δ⊗X⊗A//A⊗A⊗X⊗A
(2) τA⊗A,X⊗A
A⊗τA,X⊗X //A⊗X⊗A⊗A
(3) τA,X−1 ⊗A⊗A
xxpppppppppppppppppppppppp
A⊗X⊗m//A⊗X⊗A
τA,X−1 ⊗A
{{xxxxxxxxxxxxxxxxxx
X⊗A⊗A
(1)
X⊗δ⊗A//X⊗A⊗A⊗A
X⊗A⊗m //X⊗A⊗A ,
in which the diagrams (1) and (2) commute by naturality ofτ, while diagram (3) commutes by naturality of composition. Since each component ofτis an isomorphism,λXis an isomorphism if and only if the composite (X⊗A⊗m)(X⊗δ⊗A) is so. Since (X⊗A⊗m)(X⊗δ⊗A) = X ⊗((A⊗m)(δ⊗A)) and since (A⊗m)(δ⊗A) is an isomorphism if and only if A has an antipode, it follows that the composite (X⊗A⊗m)(X⊗δ⊗A) - and hence λX- is an isomorphism for allX ∈Vif and only ifAhas an antipode.
6 Categories with finite products and Galois objects
In the category Set of sets, for any objectG, the productG×−defines an endofunctor. This is always a comonad with the coproduct given by the diagonal map, and it is a monad provided Gis a semigroup. In this caseG× −is a (mixed) bimonad and it is a Hopf monad if and only ifGis a group. We refer to [28, 5.19] for more details.
In this final section we study similar operations in more general categories and this leads eventually to the Galois objectsin such categories as studied in Chase and Sweedler [11].
LetAbe a category with finite products. In particular,Ahas a terminal object, which is the product over the empty set. Then (A,×,1) is a symmetric monoidal category, wherea×b is some chosen product ofaandb, and1is a chosen terminal object inA, while the symmetry τa,b:a×b→b×ais the unique morphism for which the diagram
commutes. The associativity and unit constraints are defined via the universal property for products. Such a category is called a cartesian monoidal category.
Similarly, acocartesian monoidal categoryis a monoidal category whose monoidal structure is given by the categorical coproduct and whose unit object is the initial object. Any category with finite coproducts can be considered as a cocartesian monoidal category.
Given morphisms f : a → xand g :a → y in A, we write< f, g >:a → x×y for the
It is well known that every objectcofAhas a unique (cocommutative) comonoid structure in the monoidal category (A,×,1). Indeed, the counitε:c→1is the unique morphism !c to the terminal object 1, and the comultiplication δ :c → c×c is the diagonal morphism ∆c. This yields an isomorphism of categories Comon(A)'A. Given an arbitrary object c ∈A, we writecfor the corresponding comonoid in (A,×,1).
6.1 Proposition. The assignment
(a, θa:a→a×c)7−→(p2·θa :a→c) yields an isomorphism of categories
cA'A↓c,
where cA =Ac×−, while A↓c is the comma-category of objects over c, that is, objects are morphisms f :a→cwith codomainc andmorphismsare commutative diagrams
a
If the category A has pullbacks, then for any morphism f : c → d in A, the functor f∗ : A↓c →A↓dgiven by the composition with f has the right adjoint f∗ :A↓d→ A↓c given by pulling back along the morphism f. Now, identifying f :c→dwith the morphism f : c→ dof the corresponding comonoids in A, one can see the functorsf∗ and f∗ as the induction functorcA→dAand the coinduction functordA→cA, respectively. Given an object c∈A, we writePc andUc for the functors (!c)∗ and (!c)∗.
Given a symmetric monoidal categoryV= (V,⊗,I), the category Mon(V) of monoids in Vis again a monoidal category. For two V-monoids A= (A, mA, eA) and B= (B, mB, eB), their tensor product is defined as
A⊗B= (A⊗B,(mA⊗mB)(1⊗τA,B⊗1), eA⊗eB),
whereτ is the symmetry inV. The unit object for this tensor product is the trivialV-monoid I= (I, II, II). Similarly, the category Comon(V) ofV-comonoids inherits, in a canonical way, the monoidal structure fromVmaking it a monoidal category.
It is well-known that one can describe bimonoids in any symmetric monoidal categoryVas monoids in the monoidal category of comonoids inV. Thus, writing Bimon(V) for the category of bimonoids inV, then Bimon(V) = Mon(Comon(V)).In particular, since Comon(A)'Afor any cartesian monoidal categoryA, one has Bimon(A) = Mon(Comon(A))'Mon(A). Thus, for any monoidb= (b, mb, eb) in (A,×,1), the 6-tuple
bb= ((b, mb, eb),(b,∆b,!b))
is a bimonoid in (A,×,1). In particular, then the functorb×−:A→Ais a (τb,b×−)-bimonad (in the sense of [21]).
Fix now a monoid b = (b, mb, eb) in (A,×,1). Since bb is a bimonoid in (A,×,1), the category bA:=Ab×− ofb-modules is monoidal. More precisely, if (x, αx),(y, αy)∈bA, then their tensor product is the pair (x×y, αx×y), whereαx×y is the composite
b×x×y ∆b×x×y //b×b×x×y b×τb,x×y //b×x×b×y αx×αy //x×y.
It is easy to see that this monoidal structure is cartesian and coincides with the cartesian structure onbAwhich can be lifted from Aalong the forgetful functor bA→A.
Suppose now that (c, αc : b×c → c) ∈ bA. Applying the previous proposition to the comonoid (c, αc) in the cartesian monoidal categorybAgives
6.2 Proposition. If (c, αc)∈bA, then the assignment
((x, αx), θ(x,αx))−→((x, αx), p2·θ(x,αx)) yields an isomorphism of categories
(c,αc)(bA)'bA↓(c, αc).
We have seen that the data
be= (b= (b× −, mb× −, eb× −), b= (b× −,∆b× −,!b× −), τb,b× −)
define a (τb,b× −)-bimonad onAand, considering bas an object ofbAvia the multiplication mb : b×b → b, one obtains easily that the categories Abb := Abb(τb,b × −) (compare 4.1) and (b,mb)(bA) are isomorphic. Thus, by the previous proposition, the categories Abb and
bA↓(b, mb) are also isomorphic.
6.3 Theorem. Assume that (i) A has small limits, or
(ii) A has colimits and the functorb× −preserves them, or (iii) A admits equalisers andb× −has a right adjoint, or (iv) A admits coequalisers andb× −has a left adjoint.
Then the functor
K:A→bA↓(b, mb), a7→(b×a, p1:b×a→b), is an equivalence of categories if and only if bis a group.
Proof. It is easy to see that, modulo the isomorphism Abb 'bA↓(b, mb), the functor K:A→bA↓(b, mb) can be identified with the comparison functorK:A→Abb, which by 4.5 is an equivalence of categories if and only if the bimonadbehas an antipode, which is the case if and only if theA-bimonoidbb has one, i.e.,bb is a Hopf monoid in (A,×,1). Now the result follows from the fact that in any cartesian monoidal category, a Hopf algebra is nothing but
a group (see, for example, [28, 5.20]). tu
Consider now an object (c, αc)∈bA. Since (c, αc) is a comonoid in the cartesian monoidal category (bA,×,1), the composite
b×c× −∆b×c×−//b×b×c× − b×τb,c×− //b×c×b× − αc×b×− //c×b× − is an entwining from the monad Tb =b× − to the comonadGc =c× −. Then one has a liftingTfbof the monad Tbalong the forgetful functorcA=A↓c→A.It is easy to see that if (x, f:x→c)∈A↓c, then
Tfb(x, f) = (b×x, αc·(b×f) :b×x→c).
We writeb(A↓c) for the category (A↓c)
Tfb. It is also easy to see that the functor K:A→b(A↓c)
that takes an objecta∈Ato the object
(c×a, αc×a:b×c×a→c×a), makes the diagram
b(A↓c)
U
A P
c
//
K
<<
xx xx xx xx
x A↓c
commute, whereU is the evident forgetful functor. Then the correspondingTfb-module struc-ture on Pc is given by the morphism αc × − : b×c × − → c × −. Since the forgetful functor Uc : A↓ c → A that takes f : x → c to x is left adjoint to the functor Pc and since the (f : x → c)-component of the unit of the adjunction Uc a Pc is the morphism
< f, Ix>:x→c×x, the (f :x→c)-componenttf of the monad morphismt:Tfb→PcUc is the composite
b×x b×<f,Ix> //b×c×xαc×Ix//c×x.
We writeγc for the morphism tIc :b×c→c×c.
One says that a morphism f : a → b in A is an (effective) descent morphism if the corresponding functorf∗:A↓b→A↓ais precomonadic (resp. monadic).
6.4 Theorem. Let b= (b, mb, eb)be a monoid in Aand let(c, αc)∈bA. Suppose that (i) Aadmits all small limits, or
(ii) Aadmits coequalisers of reflexive pairs and the functorsb×−:A→Aandc×−:A→A both have left adjoints.
Then the functorK:A→b(A↓c)is an equivalence of categories if and only ifγc:b×c→c×c is an isomorphism and !c:c→1is an effective descent morphism.
Proof.According to Proposition 1.15, the functorKis an equivalence of categories if and only if the functor Pc is comonadic (i. e. if the morphism !c : c →1 is an effective descent morphism) andt:Tfb→PcUcis an isomorphism of monads. Since the functorsb× −:A→A
and c× −: A→ Aboth preserve those limits that exist in A, it follows from 2.8 that ifA satisfies (i) or (ii),tis an isomorphism if and only if its restriction on freePcUc-algebras is so.
But any freePcUc-algebra has the form (c×x, p1) for somex∈Aand it is not hard to see that the (c×x, p1)-component t(c×x,p1)oft is the morphismγc×x. It follows thatt(c×x,p1)is an isomorphism for all x∈Aif and only if the morphismγc is an isomorphism. This completes
the proof. tu
We say an object a ∈ A is faithful if the functor a× − : A → A is conservative. Note that an arbitrarya∈Afor which the unique morphism !a :a→1is a descent morphism is necessarily faithful.
Following Chase and Sweedler [11] we call an object (c, αc)∈bAa Galois b-object ifcis a faithful object inAsuch that the morphismγc:b×c→c×c is an isomorphism. Using this notion, we can rephrase the previous theorem as follows.
6.5 Theorem. In the situation of the previous theorem, if(c, αc)∈bA is a Galoisb-object, then the functorK:A→b(A↓c)is an equivalence of categories if and only if!c :c→1is an effective descent morphism.
If any descent morphism in Ais effective (as surely it is when Ais an exact category in the sense of Barr, see [15]), then one has
6.6 Corollary. If every descent morphism in A is effective, then for any Galois b-object (c, αc), the functorK:A→b(A↓c)is an equivalence of categories.
Note that ifg:I→G¯c is a grouplike morphism for the comonadG¯c, then the composite 1−→g1 G¯c(1) =c×1−→p2 1is the identity morphism, implying that the morphism !c :c →1 is a split epimorphism. It is then easy to see that the counit of the adjunction Uc aPc is a split epimorphism, and it follows from the dual of [19, Proposition 3.16] that the functor Pc is monadic (i.e., !c :c→1is an effective descent morphism) provided that the categoryAis Cauchy complete. In the light of the previous theorem, we get:
6.7 Theorem. In the situation of Theorem 6.4, ifAis Cauchy complete and if there exists a grouplike morphism for the comonad G¯c, then the functor K:A→b(A↓c)is an equivalence of categories if and only if (c, αc)∈bAis a Galois b-object.
Recall from [11] that an objecta∈Ais(faithfully) coflat if the functor a× −:A→A
preserves coequalisers (resp. preserves and reflects coequalisers).
6.8 Theorem. Let Abe a category with finite products and coequalisers, and b= (b, mb, eb) a monoid in the cartesian monoidal categoryAwithbcoflat and let(c, αc)∈bAbe ab-Galois object with!c :c→1an effective descent morphism. Assume
(i) Aadmits all small limits, or
(ii) the functorsb× −:A→A andc× −:A→A both have left adjoints.
Thenc is (faithfully) coflat.
Proof. Note first that sinceA↓c'cAand since the categoryAadmits coequalisers, the category A↓calso admits coequalisers and the forgetful functorUc:A↓c→Acreates them.
Now, ifb is coflat, then the functorb× −:A→Apreserves coequalisers, and it follows from the commutativity of the diagram
A↓c
Uc
Tfb //A↓c
Uc
A T
b=b×−//A
that the functorTfbalso preserves coequalisers. As in the proof of 6.4, one can show that the morphismt:Tb→PcUc is an isomorphism of monads. Thus, in particular, the monadPcUc
preserves coequalisers. Since the morphism !c :c→1is an effective descent morphism by our assumption onc, the functorPc is monadic. Applying now the dual of [19, Proposition 3.11], one gets that the functor UcPc =c× −also preserves coequalisers. Thuscis coflat. tu
As a consequence, we have:
6.9 Theorem. LetA be a category with finite products and coequalisers in which all descent morphisms are effective. Suppose that b= (b, mb, eb)is a monoid in the cartesian monoidal categoryA withb coflat and that(c, αc)∈bAis ab-Galois object. If
(i) Aadmits all small limits, or
(ii) the functorsb× −:A→A andc× −:A→A both have left adjoints, thenc is (faithfully) coflat.
6.10. Opposite category of commutative algebras. Letk be a commutative ring (with unit) and letAbe the opposite of the category of commutative unitalk-algebras.
It is well-known thatAhas finite products and coequalisers. IfA= (A, mA, eA) andB= (B, mB, eB) are objects ofA(i.e. ifAandBare commutativek-algebras), thenA⊗kBwith the obviousk-algebra structure is the product ofAandBinA: the projectionsp1:A⊗kB→A and p2 : A⊗k B → B are given by IA⊗keB : A →A⊗kB and eA⊗k IB : B →A⊗kB, respectively. Furthermore, if f, g : A → B are morphisms in A, then the pair (C, i), where C ={b∈B|f(b) =g(b)} andi:C →B is the canonical embedding ofk-algebras, defines a coequaliser inA. The terminal object inAisk.
An objectAinA(i.e. a commutativek-algebra) is (faithfully) coflat if and only ifA is a (faithfully) flatk-module (see, [11]). Moreover, a monoid in the cartesian monoidal category Ais a commutative k-bialgebra, which is a group inAif and only if it has an antipode, and if B is a commutative k-bialgebra, then (C, αC) ∈ BA if and only if C is a commutative B-comodule algebra.
Note that in the present context, (C, αC) ∈ BA is a Galois B-object if C is a faithful k-module and the composite
γC:C⊗kC αC⊗kIC //B⊗kC⊗kC B⊗kmC //B⊗kC , wheremC:C⊗kC→C is the multiplication inC, is an isomorphism.
Since the category A admits all small limits and since in A every descent morphism is effective (see [20]), one can apply Theorem 6.9 to deduce the following
6.11 Theorem. LetBbe a commutativek-bialgebra withBa flatk-module. Then any Galois B-object in Ais a faithfully flatk-module.
Note finally that whenBis a Hopf algebra which is finitely generated and projective as a k-module, the result was obtained by Chase and Sweedler, see [11, Theorem 12.5].
Acknowledgements. The work on this paper was started during a visit of the first author at the Department of Mathematics at the Heinrich Heine University of D¨usseldorf supported by the German Research Foundation (DFG) and continued with support by Volkswagen Foun-dation (Ref.: I/84 328). The authors express their thanks to all these institutions.
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Addresses:
Razmadze Mathematical Institute, 1, M. Aleksidze st., Tbilisi 0193, and
Tbilisi Centre for Mathematical Sciences, Chavchavadze Ave. 75, 3/35, Tbilisi 0168, Republic of Georgia, bachi@rmi.acnet.ge
Department of Mathematics of HHU, 40225 D¨usseldorf, Germany,wisbauer@math.uni-duesseldorf.de