. Let (L, R, α, β) be a regular pairing (see 1.4).
(1) For the product
m : RLRL
rLRℓ
−−−→ RLRL
RεL
−−→ RL,
(RL, m, η) is a weak monad on A. If α is symmetric, m = RεL.
(2) rℓ : RL → RL yields morphisms of non-unital monads respecting
the quasi-units,
(RL, RεL, η) → (RL, RεL, η) and (RL, RεL, η) → (RL, m, η).
and an endomorphism of weak monads (RL, m, η) → (RL, m, η).
Proof. One easily verifies m · ηRL = rℓ = m · RLη, the condition for a
weak monad. The other claims are shown similarly to 1.8
Combining the preceding observations we have shown:
1.10. Proposition. Let (L, R, α, β) be a regular pairing and assume the
idempotents ℓ and r to split. With the notation from 1.4, (LR, LηR, ε) is
a comonad on B and (RL, RεL, η) is monad on A. Then,
(1) the natural transformation pp′ : LR → LR induces morphisms
of non-counital comonads (LR, LηR, ε) → (LR, LηR, ε), and mor-
phisms of weak comonads (LR, δ, ε) → (LR, LηR, ε);
(2) the natural transformation p′p : RL → RL induces morphisms of
non-unital monads (RL, RεL, η) → (RL, RεL, η) and morphisms
of weak monads (RL, m, η) → (RL, RεL, η).
1.11. Regular pairings and comodules. Let (L, R, α, β) be a regular
pairing and consider the weak comonad (LR, δ, ε) defined in 1.8. Then
a non-counital (LR, δ, ε)-comodule (B, υ) is compatible (see 1.1) if υ =
εLRB · δB · υ = rℓB · υ.
Write B
LR,δ
for the full subcategory of B
−→
LR,δ formed by the compat-
ible (LR, δ, ε)-comodules. For any B ∈ B, (LR(B), δB) is a compatible
(LR, δ, ε)-comodule, and thus we have a functor
φLR,δ : B → B
LR,δ
, B 7→ (LR(B), δB).
294 Weak Frobenius monads
The obvious forgetful functor ULR,δ : B
LR,δ
→ B need not be (left) adjoint
to φLR but (φLR, ULR,δ) allows for a regular pairing (see 1.7).
Denoting by B
−→
LR,η the non-counital comodules for (LR, LηR, ε), the
natural transformation (LR, LηR, ε) → (LR, δ, ε) induced by ℓr (see 1.8)
defines a functor tℓr : B
−→
LR,η
→ B
−→
LR,δ. It is easy to see that hereby the
image of any comodule in B
−→
LR,η is a compatible comodule in B
−→
LR,δ
leading to a commutative diagram
B
φLR,η
//
φLR,δ &&
B
−→
LR,η
tℓr
��
B
LR,δ
.
In case the idempotents ℓ and r are splitting, we get the splitting
natural transformation pp′ : LR → LR (from 1.4) which induces functors
B
LR,δ
→ B
LR
and B
LR,η
→ B
LR
, also denoted by pp′, with commutative
diagram
B
−→
LR,η
pp′
''
tℓr // B
LR,δ
pp′
��
B
φLR,η
OO
φLR
// B
LR
.
Since LR is a comonad, every non-counital LR-comodule is compatible,
that is B
LR
= B
−→
LR, but need not be counital.
1.12. Remark. As pointed out by an anonymous referee, a regular pairing
(L, R, α, β) defined in 1.4 is in fact the same as an adjunction in the local
idempotemt closure Cat of the 2-category Cat of categories and hence
corresponds to a comonad in Cat. This lives on the 1-cell (LR, ℓr) with
coproduct ℓRLr · LηR and counit ε (see [3]). In this approach, similar to
Proposition 1.6, the properties of the weak comonad LR are described by
properties of a related comonad LR.
We are also interested in the modules and comodules induced directly
by RL and LR, respectively.
2. (Co)firm (co)modules
To develope further constructions for pairings of functors symmetry
conditions are needed and so we consider weak (co)monads.
R. Wisbauer 295
The notion of (co-)equalisers in categories may be modified in the
following way.
2.1. Definitions. Let K be a class of morphisms in a category A closed
under composition. A cofork
B
k // C
g //
f
// D
is said to be a K-equaliser provided k ∈ K and, for any h : Q → C in K
with f · h = g · h, there exists a unique q : Q → B in K such that h = k · q.
If this holds, then, for morphisms r, s : X → B in K, k · r = k · s implies
r = s.
Similarly, a fork
B
g //
f
// C
s // D
is said to be a K-coequaliser provided s ∈ K and, for any h : C → Q in K
with h · f = h · g, there is a unique q : D → Q in K such that h = q · s. In
this case, for morphisms t, u : D → Y in K, t · s = u · s implies t = u.
A class K of morphisms in A is called an ideal class if for any morphisms
A
f
−→ B
g
−→ C in A, f or g in K implies that g · f is in K.
Taking for K the class of all morphisms in A, the notions defined
above yield the usual equalisers and coequalisers in the category A.
2.2. K-cofirm comodules. Let (G, δ) be a non-counital comonad. Given
an ideal class K of morphisms in the category B
−→
LR of non-counital G-
comodules, a comodule (B, ω) is called K-cofirm provided the defining
cofork
B
ω // G(B)
δB //
G(ω)
// GG(B)
is a K-equaliser. If we choose for K all morphisms in B
−→
LR, a K-cofirm
comodule is just called cofirm.
2.3. Compatible comodule morphisms. Now let (G, δ, ε) be a weak
comonad and γ := Gε ·δ : G → G the idempotent comonad morphism. We
call a morphism h between G-comodules (B, ω) and (B′, ω′) γ-compatible,
provided it induces commutativity of the triangles in the diagram
B
h
��
ω //
h
))
G(B)
εB // B
h
��
B′
ω′
// G(B′) εB′
// B′.
296 Weak Frobenius monads
Clearly, since the outer diagram is always commutative for comodule
morphisms, it is enough to require commutativity for one of the triangles.
Thus one readily obtains:
(1) The class Kγ of all γ-compatible morphisms in B
G
is an ideal class.
(2) A morphism h : Q → G(B) of G-comodules is in Kγ if and only if
γB · h = h.
(3) A morphism h : G(B) → Q of G-comodules is in Kγ if and only if
h · γB = h.
Evidently, a G-comodule (B, ω) is compatible (as in 1.1) if and only
if ω ∈ Kγ , that is, ω = γB · ω.
Notice that γ = IG implies that every non-counital G-comodule is
γ-compatible, that is, B
−→
G = B
G
; in this case, however, not every G-
comodule morphism need to be γ-compatible and a G-comodule (B, ω)
need not be counital but only satisfies ω = ω · εB · ω.
2.4. Proposition. If (G, δ, ε) is a weak comonad, then any compatible
G-comodule (B, ω) is Kγ-cofirm.
Proof. We have to show that the cofork
B
ω // G(B)
δB //
G(ω)
// GG(B)
is a Kγ-equaliser. Let (Q, κ) be a G-comodule and h : Q → G(B) a
morphism in Kγ with G(ω) · h = δB · h. In the diagram
Q
h //
h
$$
κ
��
G(B)
G(ω)
��
εB // B
ω
��
G(B)
δB //
δB
��
GG(B)
εG(B)
$$
G(Q)
G(h)
// GG(B)
G εB
// G(B),
(2.1)
all inner diagrams are commutative. This shows that h̃ := εB · h : Q → B
is a G-comodule morphism with
ω · h̃ = ω · εB · h = εG(B) · G(ω) · h
= εG(B) · δB · h = γB · h = h,
εB · ω · h̃ = εB · γB · h = εB · h = h̃,
R. Wisbauer 297
thus h̃ ∈ Kγ . Moreover, for any q : Q → B in Kγ with ω · q = h, we have
εB · h = εB · ω · q = q, showing uniqueness of q.
Replacing (Q, h) in diagram (2.1) by (B, ω), we see that εB · ω is a
comodule morphism and this leads to the following observation.
2.5. Proposition. If (G, δ, ε) is a (proper) comonad, then any non-
counital G-comodule (B, ω) is cofirm if and only if it is counital.
Proof. Since we have a comonad, γ = IG, every G-comodule (B, ω) is
γ-compatible, and ω = ω · εB · ω (see 2.3).
If (B, ω) is cofirm, then ω is monomorph in B
−→
LR; since εB · ω and IB
are morphisms in B
−→
G we conclude εB · ω = IB , that is, (B, ω) is counital.
It is folklore that any counital G-comodule is cofirm.
2.6. K-firm modules. Let (F, m) be a non-unital monad on B. Given
an ideal class of morphisms in the category B
−→
F of non-unital F -modules
(see [8]), a module (B, ̺) is called K-firm provided the defining fork
FF (B)
mB //
F (̺)
// F (B)
̺ // B
is a K-coequaliser (Definitions 2.1).
2.7. Remark. Following [2, 2.3], a non-unital F -module (B, ̺) is called
firm provided it is K-firm for the class K of all morphisms in B
−→
F and ̺ is
an epimorphism in B. The term firm was coined by Quillen for non-unital
algebras A over a commutative ring k with the property that the map
A ⊗A A → A, a ⊗ b 7→ ab, is an isomorphism. Then, an A-module is firm
provided it is firm for the monad A ⊗k − on the category of k-modules.
In the category of non-unital A-modules, coequalisers are induced by
coequalisers of k-modules and hence are epimorph (in fact surjective) as
k-module morphisms (e.g. [2, 6.1]).
2.8. Compatible module morphisms. Let (F, m, η) be a weak monad
with idempotent monad morphism ϑ := m · ηF : F → F . A morphism h
between F -modules (B, ̺) and (B′, ̺′) is called ϑ-compatible, provided it
induces commutativity of the triangles in the diagram
B
h
��
ηB //
h
))
F (B)
̺ // B
h
��
B′
ηB′
// F (B′)
̺′
// B′.
298 Weak Frobenius monads
Similar to 2.3 one obtains:
(1) The class Kϑ of all ϑ-compatible morphisms in BF is an ideal class.
(2) A morphism h : Q → F (B) of F -modules is in Kϑ if and only if
ϑB · h = h.
(3) A morphism h : LR(B) → Q of F -modules is in Kϑ if and only if
h · ϑB = h.
Clearly, an F -module (B, ̺) is compatible (see 1.2) if and only if
̺ ∈ Kϑ, that is, ̺ · ϑB = ̺.
2.9. Remark. Given the assumptions in 2.8, one may consider the sub-
category of BF consisting of the same objects and as morphisms the
ϑ-compatible morphisms. Then the identity morphism on a ϑ-compatible
module (B, ̺) is ̺ · ηB : B → B and equalisers in this category are essen-
tially the Kϑ-equalisers. This situation is also addressed in [3, Remark
2.5] (with different terminology).
Dual to the Propositions 2.4 and 2.5 we now have:
2.10. Proposition. If (F, m, η) is a weak monad, then any ϑ-compatible
F -module (B, ̺) is Kϑ-firm.
2.11. Proposition. If (F, m, η) is a (proper) monad, then a non-unital
F -module (B, ̺) is firm if and only if it is unital.
3. Frobenius property and Frobenius bimodules
In the setting of 1.3, assume α and β̃ to be given, that is, there are
natural transformations η : IA → RL and ε̃ : RL → IA. Then (LR, LηR)
is a non-counital comonad, and (LR, Lε̃R) is a non-unital monad on B (see
[8]). This section is for studying the interplay between the corresponding
module and comodule structures.
Let B
−→
LR
LR denote the category of objects in B which have an LR-
module as well as an LR-comodule structure (LR-bimodules) and with
morphisms which are LR-module and LR-comodule morphisms.
By naturality, we have the commutative diagram (Frobenius property)
LRLRLR
LRLε̃R
&&
LRLR
LηRLR
88
LRLηR &&
Lε̃R // LR
LηR // LRLR
LRLRLR
Lε̃RLR
88
.
(3.1)
R. Wisbauer 299
We are interested in LR-modules and LR-comodules subject to a
reasonable compatibility condition.
3.1. Frobenius bimodules. A triple (B, ̺, ω) with an object B ∈ B and
two morphisms ̺ : LR(B) → B and ω : B → LR(B) is called a Frobenius
bimodule provided the data induce commutativity of the diagram
LRLR(B)
LR(̺)//
Lε̃R
��
(I)
LR(B)
LR(ω) //
̺
��
(II)
LRLR(B)
Lε̃R
��
LR(B)
̺ //
LηR
��
(III)
B
ω //
ω
��
(IV)
LR(B)
LηR
��
LRLR(B)
LR(̺)
// LR(B)
LR(ω)
// LRLR(B).
This implies that ̺ : LR(B) → B defines a (non-unital) LR-module and
ω : B → LR(B) a (non-counital) LR-comodule; if that is already known,
the conditions on Frobenius bimodules reduce to commutativity of the
diagrams (II) and (III), that is commutativity of (Frobenius property for
modules)
LRLR(B)
LR(̺)
&&
LR(B)
̺ //
LR(ω) &&
LηR
88
B
ω // LR(B)
LRLR(B)
Lε̃R
88
.
(3.2)
Denote by B
LR
LR the category with the Frobenius LR-bimodules as
objects and morphisms which are LR-module as well as LR-comodule
morphisms.
By the commutative diagram (3.1), for any B ∈ B, LR(B) is a Frobe-
nius bimodule with the canonical structures, that is, there is a functor
KLR
LR : B → B
LR
LR, B 7→ (LR(B), Lε̃R(B), LηR(B)).
3.2. Natural mappings. Assume again η : IA → RL and ε̃ : RL → IA
to be given (see 1.3). Then there are maps, natural in A, A′
∈ A,
ΦA,A′ : MorB(L(A), L(A′)) → MorA(A, A′), g 7→ ε̃A′ · R(g) · ηA,
LA,A′ : MorA(A, A′) → MorB(L(A), L(A′)), f 7→ L(f),
ΦA,A′ · LA,A′ : MorA(A, A′) → MorA(A, A′), f 7→ f · ε̃A · ηA = ε̃A′ · ηA′ · f.
300 Weak Frobenius monads
• If ε̃ · η = IA, then Φ · L−,− is the identity (L is separable).
• If η · ε̃ · η = η, then Φ · L−,− · Φ = Φ (Φ · L−,− is idempotent).
The natural transformation
θ : LR
LηR
−−−→ LRLR
Lε̃R
−−→ LR
is an LR-module as well as an LR-comodule morphism. From dia-
gram (3.1) one immediately obtains the equalities
LηR · θ = LR θ · LηR = θ LR · LηR,
θ · Lε̃R = Lε̃R · θ LR = Lε̃L · LR θ.
Similar relations are obtained for Frobenius bimodules.
3.3. Proposition. Given η : IA → RL and ε̃ : RL → IA, let (B, ̺, ω) be
a Frobenius LR-bimodule (see 3.1). Then
̺ · ω · ̺ = ̺ · θB and ω · ̺ · ω = θB · ω.
(1) If ε̃ · η = IA, then ̺ · ω · ̺ = ̺ and ω · ̺ · ω = ω.
Then, if ̺ is an epimorphism in B
−→
LR or ω is a monomorphism in
B
−→
LR, one gets ̺ · ω = IB.
(2) If η · ε̃ · η = η or ε̃ · η · ε̃ = ε̃, then ω · ̺ is an idempotent morphism.
Proof. The equalities claimed and (1) can be derived from the commuta-
tive diagram
LRLR(B)
LR(̺)
&&
Lε̃R(B)// LR(B)
̺
""
B
ω //
ω
""
LR(B)
̺ //
LR(ω)
&&
LηR(B)
88
B
ω // LR(B) ̺
// B
LR(B)
LηR(B)
// LRLR(B)
Lε̃R(B)
88
.
(2) To show this, extend the above diagram by ω on the right or by ̺
on the left, respectively.
3.4. Compatible bimodule morphisms. Assume η : IA → RL and ε̃ :
RL → IA to be given. A morphism h between Frobenius modules (B, ̺, ω)
R. Wisbauer 301
and (B′, ̺′, ω′) is called θ-compatible, provided it induces commutativity
of the diagram
B
h
��
ω //
h
))
LR(B)
̺ // B
h
��
B′
ω′
// LR(B′)
̺′
// B′.
One easily obtains the following.
(1) The class Kθ of all θ-compatible bimodule morphisms in B
−→
LR
LR is an
ideal class.
(2) A morphism h : Q → LR(B) of LR-bimodules is in Kθ if and only
if θB · h = h.
(3) A morphism h : LR(B) → Q of LR-bimodules is in Kθ if and only
if h · θB = h.
(4) If ε̃ · η · ε̃ = ε̃, then Lε̃R = θ · Lε̃R, that is, Lε̃R is θ-compatible.
(5) If η · ε̃ · η = η, then LηR = LηR · θ, that is, LηR is θ-compatible.
(6) For a Frobenius bimodule (B, ω, ̺), ω is θ-compatible if and only if
ω = ω · ̺ · ω, and ̺ is θ-compatible if and only if ̺ = ̺ · ω · ̺.
The next result shows how (co)firm (co)modules enter the picture.
3.5. Proposition. Let η : IA → RL and ε̃ : RL → IA be given and
consider a Frobenius LR-bimodule (B, ̺, ω).
(1) If ω is θ-compatible, then (B, ω) is Kθ-cofirm;
if ̺ is θ-compatible, then (B, ̺) is Kθ-firm.
(2) If ε̃ · η · ε̃ = ε̃, then (LR(B), Lε̃R(B)) is a Kθ-firm module;
if η · ε̃ · η = η, (LR(B), LηR(B)) is a Kθ-cofirm comodule.
Proof. (compare Proposition 2.4) (1) For a non-counital LR-comodule
(Q, κ), let h : Q → LR(B) be a comodule morphism with LηR · h =
LR(ω) · h and h = θ · h. For h̃ := ̺ · h we get
ω · h̃ = ω · ̺ · h = Lε̃R · LR(ω) · h = Lε̃R · LηR · h = h.
For any θ-compatilbe comodule morphism q : Q → B with ω · q = h, we
have g = ̺ · ω · q = ̺ · h = h̃, showing uniqueness of h̃.
The second claim is shown similarly.
(2) In view of 3.4, (4) and (5), the assertions follow from (1).
302 Weak Frobenius monads
3.6. Proposition. Assume η : IA → RL and ε̃ : RL → IA to be given.
Let K be an ideal class of LR-comodule morphisms and suppose Lε̃R(B)
in K for any B ∈ B.
(1) If (B, ω) in B
−→
LR is a K-cofirm comodule (see 2.2), there is a unique
̺ : LR(B) → B in K making (B, ̺, ω) a Frobenius bimodule.
(2) With this module structure, LR-comodule morphisms between K-
cofirm LR-comodules (B, ω) and (B′, ω′) are morphisms of the Frobe-
nius bimodules (B, ω, ̺) and (B′, ω′, ̺′).
Proof. (1) Consider the diagram (see 3.1)
LRLR(B)
Lε̃R
��
LR(̺) //
(I)
LR(B)
̺
��
LR(ω) //
(II)
LRLR(B)
Lε̃R
��
LR(B)
̺ //
LηR
��
(III)
B
ω //
ω
��
(IV)
LR(B)
LηR
��
LRLR(B)
LR(̺)
// LR(B)
LR(ω)
// LRLR(B),
where (IV) is assumed to be a K-equaliser. Since
LηR(B) · Lε̃R(B) · LR(ω) = Lε̃RLR(B) · LRLηR(B) · LR(ω)
= Lε̃RLR(B) · LRLR(ω) · LR(ω)
= LR(ω) · Lε̃R(B) · LR(ω),
and Lε̃RB · LR(ω) is in K, there exists a unique ̺ : LR(B) → B in K
leading to the commutative diagram (II), and (III) commutes since ̺ is
required to be a comodule morphism. Moreover,
ω · ̺ · Lε̃R(B) = LR(̺) · LηR(B) · Lε̃R(B)
= LR(̺) · Lε̃RLR(B) · LRLηR(B)
= Lε̃R(B) · LRLR(̺) · LRLηR(B)
= Lε̃R(B) · LR(ω) · LR(̺) = ω · ̺ · LR(̺),
and hence ̺ · Lε̃R(B) = ̺ · LR̺ since ω is a K-equaliser. This means that
the diagram (I) is also commutative.
R. Wisbauer 303
(2) Now let h : B → B′ be an LR-comodule morphism. Then
ω′
· h · ̺ = LR(h) · ω · ̺
= LR(h) · Lε̃R(B) · LR(ω)
= Lε̃R(B′) · LRLR(h) · LR(ω)
= Lε̃R(B′) · LR(ω′) · LR(h) = ω′
· ̺′
· LR(h)
and, since both h · ̺ and ̺′
· LR(h) are in K, this implies that they are
equal (see Definition 2.1), that is, h is also an LR-module morphism.
Symmetric to Proposition 3.6 we get:
3.7. Proposition. Assume η : IA → RL and ε̃ : RL → IA to be given.
Let K
′ be an ideal class of LR-module morphisms and suppose LηR(B)
belongs to K
′ for any B ∈ B.
(1) If (B, ̺) in B
−→
LR is a K
′-firm module (see 2.6), there is a unique
ω : B → LR(B) in K
′ making (B, ̺, ω) a Frobenius bimodule.
(2) With this comodule structure, LR-module morphisms between K
′-
firm LR-modules (B, ̺) and (B′, ̺′) are morphisms of the Frobenius
bimodules (B, ω, ̺) and (B′, ω′, ̺′).
So far we have only considered the case when α and β̃ (in 1.3) exist.
Now we want to include more mappings in our assumptions.
3.8. Lemma. Refer to the notation in 1.3 and 3.2.
(1) Let (L, R, α, β) be any pairing and ε̃ : RL → IA a natural transfor-
mation satisfying η · ε̃ · η = η. Then ℓr · θ = ℓr.
(2) Let (R, L, α̃, β̃) be any pairing and η : IA → RL a natural transfor-
mation satisfying ε̃ · η · ε̃ = ε̃. Then θ · ℓ̃r̃ = ℓ̃r̃ .
Proof. The assertions follow immediately from the definitions.
3.9. Theorem. Let (L, R, α, β) be a regular pairing with β symmetric
and ε̃ : RL → IA any natural transformation. Then,
(1) ε̂ := ε̃ · rℓ : RL → IA is a natural transformation with ε̂ = ε̂ · rℓ.
Furthermore, ℓr · Lε̂R = Lε̂R, that is, Lε̂R is ℓr-compatible as an
LR-comodule morphism;
(2) (LR, LηR, ε) is a weak comonad and if ω : B → LR(B) is an ℓr-
compatible LR-comodule, there is a unique ̺ : LR(B) → B in Kℓr
making (B, ̺, ω) a Frobenius (LR, η, ε̂)-module, given by
̺ : LR(B)
LR(ω)
−−−−→ LRLR(B)
Lε̂R(B)
−−−−→ LR(B)
εB
−→ B;
304 Weak Frobenius monads
(3) morphisms between ℓr-compatible LR-comodules (B, ω), (B′, ω′) are
LR-bimodule morphisms between (B, ̺, ω) and (B′, ̺′, ω′).
Proof. (1) By our symmetry assumption, ℓR = Lr and the diagram
LRL
Lε̂R
��
LrℓR
ℓRLr
//
Lε̂R ((
LRLR
Lε̂R
��
LR
ℓr
// LR
commutes, showing Lε̂R = ℓr · Lε̂R.
(2) As shown in Proposition 2.4, (B, ω) is Kℓr -cofirm and hence the
existence of ̺ follows by Proposition 3.6. For the Frobenius module
(B, ̺, ω), we have the commutative diagram
LRLR(B)
LR(̺)
&&
εLR // LR(B)
̺
""
LR(B) ̺
//
LηR
88
LR(ω) &&
B ω
// LR(B) εB
// B
LRLR(B)
Lε̂R
88
.
Since ̺ is ℓR-compatible, the upper paths yields ̺ · ℓR = ̺. The lower
path is the composite given for ̺.
(3) Since Kℓr is an ideal class, the assertion about the bimodule
morphisms follows by Proposition 3.6.
Instead of (L, R, α, β), we may require (R, L, α̃, β̃) to be a regular
pairing (see 1.3) and relate the bimodules for (LR, η, ε̃) with modules for
(LR, Lε̃R). By symmetry we obtain:
3.10. Theorem. Let (R, L, α̃, β̃) be a regular pairing of functors with α̃
symmetric and η : IA → RL any natural transformation. Then,
(1) η̂ := r̃ ℓ̃ ·η : IA → RL is a natural transformation with η̂ = r̃ ℓ̃ · η̂ and
Lη̂R = r̃ ℓ̃ · Lη̂R, that is, Lη̂R is r̃ ℓ̃-compatible as an LR-module
morphism (see 2.8);
(2) (LR, Lε̃R, η̃) is a weak monad and if ̺ : LR(B) → B is an ℓ̃r̃ -
compatible LR-module, there is a unique ω : B → LR(B) in K
ℓ̃̃r
making (B, ̺, ω) a Frobenius (LR, η̂, ε̃)-bimodule given by
ω : B
η̃B
−−→ LR(B)
Lη̂R(B)
−−−−→ LRLR(B)
LR(̺)
−−−−→ LR(B);
R. Wisbauer 305
(3) morphisms between ℓ̃r̃ -compatible LR-modules (B, ̺), (B′, ̺′) are
(LR, η̂, ε̃)-bimodule morphism between (B, ̺, ω) and (B′, ̺′, ω′).
4. Weak Frobenius monads
As we have seen in the previous section, for results on the inter-
play between (co)module and bimodule structures for Frobenius monads
symmetry conditions on our pairings were needed, that is, the intrinsic
non-(co)unital (co)monads became weak (co)monads. Hence we will con-
centrate in this section on this kind of (co)monads and also apply results
from Section 2.
4.1. Frobenius property. Let (F, m) be a non-unital monad, (F, δ)
a non-counital comonad, (B, ̺) ∈ B
−→
F and (B, ω) ∈ B
−→
F . We say that
(F, m, δ) satisfies the Frobenius property and (B, ̺, ω) is a Frobenius
bimodule, provided they induce commutativity of the respective diagrams,
FFF
F m
$$
FF
δF
;;
F δ ##
m // F
δ // FF ,
FFF
mF
::
FF (B)
F (̺)
$$
F (B)
δB
::
F (ω) $$
̺ // B
ω // F (B).
FF (B)
mB
::
The Frobenius bimodules as objects and the morphisms, which are
F -module as well as F -comodule morphisms, form a category which we
denote by B
−→
F
F . Transferring the Propositions 3.6 and 3.7 yields:
4.2. Theorem. Assume (F, m, δ) to satisfy the Frobenius property. Let
(F, δ, ε) be a weak comonad, γ := εF · δ, and assume m = γ · m. Then,
(1) for any γ-compatible F -comodule (B, ω), there is a unique γ-com-
patible F -comodule morphism
̺ : F (B)
F (ω)
−−−→ FF (B)
mB
−−→ F (B)
εB
−→ B
making (B, ̺, ω) a Frobenius bimodule;
(2) any F -comodule morphism between γ-compatible comodules (B, ω)
and (B′, ω′) becomes a morphism between the Frobenius bimodules
(B, ̺, ω) and (B′, ̺′, ω′);
306 Weak Frobenius monads
(3) there is an isomorphism of categories
Ψ : B
F
→ B
F
F , (B, ω) 7→ (B, ̺, ω),
with the forgetful functor UF : B
F
F → B
F
as inverse, where B
F
F
denotes the category of Frobenius bimodules which are γ-compatible
as F -comodules.
Proof. By our compatibility condition on m, we can apply Proposition 3.9
and the formula for ̺ given there. The assertions about the functors follow
directly from the constructions.
4.3. Theorem. Assume (F, m, δ) to satisfy the Frobenius property. Let
(F, m, η) be a weak monad, ϑ := m · Fη, and assume δ = δ · ϑ. Then,
(1) for a ϑ-compatible F -module (B, ̺), there is a unique ϑ-compatible
module morphism
ω : B
ηB
−−→ F (B)
δB
−→ FF (B)
F (̺)
−−−→ F (B)
making (B, ̺, ω) a Frobenius bimodule;
(2) any F -morphism between ϑ-compatible modules (B, ̺), (B′, ̺′) be-
comes a morphism between the Frobenius bimodules (B, ̺, ω) and
(B′, ̺′, ω′);
(3) there is an isomorphism of categories
Φ : BF → B
F
F , (B, ̺) 7→ (B, ̺, ω),
with the forgetful functor UF : B
F
F → BF as inverse, where B
F
F
denotes the category of Frobenius modules which are ϑ-compatible
as F -modules.
Proof. By Proposition 3.10 and the formula for ω given there.
4.4. Definition. We call (F, m, η; δ, ε) a weak Frobenius monad provided
(F, m, η) is a weak monad, (F, δ, ε) is a weak comonad, (F, m, δ) has the
Frobenius property (see (4.1)), and m · Fη = Fε · δ (i.e. ϑ = γ).
As a first property we observe:
4.5. Proposition. Let (F, m, η; δ, ε) be a weak Frobenius monad and
assume the idempotent m ·Fη = Fε ·δ to be split by F → F → F . Then F
has a canonical monad and comonad structure (F , m, η; δ, ε) which makes
it a Frobenius monad.
R. Wisbauer 307
Proof. The monad and comonad structures on F are obtained from 1.1
and 1.2 and a routine diagram chase shows that the Frobenius property
(see 4.1) is satisfied.
Summarising we obtain our main result for these structures.
4.6. Theorem. Let (F, m, η; δ, ε) be a weak Frobenius monad. Then the
constructions in 4.2 and 4.3 yield category isomorphisms
B
F Ψ // B
F
F
UF
// BF , BF
Φ // B
F
F
UF // B
F
.
where B
F
F denotes the category of those Frobenius F -bimodules which are
(γ-)compatible as F -comodules and (ϑ-)compatible as F -modules.
Proof. For a weak monad (F, m, η), m is ϑ-compatible and hence γ-
compatible by our assumption γ = ϑ. Similarly, δ is ϑ compatible and
hence the conditions in the preceding propositions are satisfied.
For (proper) monads and comonads the assertions simplify. For Propo-
sition 4.2 this situation is considered in [2, Section 4] and our results for
this case correspond essentially to [2, Lemma 2, Corollary 1].
4.7. Corollary. Let (F, m, δ) satisfy the Frobenius property and assume
(F, δ, ε) to be a comonad.
(1) For any counital F -comodule ω : B → F (B), there is some F -
module morphism ̺ : F (B) → B making (B, ̺, ω) a Frobenius
bimodule.
(2) If (F, m) allows for a unit, then (B, ̺) is a unital F -module.
(3) If m · δ = IF , then, for any Frobenius bimodule (B, ̺, ω), (B, ̺) is
a firm F -module.
Proof. (1), (2) hold by Theorem 4.2; (3) follows from Theorem 3.5.
4.8. Corollary. Let (F, m, δ) satisfy the Frobenius property and assume
(F, m, η) to be a monad.
(1) For any unital F -module ̺ : F (B) → B, there is some F -comodule
morphism ω : B → F (B) (given in 3.10) making (B, ̺, ω) a Frobe-
nius bimodule.
(2) If (F, δ) allows for a counit, then (B, ω) is a counital F -comodule.
(3) If m · δ = IF , then, for any Frobenius bimodule (B, ̺, ω), (B, ω) is
a firm F -comodule.
308 Weak Frobenius monads
For proper monads and comonads F , all non-unital F -modules are
compatible and all non-counital F -comodules are compatible, that is,
B
−→
F = BF and B
−→
F = B
F
. Thus we have:
4.9. Corollary. Let (F, m, η; δ, ε) be a Frobenius monad. There are cate-
gory isomorphisms
Ψ : B
−→
F
→ B
−→
F
F , Φ : B
−→
F → B
−→
F
F ,
where B
−→
F
F denotes the category of non-unital and non-counital Frobenius
F -bimodules, and
Ψ′ : BF
→ B
F
F , Φ′ : BF → B
F
F ,
where B
F
F is the category of unital and counital Frobenius F -bimodules.
It is easy to see that (by (co)restriction) these isomorphisms induce
isomorphisms between the category of unital F -modules, counital F -
comodules, and of unital and counital Frobenius bimodules, an observation
following from Eilenberg-Moore [4], which may be considered as the
starting point for the categorical treatment of Frobenius algebras.
References
[1] Böhm, G., The weak theory of monads, Adv. Math. 225(1), 1-32 (2010).
[2] Böhm, G. and Gómez-Torrecillas, J., Firm Frobenius monads and firm Frobenius
algebras, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 56(104), 281-298 (2013).
[3] Böhm, G., Lack, S. and Street, R., Idempotent splittings, colimit completion, and
weak aspects of the theory of monads, J. Pure Appl. Algebra 216, 385-403 (2012).
[4] Eilenberg, S. and Moore, J.C., Adjoint functors and triples, Ill. J. Math. 9, 381-398
(1965).
[5] Mesablishvili, B. and Wisbauer, R., QF functors and (co)monads, J. Algebra 376,
101-122 (2013).
[6] Street, R., Frobenius monads and pseudomonoids, J. Math. Phys. 45(10), 3930-3948
(2004).
[7] Wisbauer, R. On adjunction contexts and regular quasi-monads, J. Math. Sci.,
New York 186(5) (2012), 808-810; transl. from Sovrem. Mat. Prilozh. 74 (2011).
[8] Wisbauer, R., Regular pairings of functors and weak (co)monads, Algebra Discrete
Math. 15(1), 127-154 (2013).
Contact information
R. Wisbauer Department of Mathematics, HHU
40225 Düsseldorf, Germany
E-Mail(s): wisbauer@math.uni-duesseldorf.de
Web-page(s): www.math.uni-duesseldorf.de
/∼wisbauer
Received by the editors: 28.12.2015.
|
| id | nasplib_isofts_kiev_ua-123456789-155238 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T15:42:22Z |
| publishDate | 2016 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Wisbauer, R. 2019-06-16T14:26:41Z 2019-06-16T14:26:41Z 2016 Weak Frobenius monads and Frobenius bimodules / R. Wisbauer // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 2. — С. 287–308. — Бібліогр.: 8 назв. — англ. 1726-3255 2010 MSC:18A40, 18C20, 16T1. https://nasplib.isofts.kiev.ua/handle/123456789/155238 As observed by Eilenberg and Moore (1965), for a monad F with right adjoint comonad G on any category A, the category of unital F-modules AF is isomorphic to the category of counital G-comodules AG. The monad F is Frobenius provided we have F=G and then AF≃AF. Here we investigate which kind of isomorphisms can be obtained for non-unital monads and non-counital comonads. For this we observe that the mentioned isomorphism is in fact an isomorphisms between AF and the category of bimodules AFF subject to certain compatibility conditions (Frobenius bimodules). Eventually we obtain that for a weak monad (F,m,η) and a weak comonad (F,δ,ε) satisfying Fm⋅δF=δ⋅m=mF⋅Fδ and m⋅Fη=Fε⋅δ, the category of compatible F-modules is isomorphic to the category of compatible Frobenius bimodules and the category of compatible F-comodules. The author wants to thank Bachuki Mesablishvili for proofreading. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Weak Frobenius monads and Frobenius bimodules Article published earlier |
| spellingShingle | Weak Frobenius monads and Frobenius bimodules Wisbauer, R. |
| title | Weak Frobenius monads and Frobenius bimodules |
| title_full | Weak Frobenius monads and Frobenius bimodules |
| title_fullStr | Weak Frobenius monads and Frobenius bimodules |
| title_full_unstemmed | Weak Frobenius monads and Frobenius bimodules |
| title_short | Weak Frobenius monads and Frobenius bimodules |
| title_sort | weak frobenius monads and frobenius bimodules |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/155238 |
| work_keys_str_mv | AT wisbauerr weakfrobeniusmonadsandfrobeniusbimodules |