MMAPPING COALGEBRAS ICOMONADS
BRICE LE GRIGNOUAbstract. In this article we describe properties of the 2-functor from the 2-category of comonads tothe 2-category of functors that sends a comonad to its forgetful functor. This allows us to describecontexts where algebras over a monad are enriched tensored and cotensored over coalgebras overa comonad.
ContentsIntroduction 11. Monoidal context 22. Comonads and monoidal structures 163. Mapping coalgebras 23Appendix A. Category enriched tensored and cotensored over a monoidal category 27Appendix B. Monads, comonads, limits and colimits 32References 33IntroductionThis is the first of a series of articles about categories enriched over a category describing somenotion of a coalgebra. This article is devoted to coalgebras over a comonad. The next article will dealwith coalgebras over an operad and the last article will focus on coalgebras in the context of chaincomplexes.Let us start with Sweedler’s theory ([Swe69], [AJ13]). For any two differential graded coassocia-tive coalgebras (V , w V , τ V ) and (W , w W , τ W ) , the tensor product V ⊗ W inherits the structure of acoassociative coalgebra as follows V ⊗ W w V ⊗ w W −−−−→ V ⊗ V ⊗ W ⊗ W (cid:39) V ⊗ W ⊗ V ⊗ W. This gives a symmetric monoidal structure on the category of differential graded coalgebras. Moreover,from the existence of cofree coalgebras [Ane14], one can show that this monoidal structure is closed.Besides, for any differential graded associative algebra (A , m ) and any differential graded coalgebra (V , w ) , the mapping chain complex [V , A] has the canonical structure of an algebra [V , A] ⊗ [V , A] (cid:44) → [V ⊗ V , A ⊗ A] [ w,m ] −−−→ [V , A] . It is usually called the convolution algebra of V and A . From the existence of free algebras and cofreecoalgebras, one can build a left adjoint V (cid:2) − to the functor [V , − ] and a left adjoint {− , A } to thefunctor [ − , A] . These three bifunctors [ − , − ] : Coalgebras op × Algebras → Algebras − (cid:2) − : Coalgebras × Algebras → Algebras {− , −} : Algebras op × Algebras → Coalgebras make the category of algebras tensored, cotensored and enriched over the category of coalgebras.This may be reinterpreted in the language of monads and comonads since associative dg algebrasand coassociative dg coalgebras are respectively algebras over a monad M and coalgebras over a Date : September 22, 2020. a r X i v : . [ m a t h . C T ] S e p BRICE LE GRIGNOU comonad Q on chain complexes (see [Ane14] for a description of Q ). Indeed, the structure of amonoidal category on coalgebras is related to the structure of a Hopf comonad on Q , that is the dataof a natural map Q ( X ) ⊗ Q ( Y ) → Q ( X ⊗ Y ) satisfying dual conditions as those of a Hopf monad (see [Moe02]). Moreover, the lifting of themapping chain complex bifunctor X, Y (cid:55)→ [ X, Y ] to a bifunctor from Coalgebras op × Algebras to Algebras is related to the structure of a Hopf module monad on M with respect to the Hopf comonad Q , that is the data of a natural map M ([ X, Y ]) → [ Q ( X ) , M ( Y )] satisfying again some conditions. All these relations between monoidal structures on categories ofalgebras and coalgebras and structures on monads and comonads are encoded in a single 2-functorfrom the 2-category Comonads of categories with comonads and oplax morphisms to the 2-category
Functors made up of functors between two categories.
Theorem.
The construction that sends a category C with a comonad Q to the forgetful functor U Q from Q -coalgebras to C induces a 2-functor from the two category Comonads to the 2-category
Functors that is strictly fully faithful and preserves strict finite products.
This theorem shows in particular that these relations between monoidal structures on categoriesof algebras and coalgebras and structures on monads and comonads are bijections. For instance,a monoidal structure on the category of M -algebras that lifts that of chain complexes is equivalentto the structure of a Hopf monad on M (see again [Moe02]). Moreover, the existence of the twobifunctors − (cid:2) − and {− , −} described above is then just a consequence of Johnstone’s adjoint lifitingtheorem ([Joh75]). Layout.
In the first section, we describe a strict version of a monoidal symmetric monoidal categorythat we call a monoidal context. In the second section, we describe the monoidal context of comonadsand relate it to the monoidal context of functors. In the third section, we apply this relation and theadjoint lifting theorem to describe mapping coalgebras.
Acknowledgement.
I would like to thank Damien Lejay and Mathieu Anel for inspiring discussions.The idea of this work came when reading the book [AJ13] by Mathieu Anel and André Joyal. Thiswork was supported in part by the NWO grant of Ieke Moerdijk.
Notations and conventions.
Let us consider three universes U < V < W . We have hence a hierarchyof sizes of sets as we will call a set (cid:46) small if it is U -small; (cid:46) large if it is V -small; (cid:46) very large if it is W -small; (cid:46) very very large if it is W -large.While working inside a category, we assume that objects and structures are built using small sets. Onthe contrary sets of objects of a category are large. These large categories are organized into a verylarge 2-categories whose collection is very very large.Notation. For a monad M on a category E , the induced monadic adjunction relating M -algebras to E will be denoted T M (cid:97) U M . Similarly, for a comonad Q , the induced comonadic adjunction relating Q -coalgebras to E will be denoted U Q (cid:97) L M .1. Monoidal contextIn this section, we describe a semi-strict notion of a symmetric monoidal 2-category that we call amonoidal context. Then, we describe some usual algebraic notions in a monoidal context.We refer to [DS97] and [SP09] for broader notions of symmetric monoidal 2-categories. In par-ticular, the PhD thesis of Chris Schommer-Pries describes in details a "full" version of a symmetricmonoidal bicategory.1.1. Strict 2-categories.
APPING COALGEBRAS I COMONADS 3
The definition of a 2-category.
Definition 1.
A strict 2-category C is a category enriched in categories, that is the data of (cid:46) a set of objects Ob( C ) ; (cid:46) for any two objects X, Y , a category C ( X, Y ) ; (cid:46) a unital associative composition C ( Y, Z ) × C ( X, Z ) → C ( Y, Z ) whose units are elements X ∈ C ( X, X ) .A strict 2-functor F between two strict 2-categories C and D is a morphism of categories enriched incategories , that is the data of (cid:46) a function F ( − ) : Ob( C ) → Ob( D ) ; (cid:46) functors F ( X, Y ) : C ( X, Y ) → D ( F ( X ) , F ( Y )) that commute with the composition and sendunits to units.This forms the category − Cats of strict 2-categories.Remark 1. From now on, we will call strict 2-functors just 2-functors.
Definition 2.
Given a strict 2-category C and two objects X, Y ∈ Ob( C ) , (cid:46) an object of C ( X, Y ) will be called a morphism of C ; (cid:46) a morphism in the category C ( X, Y ) will be called a 2-morphism of C . Definition 3.
Let C , be a strict 2-category and X, Y, Z be three objects. (cid:46)
Given morphisms f , f : X → Y and g , g : Y → Z and 2-morphisms a : f → f and b : g → g , one can compose a and b using the functor C ( Y, Z ) × C ( X, Y ) → C ( X, Z ) to obtain a 2-morphism from g ◦ f to g ◦ f that is denoted b ◦ h a and is called the horizontalcomposition of a and b . (cid:46) Given morphisms f , f , f : X → Y and 2-morphisms a : f → f and a : f → f , one cancompose a and a as morphisms in the category C ( X, Y ) to obtain a 2-morphism from f to f , denoted a ◦ v a and called the vertical composition of a and a .Notation. Let us consider a 2-morphism a of C , that is a morphism in C ( X, Y ) for two objects X, Y .For any morphism f : Y → Z , we will usually denote the horizontal composition Id f ◦ h a as f ◦ h a. Similarly, for any morphism g : Z → X , we will denote a ◦ h Id g as a ◦ h g . Definition 4.
An isomorphism in a strict 2-category C is an isomorphism of the underlying category(that is the skeleton of C denoted sk( C )) . A 2-isomorphism is an invertible 2-morphism. Finally,an equivalence in C is a morphism f : X → Y so that there exists a morphism g : Y → X and2-isomorphisms a : Id X (cid:39) g ◦ f ; b : Id Y (cid:39) f ◦ g. Then, g is a pseudo-inverse of f . Definition 5.
An adjunction in C is the data of two objects X, Y together with morphisms l : X → Yr : Y → X and 2-morphisms η : Id X → r l and (cid:15) : lr → Id Y so that ( r ◦ h (cid:15) ) ◦ v ( η ◦ h r ) = Id R ;( (cid:15) ◦ h l ) ◦ v ( l ◦ h η ) = Id R . Definition 6.
An adjoint equivalence is an adjunction whose unit and counit are isomorphisms.
Proposition 1. If f : X → Y is an equivalence of a strict 2-category with pseudo-inverse g , then, f and g are part of an adjoint equivalence. BRICE LE GRIGNOU
Proof.
Let us consider two 2-isomorphisms η : Id Y (cid:39) f g ; ζ : gf (cid:39) Id X . We can notice that the two maps gf ◦ h ζ and ζ ◦ h gf from gf gf to gf are equal. If we define the2-isomorphism (cid:15) : gf (cid:39) Id X as the composition gf gf ◦ h ζ − = ζ − ◦ h gf −−−−−−−−−−→ gf gf g ◦ h η − ◦ h f −−−−−−→ gf ζ − −−→ Id X , then, the tuple ( f , g, η, (cid:15) ) is an adjunction. (cid:3) The strict 2-category of strict 2-categories.
Categories are organised into a 2-category. Simi-larly, 2-categories form a 3-category. But in the same way one can consider the category of categories,one can also consider a 2-category of strict 2-categories. This implies that we will not consider map-ping categories up to equivalences but up to isomorphisms.
Definition 7.
Categories form a coreflexive full subcategory of strict 2-categories. We denote sk thecorresponding idempotent comonads that sends a strict 2-category to its underlying category (that iswith the same objects and morphisms) called its skeleton. Definition 8.
Let
Mor be the category with two objects and and a non trivial morphism from to . Let Nat be the strict 2-category with two objects and and so that Nat (0 ,
1) =
Mor ; Nat (0 ,
0) =
Nat (1 ,
1) = ∗ ; Nat (1 ,
0) = ∅ . Given two strict 2-categories C , D , their product C × D is the strict 2-categories whose (cid:46) set of objects is Ob( C × D ) = Ob( C ) × Ob( D ) ; (cid:46) categories of morphisms are C × D (( X, X (cid:48) ) , ( Y, Y (cid:48) )) = C ( X, Y ) × D ( X (cid:48) , Y (cid:48) ) . Proposition 2.
The category of strict 2-categories is a cartesian closed monoidal category.Proof.
It suffices to show that for any 2-category C , the endofunctor − × C of − Cats has a rightadjoint. It is given by the functor 2-Fun ( C , − ) that sends a strict 2-category D to the 2-categorywhose (cid:46) objects are 2-functor from C → D ; (cid:46) morphisms are 2-functors from Mor × C to D ; (cid:46) Nat × C to D ; (cid:3) Remark 2. This product of strict 2-categories or even of bicategories does not have good homotopicalproperties. For instance, the product
Mor × Mor is the commutative square and not the squarecommutative up to a natural isomorphism.
Definition 9.
Given two 2-functors
F, G from C to D , a strict natural transformation from F to G isjust a morphism from F to G in the category 2-Fun ( C , D ) . Proposition 3.
A strict natural transformation A from F to G (that share the same source C andthe same target D ) is equivalent to the data of morphisms in D A ( X ) : F ( X ) → G ( X ) for any object X ∈ C so that (cid:46) for any morphism f : X → Y in C , the following square diagram commutes in D F ( X ) G ( X ) F ( Y ) G ( Y ); A ( X ) F ( f ) G ( f ) A ( Y ) APPING COALGEBRAS I COMONADS 5 (cid:46) for any 2-morphism a : f → g in C where f , g : X → Y , we have A ( Y ) ◦ h F ( a ) = G ( a ) ◦ h A ( X ) . In other words, this is a natural transformation from the functor sk( F ) to the functor sk( G ) thatsatisfies the last condition.Proof. Straightforward. (cid:3)
Definition 10.
Given two 2-functors
F, G from C to D and two strict natural transformations A and A (cid:48) from F to G , a modification from A to A (cid:48) is a 2-morphism from A to A (cid:48) in the 2-category2-Fun ( C , D ) .Remark 3. We will work in a framework strict enough to avoid modifications. Definition 11.
A 2-functor F : C → D is strictly fully faithful if for any objects X, Y ∈ Ob( C ) , thefunctor F ( X, Y ) : C ( X, Y ) → D ( F ( X ) , F ( Y )) is an isomorphism of categories. Definition 12.
A 2-functor F : C → D is strictly essentially surjective if the underlying functor sk( F ) is essentially surjective. Definition 13.
A 2-functor F : C → D is an iso-equivalence if there exists a 2-functor G : D → C and strict natural isomorphisms Id C (cid:39) G ◦ F ;Id D (cid:39) F ◦ G. Remark 4. In other words F is an iso-equivalence if it is an equivalence in the 2-category made up of2-categories, 2-functors and strict natural transformations. Proposition 4.
A 2-functor F is an iso-equivalence if and only if it is strictly fully faithful and strictlyessentially surjective.Proof. This follows from the same arguments as for categories. (cid:3)
Monoidal context.
We give here the definition of a monoidal context, which could also be calledan almost strict symmetric monoidal strict 2-category. Indeed, this is the data of a strict 2-categorytogether with a symmetric monoidal structure which is as strict as symmetric monoidal structures oncategories.Any 2-category is equivalent to a strict 2-category. But this is not true for 3-categories. Likewisewe do not expect our notion of monoidal context to encompass all symmetric monoidal 2-categories.1.2.1.
The definition of a monoidal context.
Definition 14.
A monoidal context is the data a strict 2-category C together with (cid:46) a 2-functor − ⊗ − : C × C → C ; (cid:46) a 2-functor (cid:49) : ∗ → C ; (cid:46) a strict natural isomorphism call the associator ( X ⊗ Y ) ⊗ Z (cid:39) X ⊗ ( Y ⊗ Z ) (cid:46) a strict natural isomorphism called the commutator X ⊗ Y (cid:39) Y ⊗ X ; (cid:46) two strict natural isomorphisms called the unitors (cid:49) ⊗ X (cid:39) X (cid:39) X ⊗ (cid:49) that makes the skeleton sk( C ) a symmetric monoidal category (for instance they satisfy the pentagonidentity and the triangle identity).We now define morphisms between monoidal contexts that we call context functors. This is analmost strict version of symmetric monoidal 2-functor. Definition 15.
A context functor between two monoidal contexts C , D is the data of BRICE LE GRIGNOU (cid:46) a 2-functor F : C → D ; (cid:46) a strict natural transformation F ( X ) ⊗ F ( Y ) → F ( X ⊗ Y ) ; (cid:46) a strict natural transformation (cid:49) → F ( (cid:49) ) ;that makes the functor between skeletons sk( F ) a symmetric monoidal functor (in particular thestructural strict natural transformations are isomorphisms). Definition 16.
Given two context functors
F, G : C → D , a monoidal natural transformation betweenthem is a strict natural transformation A : F → G so that the induced natural transformation from sk( F ) to sk( G ) is monoidal. In other words, A commutes with the structural natural transformationsof the two context functors.1.2.2. Multiple tensor.
Let us consider a monoidal context C and a sequence of objects X . . . , X n ∈ Ob( C ) . One can define the n -tensor X ⊗ · · · ⊗ X n as follows. (cid:46) if n = 0 , this is just (cid:49) ; (cid:46) if n = 2 , this is just X ⊗ X ; (cid:46) if n > , there are many ways to introduce parenthesis in the expression X ⊗ · · · ⊗ X n sothat it only uses tensors of two elements (actually, this ways are canonically in bijection withplanar binary trees with n-leaves). All of these ways yields objects in C which are canonicallyisomorphic to each other. We obtain thus a diagram in sk( C ) which is equivalent to the oneobject diagram. Then, we define X ⊗ · · · ⊗ X n as the colimit of this diagram.Moreover, if we have a decomposition n = i + · · · + i k we have a canonical isomorphism X ⊗ · · · ⊗ X n (cid:39) ( X ⊗ · · · ⊗ X i ) ⊗ · · · ⊗ ( X n − i k +1 ⊗ · · · ⊗ X n ) . We will usually identify these two objects in the notations.1.2.3.
Cartesian monoidal structures.
Definition 17.
Let C be a strict 2-category and X, Y be two objects. The strict product of X and Y if it exists is an object X, × Y defined up to a unique isomorphism equipped with two morphisms X × Y → X ; X × Y → Y ; so that for any object Z the functor C ( Z, X × Y ) → C ( Z, X ) × C ( Z, Y ) is an isomorphism of categories. Definition 18.
A strict final element in a strict 2-category C is an object ∗ defined up to a uniqueisomorphism so that for any object Z the functor C ( Z, ∗ ) → ∗ is an isomorphism of categories. Definition 19.
A strict 2-category is said to have strict finite products if it has strict products of anypair of objects and if it as a strict final element.
Proposition 5.
Suppose that the strict 2-category C has strict finite products. Then C gets from theproduct and the final element the structure of a monoidal context.Proof. The associator is the canonical natural transformation ( X × Y ) × Z (cid:39) X × ( Y × Z ) and the unitors are the canonical natural transformations X × ∗ (cid:39) X (cid:39) ∗ × X. (cid:3) Definition 20.
A cartesian monoidal context is a strict 2-category that has strict finite products andthat is equipped with the induced structure of a monoidal context.
APPING COALGEBRAS I COMONADS 7
Definition 21.
Let C , D be two strict 2-categories with strict finite products. A 2-functor F : C → D preserve strict products if the canonical morphisms F ( X × Y ) → F ( X ) × F ( Y ); F ( ∗ ) → ∗ ; are isomorphisms. Proposition 6.
Let C , D be two cartesian monoidal contexts. A 2-functor F : C → D that preservesstrict finite products has the canonical structure of a context functor.Proof. This structure is given by the strict natural transformation F ( X × Y ) (cid:39) F ( X ) × F ( Y ) . (cid:3) Categorical operads acting on a monoidal context.
To describe algebraic structures in amonoidal context C , one general way is to consider actions of another monoidal context on C , thatis context functors from some monoidal context A to C . We choose here to deal with the restrictedframework of action of operads enriched in categories that we call categorical operads.1.3.1. Operads.
Definition 22.
A categorical planar operad P is a planar coloured operad enriched in categories, thatis the data of a set of objects (also called colours) Ob( P ) , together with categories P ( c , . . . , c n ; c ) for any n ≥ and objects c , . . . , c n , c ∈ Ob( P ) and together with compositions P ( c ; c ) × P ( d ; c i ) (cid:47) i −→ P ( c (cid:47) i d ; c )1 c ∈ P ( c ; c ) that satisfy associativity and unitality conditions and where c = ( c , . . . , c n ); d = ( d , . . . , d m ); c (cid:47) i d = ( c , . . . , c i − , d , . . . , d m , c i +1 , . . . , c n ) . Definition 23.
A categorical operad P is a coloured operad enriched in categories, that is the dataof categorical planar operad together with isomorphisms σ ∗ : P ( c ; c ) → P ( c σ ; c ) for any permutation σ ∈ Σ n , where c = ( c , . . . , c n ); c σ = ( c σ (1) , . . . , c σ ( n ) ); so that ( µ ◦ σ ) ∗ = σ ∗ ◦ µ ∗ and that satisfy coherence conditions with respect to the operadic compo-sition. Definition 24.
A morphism of categorical (planar) operads from P to Q is the data of a function f ( − ) : Ob( P ) → Ob( Q ) together with functors f ( c , . . . , c n ; c ) : P ( c , . . . , c n ; c ) → Q ( c , . . . , c n ; c ) that commute strictly with the operadic compositions, units and actions of the symmetric groups (inthe non planar case). This defines the category of categorical planar operads and the category ofcategorical operads. We also have a forgetful functor from categorical operads to categorical planaroperads.Remark 5. As for strict 2-categories, categorical operads are actually organised into a 3-category. BRICE LE GRIGNOU
Algebras.
Definition 25.
Let us denote
End the functor from the category of monoidal contexts to the categoryof (very large) categorical operads that sends a monoidal context C to the operad End( C ) so that Ob(End( C )) = Ob( C ) and End( C )( X , . . . , x n ; x ) = C ( X ⊗ · · · ⊗ X n , X ) . Definition 26.
Given a categorical (planar) operad P and a monoidal context C , a P -algebra is thedata of a morphism of categorical (planar) operads from P to End( C ) .More concretely, such an algebra is the data of an object A c ∈ C for any colour c ∈ Ob( P ) togetherwith functors P ( c , · · · c n ; c ) → C ( A c ⊗ · · · ⊗ A c n , A c ) p (cid:55)→ A ( p ) that commute with the units, compositions and actions of the symmetric groups (for the non planarcase). Definition 27.
Let P be a categorical (planar) operad and let C be a monoidal context. Let A and B be two P -algebras in C . A lax P -morphism from A to B is the data of morphisms in C f c : A c → B c for any colour c ∈ Ob( P ) , and 2-morphisms in C B ( p ) ◦ f ⊗ n → f ◦ A ( p ) for any element p ∈ P ( c , . . . , c n ; c ) , where f ⊗ n and f actually stands for f c ⊗ · · · ⊗ f c n (or Id (cid:49) if n = 0 ) and f c . We also require that (cid:46) these 2-morphisms are trivial on units c ∈ P ( c ; c ) ; (cid:46) they are natural with respect to p in the sense that the following diagram commutes B ( p ) ◦ f ⊗ n f ◦ A ( p ) B ( p (cid:48) ) ◦ f ⊗ n f ◦ A ( p (cid:48) ) for any morphism p → p (cid:48) in P ( c , . . . , c n ; c ) (cid:46) they behave in a coherent way with respect to composition in the sense that the followingdiagram commutes B ( p ) ◦ (Id ⊗ i − ⊗ B ( p (cid:48) ) ⊗ Id ⊗ n − i ) ◦ f n + m − B ( p ) ◦ f ⊗ n ◦ (Id ⊗ i − ⊗ A ( p (cid:48) ) ⊗ Id ⊗ n − i ) f ◦ A ( p ) ◦ (Id ⊗ i − ⊗ A ( p (cid:48) ) ⊗ Id ⊗ n − i ) B ( p (cid:47) i p (cid:48) ) ◦ f ⊗ n + m − f ◦ A ( p (cid:47) i p (cid:48) ); (cid:46) in the non planar context, they behave in a coherent way with respect to the actions of symmetricgroups in the sense that the following diagram commutes B ( p σ ) ◦ f ⊗ n f ◦ A ( p σ ) B ( p ) ◦ f ⊗ n ◦ σ f ◦ A ( p ) ◦ σ. APPING COALGEBRAS I COMONADS 9
Definition 28.
Using the same notation as in Definition 27, an oplax P -morphism from A to B is thedata of morphisms in C f c : A c → B c for any colour c ∈ Ob( P ) , and 2-morphisms f ◦ A ( p ) → B ( p ) ◦ f ⊗ n for any element p ∈ P ( c , · · · , c n ; c ) . We require that these data satisfy mutatis mutandis the sameconditions as in the definition of lax morphisms. Definition 29. A P -morphism (resp. strict P -morphism) is a lax P -morphism f so that the structural2-morphisms are 2-isomorphisms (resp. identities). In particular, it is also an oplax P -morphism. Definition 30.
Given a two lax P -morphisms f , g : A → B , a P f to g is the dataof 2-morphisms in C a c : f c → g c for any colour c ∈ Ob( P ) so that the following diagram commutes B ( p ) ◦ f ⊗ n f ◦ A ( p ) B ( p ) ◦ g ⊗ n g ◦ A ( p ) for any element p of P ( c , . . . , c n ; c ) . One can define similarly a P P -morphisms. Proposition 7.
Let C be a monoidal context. The data of P -algebras in C , lax P -morphisms and P The composition of two lax P -morphisms f : A → A (cid:48) and g : A (cid:48) → A (cid:48)(cid:48) is given by thecomposition of morphisms in C ( g ◦ f ) c = g c ◦ f c and the 2-morphism A (cid:48)(cid:48) ( p ) ◦ ( g ◦ f ) ⊗ n = A (cid:48)(cid:48) ( p ) ◦ g ⊗ n ◦ f ⊗ n → g ◦ A (cid:48) ( p ) ◦ f ⊗ n → g ◦ f ◦ A ( p ) . The vertical composition and the horizontal composition of P C . (cid:3) Corollary 1.
Let C be a monoidal context and let P be a categorical operad. The data of P -algebrasin C , oplax P -morphisms and P Remark 6. Actually, one can show that these 2-categories yield monoidal contexts, using the structureof a Hopf operad on any categorical operads.One cannot compose a lax P -morphism with an oplax P -morphism, but one can rewrite a sequenceof a lax P -morphism followed by an oplax P -morphism into a sequence of an oplax P -morphism followedby a lax P -morphism. Definition 31.
Let C be a monoidal context and let P be a categorical operad. Let us consider thefollowing square diagram in C A A (cid:48)
B B (cid:48) ff (cid:48) gg (cid:48) where the objects are equipped with structures of P -algebras, where f and g (cid:48) are equipped withstructures of lax P -morphisms and where f (cid:48) and g are equipped with structures of oplax P -morphisms. Then, the 2-morphism g ◦ f → g (cid:48) ◦ f (cid:48) is a rewriting if the following diagram commutes B (cid:48) ( p ) ◦ g ⊗ n ◦ f ⊗ n g ◦ A (cid:48) ( p ) ◦ f ⊗ n , g ◦ f ◦ A ( p ) B (cid:48) ( p ) ◦ g (cid:48)⊗ n ◦ f (cid:48)⊗ n g (cid:48) ◦ B ( p ) ◦ f (cid:48)⊗ n g (cid:48) ◦ f (cid:48) ◦ A ( p ) One can define in a similar way a rewriting from g (cid:48) ◦ f (cid:48) to g ◦ f .1.3.3. Naturality of algebras.
Proposition 8.
The construction that sends a categorical operad P and a monoidal context C to the2-category made up of P -algebras in C , lax P -morphisms and P Alg − ( − ) lax : Categorical operads op × Monoidal contexts → . One obtains similar functors by replacing lax morphisms by oplax morphisms and categorical operadsby categorical planar operads.Proof.
Given a morphism of operads f : Q → P and a context functor F : C → D , one gets a2-functor Alg C ( P ) lax → Alg D ( Q ) lax as follows: (cid:46) such a 2-functor sends an object, that is a morphism of operads A : P → End( C ) to themorphism End( F ) ◦ A ◦ f ; (cid:46) its sends a lax morphism given by morphisms ( g c : A c → B c ) c ∈ Ob( P ) and 2-morphisms B ( p ) ◦ g ⊗ n → g ◦ A ( p ) to the lax morphism given by the morphisms ( F ( g f ( d ) ) : F ( A f ( d ) ) → F ( B f ( d ) )) d ∈ Ob( Q ) and the 2-morphisms F ( B ( f ( q ))) ◦ F ( g ) ⊗ n = F ( B ( f ( q )) ◦ g ⊗ n ) → F ( g ◦ A ( f ( q ))) = F ( g ) ◦ F ( A ( f ( q ))); (cid:46) it sends a P ( g c → g (cid:48) c ) c ∈ Ob( P ) to the Q ( F ( g f ( d ) ) → F ( g (cid:48) f ( d ) )) d ∈ Ob( Q ) . A straightforward but long checking shows that this construction does define a 2-functor and that itcommutes with composition. (cid:3)
Remark 7. This functor actually factor through the product categoryCategorical operads op × Categorical operads . Indeed, one can define transformations between morphisms of operads, that specialise into lax mor-phisms of algebras when the target operad is an End operad.Remark 8. Again, by only considering a functor (and not a 3-functor here), we are discarding a lot ofhigher information.Let us consider now a context functor F : C → D and a categorical (planar) operad P . APPING COALGEBRAS I COMONADS 11
Proposition 9.
The 2-functors
Alg C ( P ) lax → Alg D ( P ) lax ; Alg C ( P ) oplax ; → Alg D ( P ) oplax ; induced by F send (strict) P morphisms to (strict) P morphisms.Proof. This just follows from the fact that F sends 2-isomorphisms to 2-isomorphisms and identitiesof morphisms to identities of morphisms. (cid:3) Proposition 10.
Let us suppose that F is strictly fully faithful. Then, the square Alg C ( P ) lax Alg D ( P ) lax C Ob( P ) D Ob( P ) is a pullback square in the category of strict 2-categories. In particular, the 2-functor Alg C ( P ) lax → Alg D ( P ) lax is strictly fully faithful.Proof. It amounts to prove that the underlying square of sets given by objects is a pullback (whichmeans that a P -algebra in C is the same thing as objects A = ( A c ) c ∈ Ob( P ) in C together with thestructure of a P -algebra on F ( A ) ) and that the 2-functor Alg C ( P ) lax → Alg D ( P ) lax is strictly fullyfaithful. (cid:3) Corollary 2.
Let us suppose that F is an iso-equivalence. Then, the 2-functors induced by F between2-categories of P -algebras (consistently with lax morphisms or with oplax morphisms) are isoequiva-lencesProof. We already know that these 2-functors are strictly fully faithful. It suffices then to show thatthey are strictly essentially surjective. Let us consider a P -algebra B in D . Since F is strictly essentiallysurjective, let us consider objects ( A c ) c ∈ Ob( P ) in C together with isomorphisms F ( A c ) (cid:39) B c , c ∈ Ob( P ) . Using these isomorphisms, one can build a structure of a P -algebra A (cid:48) on the objects ( F ( A c )) c ∈ Ob( P ) so that these isomorphisms form an isomorphism of P -algebras A (cid:48) (cid:39) B in D . By Proposition 10, thisnew P -algebra A (cid:48) is the image of a P -algebra in C . (cid:3) Example of monoidal structures in a monoidal context.
Let us consider a monoidal context C . Our goal in this subsection is to describe some types of algebras encoded by categorical operads.1.4.1. Pairing.
Definition 32. A n -pairing in the monoidal context C is the data of n + 1 objects ( X , · · · X n , Y ) anda morphism p : X ⊗ · · · ⊗ X n → Y. A n-pairing will be denoted in this subsection using a corolla with n -input · · · .1.4.2. Pseudo-monoids.
Definition 33.
A pseudo monoid in C is an object A together with pairings m : A ⊗ A → Au : (cid:49) → A which we can represent as corollas m = , u = and together with 2-isomorphisms (cid:39) ; (cid:39) Id A (cid:39) ; called respectively the associator, the left unitor and the right unitor. so that (cid:46) the associator satisfies the pentagon identity, that is the following diagram commutes ; (cid:46) the unitors satisfy the triangle identity,that is the following diagram commutes . Pseudo monoids are algebras over the categorical operad A ∞ with one colour and so that A ∞ ( n ) is the groupoid equivalent to the point category whose objects are (isomorphism classes of) planartrees with only arity 2 and arity 0 vertices.1.4.3. Pseudo commutative monoids.
Let us denote κ ( X, Y ) : X ⊗ Y (cid:39) Y ⊗ X the commutator of the monoidal context C . Definition 34.
A pseudo commutative monoid is a pseudo-monoid A equipped with a 2-isomorphism m A (cid:39) m A ◦ κ ( A, A ) also written (cid:39) and called the commutator so that the following diagrams commute | Pseudo commutative monoids are algebras over the categorical operad E ∞ with one colour and sothat E ∞ ( n ) is the groupoid equivalent to the point category whose objects are pairs of a planar treeswith only arity 2 and arity 0 vertices and a permutation σ ∈ Σ n .1.4.4. Modules.
Definition 35.
Given a pseudo monoid A ∈ C , a lax left A -module is an object M ∈ C equipped with (cid:46) a pairing m M : A ⊗ M → M also written (cid:46) two 2-morphisms m M ◦ ( m A ⊗ Id M ) → m M ◦ (Id A ⊗ m M ) and m M ◦ ( u A ⊗ Id M ) → Id M alsowritten → ; → Id M ; so that the following diagrams commute ; APPING COALGEBRAS I COMONADS 13 ; . Pairs of a pseudo-monoid A and a lax left module M and algebras over an operad LM lax with twocolours. Definition 36.
One can define similarly (cid:46) an oplax left module with a pairing m M : A ⊗ M → M and 2-morphisms m M ◦ (Id A ⊗ m M ) → m M ◦ ( m A ⊗ Id M ) and Id M → m M ◦ ( u A ⊗ Id M ) ; (cid:46) a lax right module with a pairing m M : M ⊗ A → M and 2-morphisms m M ◦ (Id M ⊗ m A ) → m M ◦ ( m M ⊗ Id A ) and m M ◦ ( id M ⊗ u A ) → Id M ; (cid:46) an oplax right module with a pairing m M : M ⊗ A → M and 2-morphisms m M ◦ ( m M ⊗ Id A ) → m M ◦ (Id M ⊗ m A ) and Id M → m M ◦ ( id M ⊗ u A ) ;where the 2-morphisms are required to satisfy mutatis mutandis the same conditions as those in thedefinition of a lax left module.1.5. Opposite structures.
Let ( C , ⊗ , (cid:49) ) be a monoidal context. Since in some sense this monoidalcontext is a 3-categorical structure, there are three ways to inverse structures. (cid:46) one can take opposite morphisms; (cid:46) or take opposite 2-morphisms; (cid:46) or take the opposite monoidal structure. Definition 37.
Let C op be the monoidal context with the same object as C and so that C op ( X, Y ) = C ( Y, X ) for any two objects X, Y . Definition 38.
Let C co be the monoidal context with the same object as C and so that C co ( X, Y ) = C ( X, Y ) op for any two objects X, Y . Definition 39.
Let C tr = ( C , ⊗ tr , (cid:49) ) (where tr stands for "transposition") be the monoidal contextwith the same underlying 2-category C but whose monoidal structure is defined by X ⊗ tr Y = Y ⊗ X for any two objects X, Y . The associator, unitors and commutator are given by that of the monoidalcontext ( C , ⊗ , (cid:49) ) .It is possible to apply more than one of these transformation to obtain C coop , C optr , C cotr and C cooptr . Definition 40.
Let P be a categorical operad. A P -coalgebra in C is the data of a P -algebra in C op .Then, one can notice that (cid:46) the opposite "op" construction swaps algebras and coalgebras; (cid:46) the "co" construction swaps lax morphisms/modules into oplax morphisms/modules; (cid:46) the transposition "tr" construction swaps left modules and right modules.1.6. Doctrinal adjunction.
Let C be a monoidal context and let P be categorical operad. Such anoperad often induces a doctrine (that is a 2-monad) on C (see [Kel74]) whose definition of (op)laxmorphisms matches with that of Definition 27. Along these lines of thought, Kelly’s results ondoctrinal adjunctions apply in the framework of operads. Theorem 1. [Kel74]
Let us consider two P -algebras A, B in the monoidal context C and for any colour c ∈ Ob( P ) an adjunction A c B cl c r c in C whose unit and counit are denoted respectively η c and (cid:15) c . Then, (1) the set of structures of an oplax P -morphism on l and the set of structures of a lax P -morphismon r are canonically related by a bijection that we call Kelly’s bijection; (2) given a structure of an oplax P -morphism on l and a structure of a lax P -morphism on r , theyare related to each other through Kelly’s bijection if and only if the 2-morphisms Id ◦ Id → r c ◦ l c and l c ◦ r c → Id ◦ Id are rewritings; (3) if l and r are equipped with structures of lax P -morphisms then the 2-morphisms η c and (cid:15) c are P l is a P -morphism (hence and oplax P -morphism) and itsoplax structure is related to the lax structure on r through Kelly’s bijection; (4) if l and r are equipped with structures of oplax P -morphisms then the 2-morphisms η c and (cid:15) c are P r is a P -morphism (hence a lax P -morphism) and its laxstructure is related to the oplax structure on l through Kelly’s bijection.Proof. Let us first prove (1). Given the structure of an oplax P morphism on l , the related structureof a lax P -morphism on r is given by morphisms of the form A ( p ) ◦ r ⊗ n → r ◦ l ◦ A ( p ) ◦ r ⊗ n → r ◦ A ( p ) ◦ l ⊗ n ◦ r ⊗ n → r ◦ A ( p ) . Conversely, given the structure of a lax P morphism on r , the related structure of an oplax P -morphismon l is given by morphisms of the form l ◦ A ( p ) → l ◦ A ( p ) ◦ r ⊗ n ◦ l ⊗ n → l ◦ r ◦ A ( p ) ◦ l ⊗ n → A ( p ) ◦ l ⊗ n . A straightforward check shows that these formulas do define respectively a lax and an oplax structureand that they are inverse to each other. This defines Kelly’s bijection.Now, let us prove (2). The fact that the 2-morphisms Id ◦ Id → r c ◦ l c and l c ◦ r c → Id ◦ Id arerewritings means that the following squares are commutative l ◦ A ( p ) ◦ r ⊗ n l ◦ r ◦ B ( p ) B ( p ) ◦ ( l ◦ r ) ⊗ n B ( p ) A ( p ) r ◦ l ◦ A ( p ) A ( p ) ◦ ( r ◦ l ) ⊗ n r ◦ B ( p ) ◦ l ⊗ n for any operation p of the operad P (actually, if one square is commutative, then the other one is alsocommutative). A straightforward diagram chasing shows that the commutation of these diagrams isequivalent to the fact that the oplax structure on l and the lax structure on r are related throughKelly’s bijection.Then, let us prove (3). In that context, l is equipped with the structure of a lax P -morphism andwith the structure of an oplax P -morphism. Then, using the right commutative square just above, it isstraightforward to check that the fact that the natural transformations Id → r c l c form a P l ◦ A ( p ) → A ( p ) ◦ l ⊗ n → l ◦ A ( p ) is the identity of l ◦ A ( p ) for any p . Similarly, using the left commutative square just above, it isstraightforward to check that the fact that the natural transformations l c r c → Id form a P A ( p ) ◦ l ⊗ n → l ◦ A ( p ) → A ( p ) ◦ l ⊗ n is the identity of A ( p ) ◦ l ⊗ n for any p .Finally, (4) may be proven using the same arguments as (3). (cid:3) Proposition 11.
A lax P morphism is an isomorphism (resp. an equivalence) in the 2-category of P -algebras, lax P -morphisms and P P -morphism and the underlyingmorphism in C is an isomorphism (resp. an equivalence). We have the same result when consideringoplax morphisms instead of lax morphisms. APPING COALGEBRAS I COMONADS 15
Proof.
Let f : A → B be a lax P -morphism.Let us suppose that f is an equivalence in the 2-category of P -algebras, with right adjoint g . Itis clear that the underlying morphism of f in C is an equivalence (with pseudo-inverse the underlyingmorphism of g ). Moreover, by doctrinal adjunction, f is a P -morphism.Conversely, let f : A → B be a P -morphism whose underlying morphism in C is an equivalence,with right adjoint g : B → A . Then, again by doctrinal adjunction, g inherits the structure of alax P -morphism so that the adjunction in C relating f and g lifts to an adjunction in the 2-categoryof algebras. Since the unit and the counit of this adjunction are 2-isomorphisms, this is an adjointequivalence. (cid:3) Some monoidal contexts.
The monoidal context of strict 2-categories.
Strict 2-categories themselves form a very largecartesian monoidal context, whose pseudo commutative monoids are precisely monoidal contexts andwhose E ∞ -morphisms are context functors.1.7.2. The monoidal context of categories.
The 2-category
Cats of categories form a large cartesianmonoidal context.On can check that (cid:46) pseudo (commutative) monoids are (symmetric) monoidal categories; (cid:46) (op)lax morphisms of pseudo monoids are (op)lax monoidal functors; (cid:46) A ∞ (cid:46) lax left modules are categories tensored over a monoidal category; (cid:46) lax morphisms of lax left modules over a lax monoidal functor are functors equipped with astrength; (cid:46) a cotensorisation of a category D by a monoidal category C is the structure of a oplax right C op -module on D , or equivalently, the structure of a lax right C -module on D op .1.7.3. The monoidal context of functors.
Definition 41.
Let
Functors be the strict 2-category 2-Fun ( Mor , Cats ) , that is (cid:46) its objects are functors F : C → D ; (cid:46) its morphisms from F to F are pairs of functors ( S, T ) so that the following square diagramof categories is strictly commutative C C D D ; SF F T (cid:46) its 2-morphisms from ( S , T ) to ( S , T ) are natural transformations A S : S → S and A T : T → T so that A T ◦ h F = F ◦ h A S . Proposition 12.
The 2-category
Functors form a cartesian monoidal context. Moreover, the two2-functors to categories (target and source) preserve strict finite products.Proof.
Straightforward. (cid:3)
Let us now describe some monoidal structures in the monoidal context
Functors . (cid:46) the pseudo (commutative) monoids are given by pairs of (symmetric) monoidal categoriesand strict monoidal functors. (cid:46) the (op)lax monoidal functors from F : C → D to F : C → D are pairs of (op)laxmonoidal functors S : C → C and T : D → D so that F ◦ S = T ◦ F as (op)lax monoidalfunctors; (cid:46) the lax left modules over a pseudo monoid A → A are the data of a category M tensoredover A , a category M tensored over A and a functor M → M that commute with all thetensoring structures.
2. Comonads and monoidal structuresIn this section, we describe the monoidal context of comonads and show how it is related to themonoidal context
Functors .2.1.
The monoidal context of comonads.Definition 42.
Let ∆ act be the sub category of ∆ made up of active morphisms. More precisely, itsobjects are the posets [ n ] := (0 < < · · · < n ) for n ∈ (cid:78) and its morphisms from [ n ] to [ m ] are the morphisms of posets (that is functors) f so that f (0) = 0 and f ( n ) = m . This is a strict monoidal category with tensor product [ n ] ⊗ [ m ] = [ n + m ] . Moreover, let B (∆ act ) be the delooping of the monoidal category ∆ act , that is the strict 2-categorywith one object ∗ and so that B (∆ act )( ∗ , ∗ ) = ∆ act . Definition 43.
A category with comonad is a pair ( C , Q ) of a category equipped with a comonad.Equivalently, this is a 2-functor from B (∆ act ) to Cats .In particular, categories with comonads are algebras over a categorical operad (this is just the2-category B (∆ act ) seen as an operad) in the monoidal context Cats . Thus, one can then define(op)lax morphisms between categories with comonads.
Definition 44.
A (op)lax comonad functor between two categories with comonads ( C , Q ) and ( D , R ) is a (op)lax morphism of B (∆ act ) -algebras. For instance an oplax comonad functor is the data of afunctor F : C → D and a natural transformation F ◦ Q A −→ R ◦ F so that the the following diagrams commute F Q RFF QQ RF Q F RR F Q RFF
Id Id F. Definition 45.
Let
Comonads be the strict 2-category made up of B (∆ act ) -algebras, oplax B (∆ act ) -morphisms and B (∆ act ) (cid:46) object are categories with comonads ( C , Q ) ; (cid:46) morphisms are oplax comonad functors; (cid:46) ( F, A ) and ( F (cid:48) , A (cid:48) ) from ( C , Q ) to ( D , R ) are natural trans-formations F → F (cid:48) so that the following diagram commutes F Q RFF (cid:48)
Q RF (cid:48) . Proposition 13.
The 2-category
Comonads has strict finite products and hence is a cartesian monoidalcontext.Proof.
Straightforward. (cid:3)
Definition 46.
Given a category C and a comonad Q on it, let Cog C ( Q ) be the category of Q -coalgebras. Equivalently, this is the mapping category Cog C ( M ) = Comonads ( ∗ Monads , ( C , M )) . This defines a 2-functor from
Comonads to Cats . Proposition 14.
The 2- functor
Cog preserves strict finite products and hence is a context functor.
APPING COALGEBRAS I COMONADS 17
Proof.
This follows from the fact that ∗ Monads is a cocommutative coalgebra. (cid:3)
Proposition 15.
The forgetful 2-functor
Comonads → Cats preserves strict finite products and ishence a context functor.Proof.
Straigthforward. (cid:3)
Monoidal structures in the monoidal context of comonads.
Our goal in this subsection is todescribe pairings, pseudo-monoids and their modules in the monoidal context of comonads.2.2.1.
Pairings.
A 2-pairing in the monoidal context of comonads from the pair ( C , Q ) , ( D , O ) to ( E , R ) consists in a bifunctor C × D → E ( X, Y ) (cid:55)→ X ⊗ Y ; together with a natural transformation Q ( X ) ⊗ O ( Y ) → R ( X ⊗ Y ) so that the following diagrams commute Q ( X ) ⊗ O ( Y ) R ( X ⊗ Y ) QQ ( X ) ⊗ OO ( Y ) R ( Q ( X ) ⊗ O ( Y )) RR ( X ⊗ Y ) Q ( X ) ⊗ O ( Y ) R ( X ⊗ Y ) X ⊗ Y X ⊗ Y. Hopf comonads.
A pseudo-monoid in the monoidal context of comonads consists in a monoidalcategory C together with a Hopf comonad; this notion is dual to that of a Hopf monad (see [Moe02]). Definition 47.
A Hopf comonad on a monoidal category ( C , ⊗ , (cid:49) ) is the data of a comonad ( Q, w , n ) on C together with a structure of a lax monoidal functor on QQ ( X ) ⊗ Q ( Y ) → Q ( X ⊗ Y ); (cid:49) → Q ( (cid:49) ); so that the natural transformations w : Q → QQ and n : Q → Id are monoidal natural transformations.Then, a pseudo commutative monoid in the monoidal context of comonads is given by a symmetricmonoidal category equipped with a commutative Hopf comonad. Definition 48.
Let Q be a Hopf comonad in a symmetric monoidal category C . It is said to becommutative if the structure of a lax monoidal functor on Q is symmetric.Then, one can notice that (cid:46) a lax A ∞ -morphism in the context of comonads from ( C , Q ) to ( D , R ) is given by a laxmonoidal functor F : C → D and a natural transformation F Q → RF that is a monoidal natural transformation and that makes F an oplax comonad functor; (cid:46) an oplax A ∞ -morphism in the context of comonads from ( C , Q ) to ( D , R ) is given by an oplaxmonoidal functor F : C → D and a natural transformation F Q → RF that makes F an oplax comonad functor and that is also a rewriting. Comonads comodules.
A pair of a pseudo monoid together with a left lax module in themonoidal context of comonads is the data of a Hopf comonad ( Q, w , τ ) on a monoidal category E together with a category C tensored over E and a comonad R on C equipped with the structure of aHopf Q -comodule as defined in the following definition. Definition 49.
A structure of Hopf comodule comonad on the comonad ( R, w (cid:48) , τ (cid:48) ) is the data of astrength on the functor R with respect to the lax monoidal functor QQ ( X ) (cid:2) B ( Y ) → B ( X (cid:2) Y ) so that the natural transformation w (cid:48) is strong with respect to the monoidal natural transformation w and so that τ (cid:48) is strong with respect to τ .2.3. Monads.
We have a canonical isomorphism of monoidal contexts
Cats (cid:39)
Cats co that sends a category C to its opposite category C op . This isomorphism lifts to an isomorphismbetween the monoidal context Comonads and the monoidal context of monads.2.3.1.
The monoidal context of monads.
Definition 50.
A category with monad ( C , M ) is the data of a category C and a monad M on C .Equivalently, this is a 2-functor from B (∆ op act ) to Cats . Moreover, a lax monad functor between twocategories with monads ( C , M ) and ( D , N ) is a lax morphism of algebras over B (∆ op act ) , that is thedata of a functor F : C → D and a natural transformation N ◦ F A −→ F ◦ M so that the following diagrams commute NNF NF M F MMNF F M Id F F Id NF F M.
Definition 51.
Let
Monads be the strict 2-category of B (∆ op act ) -algebras, lax B (∆ op act ) -morphisms and B (∆ op act ) Proposition 16.
The strict 2-category
Monads form a cartesian monoidal context and the construc-tion C ∈ Cats (cid:55)→ C op induces a canonical isomorphism of monoidal contexts Monads (cid:39)
Comonads co . Proof.
Straightforward. (cid:3)
Hopf monads and module monad over a Hopf comonad.
A pseudo-monoid in the monoidalcontext of monads is a monoidal category equipped with a Hopf monad.
Definition 52. [Moe02] A Hopf monad on a monoidal category ( C , ⊗ , (cid:49) ) is the data of a monad M together with the structure of a Hopf comonad on the related comonad on C op . Equivalently, this isthe data of a monad M on C together with a structure of an oplax monoidal functor on MM ( X ⊗ Y ) → M ( X ) ⊗ M ( Y ) M ( (cid:49) ) → (cid:49) so that the natural transformations m : MM → M and u : Id → M are monoidal natural transforma-tions. APPING COALGEBRAS I COMONADS 19
Let us consider a Hopf comonad Q on a monoidal category C , a category D cotensored over C through a bifunctor D × C op → D ( X, Y ) (cid:55)→ (cid:104) X, Y (cid:105) ; and a monad M on D . A structure of a lax right ( C , Q ) -module on ( D op , M ) in the monoidal contextof comonads that enhances the cotensorisation of D by C corresponds to the structure of a Hopf Q -module monad on M . Definition 53.
A Hopf Q -module monad is the data of a monad ( M, m, u ) on D together with astrength on the functor M with respect to the lax monoidal functor QM ( (cid:104) X, Y (cid:105) ) → (cid:104) M ( X ) , Q ( Y ) (cid:105) so that the following diagrams commute MM (cid:104) X, Y (cid:105) M (cid:104) M ( X ) , Q ( Y ) (cid:105) (cid:104) MM ( X ) , QQ ( Y ) (cid:105) M (cid:104) X, Y (cid:105) (cid:104) MM ( X ) , QQ ( Y ) (cid:105) ; (cid:104) X, Y (cid:105) (cid:104)
X, Y (cid:105) M (cid:104) X, Y (cid:105) (cid:104) M ( X ) , Q ( Y ) (cid:105) . From comonads to coalgebras and back to comonads.
Let us consider two pairs ( C , Q ) and ( D , R ) of a categories with comonads (that is objects in Comonads ). Proposition 17.
Let F : C → D be a functor. Then there is a canonical bijection between (1) the set of functors F cog : Cog C ( Q ) → Cog D ( R ) that lifts F : C → D (that is F U Q = U R F cog ); (2) the set of oplax comonad structures on FF ◦ Q β −→ R ◦ F, with respect to Q and R .Proof. Given a functor F cog : Cog C ( Q ) → Cog D ( R ) that lifts F , the equality F U Q = U R F cog gives usby adjunction a morphism F cog L Q → L R F and then a morphism F Q = F U Q L Q = U R F cog L Q → U R L R F = RF.
One can check that the resulting map
F Q → RF is an oplax comonad structure on F .Conversely, given an oplax comonad structure F Q → RF on F , then for any Q -algebra V , theobject F ( V ) has the structure of a R -coalgebra given by the map F ( V ) → F Q ( V ) → RF ( V ) . This construction is natural and defines the expected lifting functor.A straightforward check shows that the two constructions are inverse to each other. (cid:3)
Corollary 3.
Given a monad M on C and a monad N on D , there is a canonical bijection between (1) the set of functors F alg : Alg C (() M ) → Alg D (() N ) that lifts F : C → D (that is F U M = U N F alg ); (2) the set of lax monad structures on FN ◦ F α −→ F ◦ M, with respect to M and N . Proof.
This follows from the same arguments as those used to prove Proposition 17. In particular,for any M -algebra A , the object F ( A ) has the structure of a N algebra given by the map NF ( A ) → F M ( A ) → F ( A ) . (cid:3) Proposition 18.
Let us consider two functors
F, G : C → D together with liftings F cog , G cog : Cog C ( Q ) → Cog D ( R ) to the categories of coalgebras, that correspond to oplax comonad structureson F respectively denoted α and β . Moreover, let A : F → G be a natural transformation. Then, thefollowing assertions are equivalent: (1) the natural transformation A lifts to a 2-morphism in Comonads from ( F, α ) to ( G, β ) ; (2) the natural transformation A lifts to a 2-morphism in Functors from ( F, F cog ) to ( G, G cog ) ; (3) for any Q -coalgebra V , the map A ( V ) : F ( V ) → G ( V ) is a morphism of R -coalgebras.Proof. The assertion (2) is clearly equivalent to (3). Let us prove that (3) is equivalent to (1).On the one hand, let us assume (1). Then for any Q -coalgebra V , the following diagram commutes F ( V ) F Q ( V ) RF ( V ) G ( V ) GQ ( V ) RG ( V ) A ( V ) A ( Q ( V )) R ( A ( V )) Thus, the map A ( V ) : F ( V ) → G ( V ) is a morphism of R -coalgebra.Conversely, let us assume (3). Then, the square diagram F Q ( X ) RF ( X ) GQ ( X ) RG ( X ) A ( Q ( X )) R ( A ( X )) decomposes as F Q ( X ) F QQ ( X ) RF Q ( X ) RF ( X ) GQ ( X ) GQQ ( X ) RGQ ( X ) RG ( X ) A ( Q ( X )) A ( QQ ( X )) R ( A ( Q ( X ))) R ( A ( X )) The left square and the right square are commutative by naturality. The middle square is commutativesince Q ( X ) is a Q -coalgebra. Hence, the whole square is commutative, which shows (1). (cid:3) Theorem 2.
The construction that sends a category with a comonad ( C , Q ) to the functor Cog C ( Q ) → C canonically induces a 2-functor from Comonads to Functors that is strictly fully faithful and thatpreserves strict finite products.Proof.
Such a 2-functor sends a morphism (that is an oplax comonad functor) ( F, A ) to the pairof functors ( F, F cog ) defined in Proposition 17, and a 2-morphism (that is a natural transformation) A (cid:48) : ( F, A ) → ( G, B ) to the pair of natural transformation ( A (cid:48)(cid:48) , A (cid:48) ) whose first component is definedin Proposition 18.It is strictly fully faithful by Proposition 17 and Proposition 18 and it preserves strict finite productsbecause both 2-functors ( C , Q ) (cid:55)→ Cog C ( Q ) and ( C , Q ) (cid:55)→ C do. (cid:3) Hence, any algebraic structure inside the 2-category
Comonads may equivalently be described usingforgetful functors from categories of coalgebras to the ground category.
Corollary 4.
The construction that sends a category with a monad ( C , M ) to the functor Alg C ( M ) → C canonically induces a 2-functor from Monads to Functors that is strictly fully faithful and thatpreserves strict finite products.
APPING COALGEBRAS I COMONADS 21
Consequences.
One can draw several consequences from Theorem 2.2.5.1.
Monoidal categories.
Given a monoidal category ( C , ⊗ , (cid:49) ) and a comonad ( Q, w , n ) (resp. amonad ( M, m, u ) ), there is a canonical bijection between(1) the set of structures of a monoidal category on Q -algebras (resp. M -algebras) that lift thatof C (that is the forgetful functor Cog C ( Q ) → C is strict monoidal);(2) the set of structures of a Hopf comonad on Q (resp. structures of a Hopf monad on M ).Indeed, given a structure of a Hopf comonad on Q , the tensor product of two Q -coalgebras V, W and the unit (cid:49) inherit structures of Q -coalgebras through the formulas V ⊗ W → Q ( V ) ⊗ Q ( W ) → Q ( V ⊗ W ); (cid:49) → Q ( (cid:49) ) . Conversely, from a structure of a monoidal category on Q -coalgebras that lifts that of C , one obtainthe structure of a Hopf comonad on Q by lifting the natural map Q ( X ) ⊗ B ( Y ) τ ( X ) ,τ ( Y ) −−−−−−→ X ⊗ Y to Q ( X ⊗ Y ) Lax monoidal functors.
Now, let us consider Hopf comonads Q and O on monoidal categoriesrespectively C and D and a lax monoidal functor F : C → D . The two following assertions areequivalent(1) the natural transformation F Q → OF is monoidal;(2) the natural map in F F ( V ) ⊗ F ( W ) → F ( V ⊗ W ) induced by the structure of a lax monoidal functor on F is a morphism of O -coalgebras forany two Q -coalgebras V, W .If these assertions are true, then the structure of a lax monoidal functor on F : C → D induces astructure of a lax monoidal functor on F cog : Cog C ( Q ) → Cog O ( O ) .2.5.3. Modules.
Let ( C , Q ) be monoidal category and a Hopf comonad and let R be a comonad ona category D tensored by C . Then, there is a canonical bijection between(1) the set of tensorisations of the category of R -coalgebras by the monoidal category of Q -coalgebras that lifts the tensorisation of D by C ;(2) the set of structures of a Hopf Q -module comonad on R .Similarly, if M is a monad on E which is cotensored by C , then there is a canonical bijection between(1) the set of cotensorisations of the category of M -algebras by the monoidal category of Q -coalgebras that lifts the cotensorisation of E by C ;(2) the set of structures of a Hopf Q -module monad on M .2.6. The adjoint lifting theorem.
In this subsection, we recall the adjoint lifting theorem and itslink with (op)lax (co)monad functors.Let us consider an oplax comonad functor ( L, A ) : ( C , Q ) → ( D , O ) . Let us assume that thefunctor L has a right adjoint R . Then, the structure of an oplax comonad functor LQ → OL on L induces by doctrinal adjunction the structure of a lax comonad functor on the right adjoint RQR → RLQR → ROLR → RO.
Thus, for any O -coalgebra W , let us consider the two following morphisms of Q -coalgebras from L Q R ( W ) to L Q RO ( W ) : (cid:46) on the one hand, the morphism induced by the map W → O ( W ) ; (cid:46) on the other hand, the composite morphism L Q R ( W ) → L Q QR ( W ) → L Q RO ( W ) . This gives us a coreflexive pair of maps(1) L Q R ( W ) ⇒ L Q RO ( W ) with common left inverse induced by the map O ( W ) → W . Theorem 3 (Adjoint lifting theorem, [Joh75]) . The functor L cog has a right adjoint if and only if thepair of maps just above in diagram 1 has an equaliser for any O -coalgebra W . Then, such a limitdefines the value of this right adjoint functor on W .Proof. Straightforward. (cid:3)
Besides, one can factorise the oplax comonad functor ( L, A ) from ( C , Q ) to ( D , O ) as follows. Proposition 19.
The endofunctor
LQR of D has the canonical structure of a comonad. Moreover,the oplax comonad functor ( L, A ) factorises as ( C , Q ) ( L,A (cid:48) ) −−−→ ( D , LQR ) (Id ,A (cid:48)(cid:48) ) −−−−→ ( D , O ) where A (cid:48) and A (cid:48)(cid:48) are respectively the natural maps LQ LQη −−→
LQRL ; LQR A −→ OLR
O(cid:15) −→ O. Proof.
The structure of a comonad on
LQR is given by the maps
LQR → LQQR → LQRLQL ; LQR → LR → Id . Proving that these maps do define a comonad and the rest of the proposition follow from a straight-forward checking. (cid:3)
More on comonad functors.
This subsection with the subset of oplax comonad morphismsspanned by "strong" morphisms.
Definition 54.
A comonad functor between categories with comonads is an oplax comonad functor ( F, A ) : ( C , Q ) → ( D , R ) so that A is a natural isomorphism.Remark 9. By the result of Kelly on doctrinal adjunctions, for any adjunction in the 2-category Comonads , the right adjoint is a comonad functor.
Proposition 20.
An oplax comonad functor ( F, A ) : ( C , Q ) → ( D , R ) is a comonad functor if andonly if the induced natural transformation F cog L Q → L R F is an isomorphism.Proof. This is a direct consequence of the way the morphisms F cog L Q → L R F and F Q → RF arerelated in the proof of Proposition 17. (cid:3) Let us consider an oplax comonad functor ( F, A ) : ( C , Q ) → ( D , R ) . We suppose that the categories C and D are complete and that the comonads Q, R preserve coreflexiveequalisers. Hence, the categories of coalgebras over these comonads are complete (see Appendix B).
Proposition 21.
Suppose that F preserves limits and that A is a natural isomorphism (hence, ( F, A ) is a comonad functor). Then, F cog preserves limits.Proof. Since U R ◦ F cog = F ◦ U Q , U Q , F preserve coreflexive equalisers and U R create coreflexiveequalisers, then F cog also preserve coreflexive equalisers.Let us consider a family of Q -coalgebras ( V i ) i ∈ I and the following diagram F cog ( (cid:81) i V i ) (cid:81) i F cog ( V i ) F cog L Q ( (cid:81) i U Q ( V i )) L R ( (cid:81) i U R F cog ( V i )) F cog L Q ( (cid:81) i QU Q ( V i )) L R ( (cid:81) i RU R F cog ( V i )) APPING COALGEBRAS I COMONADS 23 which represents a natural transformation between the two vertical subdiagrams. The middle hori-zontal arrow and the bottom horizontal arrow decompose respectively as F cog L Q ( (cid:89) i U Q ( V i )) → L R F ( (cid:89) i U Q ( V i )) → L R ( (cid:89) i F U Q ( V i )) = L R ( (cid:89) i U R F cog ( V i )); F cog L Q ( (cid:89) i QU Q ( V i )) → L R F ( (cid:89) i QU Q ( V i )) → L R ( (cid:89) i F QU Q ( V i )) → L R ( (cid:89) i RU R F cog ( V i )); all these maps are isomorphisms since F preserves products and the morphism F Q → RF is an isomor-phism. Hence, the middle horizontal arrow and the bottom horizontal arrow are also isomorphisms.Since the two vertical subdiagrams are limiting, then the top horizontal arrow is also an isomorphism.To conclude, F cog preserves coreflexive equalisers and products and hence preserves all limits. (cid:3) Corollary 5.
Let us consider a lax monad functor between categories with monads ( G, B ) : ( C , M ) → ( D , N ) where C and D are cocomplete and M and N preserve reflexive coequalisers. If F preservescolimits and B is a natural isomorphism, then the induced functor between algebras F alg preservescolimits.
3. Mapping coalgebrasIn this section, we use the adjoint lifting theorem to describe contexts where some categories of(co)algebras over a (co)monad are enriched tensored and cotensored over the category of coalgebrasover another comonad.3.1.
Pairing adjoints.
The situation.
Let us consider a 2-pairing in the monoidal context of comonads − (cid:2) − : ( C , Q ) × ( D , R ) → ( E , O ) . It is given by a bifunctor − (cid:2) − : C × D → E together with a natural map Q ( X ) (cid:2) R ( Y ) → O ( X (cid:2) Y ) thatsatisfy some commutation conditions with respect to the counits and decompositions of comonads.We know that such a natural map satisfying such conditions is equivalent to the data of a bifunctor − (cid:2) cog − : Cog C ( Q ) × Cog D ( R ) → Cog E ( O ) that lifts − (cid:2) − .Let us suppose that for any X ∈ C , the functor X (cid:2) − : D → E has a right adjoint that is denoted (cid:104)− , X (cid:105) : E → D . Then, by naturality, we obtain a bifunctor (cid:104)− , −(cid:105) : E × C op → D ; together with a natural isomorphism hom E ( X (cid:2) Y, Z ) (cid:39) hom D ( Y, (cid:104) Z, X (cid:105) ) for any ( X, Y, Z ) ∈ C × D × E .3.1.2. The adjoint.
Let V be a Q -coalgebra. The functor V (cid:2) cog − from R -coalgebras to O -coalgebraslifts the functor U Q ( V ) (cid:2) − from D to E . This corresponds to the structure of an oplax comonadfunctor on U Q ( V ) (cid:2) − with respect to R and O given by the map U Q ( V ) (cid:2) R ( − ) → QU Q ( V ) (cid:2) R ( − ) → O ( U Q ( V ) (cid:2) − ) . By doctrinal adjunction, the adjoint functor (cid:104)− , U Q ( V ) (cid:105) inherits the structure of a lax comonad functorgiven by a natural map R ( (cid:104)− , U Q ( V ) (cid:105) ) → (cid:104) O ( − ) , U Q ( V ) (cid:105) . (See Subsection 2.6) For any O -coalgebra Z, let us consider the following two R -coalgebra morphismsfrom L R ( (cid:104) Z, V (cid:105) ) to L R ( O ( (cid:104) Z ) , V (cid:105) ) :(1) on the one hand, the morphism induced by the structural morphism Z → O ( Z ) ;(2) on the other hand the composite morphism L R ( (cid:104) Z, V (cid:105) ) → L R R ( (cid:104) Z, V (cid:105) ) → L R ( (cid:104) O ( Z ) , V (cid:105) ) . They also share a left inverse induced by the counit map O ( Z ) → Z . We thus obtain a coreflexivepair of morphisms(2) L R ( (cid:104) Z, V (cid:105) ) ⇒ L R ( (cid:104) O ( Z ) , V (cid:105) ) . Proposition 22.
The functor V (cid:2) cog − : Cog D ( R ) → Cog E ( O ) admits a right adjoint, that we denote (cid:104)− , V (cid:105) R , if and only if the category of R -coalgebras admitslimits of the diagram (2) for any O -coalgebra Z .Proof. This is an application of the adjoint lifting theorem (Theorem 3). (cid:3)
Corollary 6.
If such an adjoint (cid:104)− , V (cid:105) R exists for any Q -coalgebra V , then it yields a bifunctor (cid:104)− , −(cid:105) R : Cog E ( O ) × Cog C ( Q ) op → Cog D ( R ) . and a natural isomorphism hom Cog E ( O ) ( V (cid:2) cog W, Z ) (cid:39) hom Cog D ( R ) ( W, (cid:104) Z, V (cid:105) R ) . Corollary 7.
If the category D admits coreflexive equalisers and if they are preserved by R , then thefunctor V (cid:2) cog − has a right adjoint for any Q -coalgebra V . Remark 10. If the functor − (cid:2) Y has a right adjoint [ Y, − ] for any Y ∈ D , one gets the samephenomenon, that is the functor − (cid:2) cog W has a right adjoint if and only if equalisers of pairs of mapsof the form L Q ([ W, Z ]) ⇒ L Q ([ W, O ( Z )]) exist in Q -coalgebras. This just follows from considering the composite pairing ( D , R ) × ( C , Q ) (cid:39) ( C , Q ) × ( D , R ) − (cid:2) − −−−→ ( E , O ) and applying the same results.3.2. Enrichment.
Let C be a monoidal category and let D be a category enriched, tensored andcotensored over C . Let us suppose that C have all coreflexive equalisers and that D have all coreflexiveequalisers and all reflexive coequalisers.Let us consider a Hopf comonad Q on C that preserves coreflexive equalisers, a comonad R on D that preserves coreflexive equalisers and a monad M on D that preserves reflexive coequalisers. Theorem 4.
Given a tensorisation
Cog C ( Q ) × Cog D ( R ) → Cog D ( R ) that lifts that of C on D , then R -coalgebras are enriched, tensored and cotensored over Q -coalgebras. Theorem 5.
Given a cotensorisation
Alg D ( M ) × Cog C ( Q ) op → Cog D ( M ) that lifts that of C on D , then M -algebras are enriched, tensored and cotensored over Q -coalgebras. Pairing transfer.
Let us consider two 2-pairings in the monoidal context of comonads − (cid:2) − : ( C , Q ) × ( D , R ) → ( E , O ) − (cid:2) (cid:48) − : ( C (cid:48) , Q (cid:48) ) × ( D (cid:48) , R (cid:48) ) → ( E (cid:48) , O (cid:48) ) together with a lax morphism of pairings, that is the data of functors F C : C → C (cid:48) ; F D : D → D (cid:48) ; F E : E → E (cid:48) ; that are lifted to the level of coalgebras by functors respectively denoted F C,cog , F
D,cog and F E,cog and together with a natural morphism F E ( X (cid:2) Y ) → F C X (cid:2) (cid:48) F D Y APPING COALGEBRAS I COMONADS 25 that also lifts to the level of coalgebras.Let us suppose that the functors X (cid:2) − : D → E ; X (cid:48) (cid:2) (cid:48) − : D (cid:48) → E (cid:48) ; V (cid:2) cog − : Cog D ( R ) → Cog E ( O ) ; V (cid:48) (cid:2) (cid:48) cog − : Cog D (cid:48) ( R (cid:48) ) → Cog E (cid:48) ( O (cid:48) ) ; all have right adjoints for any objects X, X (cid:48) ∈ C × C (cid:48) , any Q -coalgebras V and any Q (cid:48) -coalgebra V (cid:48) .We denote these right adjoints (cid:104)− , X (cid:105) , (cid:104)− , X (cid:48) (cid:105) (cid:48) , (cid:104)− , V (cid:105) R and (cid:104)− , V (cid:48) (cid:105) R (cid:48) . Proposition 23.
Let us suppose that (cid:46) the natural transformations F D R → R (cid:48) F D and F E O → O (cid:48) F E are isomorphisms; (cid:46) the canonical morphism F D ( (cid:104) Z, X (cid:105) ) → (cid:104) F E ( Z ) , F C ( X ) (cid:105) (cid:48) is an isomorphism for any objects X, Z ∈ C × E , ; (cid:46) the image through the functor F D,cog of the limiting diagram (cid:104)
Z, V (cid:105) R → (cid:104) L O Z, V (cid:105) R ⇒ (cid:104) L O OZ, V (cid:105) R is limiting for any O -coalgebra Z and any Q -coalgebra V .Then, for any two coalgebras Z, V ∈ Cog E ( O ) × Cog C ( Q ) , the canonical morphism F D,cog ( (cid:104) Z, V (cid:105) R ) → (cid:104) F E,cog ( Z ) , F C,cog ( V ) (cid:105) R (cid:48) is an isomorphism.Proof. First, let us prove the result in the case where Z is cofree, that is Z = L O K. From the lifting property of the functor F C , F D , F E and of the natural transformation F E ( X (cid:2) Y ) → F C X (cid:2) (cid:48) F D Y we get a commutative diagram of functors U O (cid:48) ◦ ( F C,cog ( V ) (cid:2) (cid:48) cog − ) ◦ F D,cog ( F C U Q ( V ) (cid:2) (cid:48) − ) ◦ U R (cid:48) ◦ F D,cog U O (cid:48) ◦ F E,cog ◦ ( V (cid:2) cog − ) ( F C U Q ( V ) (cid:2) (cid:48) − ) ◦ F R ◦ U R F E ◦ U O ◦ ( V (cid:2) cog − ) F E ◦ ( U Q ( V ) (cid:2) − ) ◦ U R . By adjunction, we thus get a commutative diagram of functors F D,cog ◦ L R ◦ (cid:104)− , U Q ( V ) (cid:105) F D,cog ◦ (cid:104)− , V (cid:105) R ◦ L O L R (cid:48) ◦ F D ◦ (cid:104)− , U Q ( V ) (cid:105) (cid:104)− , F C,cog ( V ) (cid:105) R (cid:48) ◦ F E,cog ◦ L O L R (cid:48) ◦ (cid:104)− , F C U Q ( V ) (cid:105) (cid:48) ◦ F E (cid:104)− , F C,cog ( V ) (cid:105) R (cid:48) ◦ L O (cid:48) ◦ F E The horizontal arrows are isomorphisms since for any isomorphism of left adjoint functors the inducedmorphism of right adjoint functors is also an isomorphism. Then, by the hypothesis, the two leftvertical arrows and the bottom right vertical arrow are all isomorphisms. Hence the map F D,cog ◦ (cid:104)− , V (cid:105) R ◦ L O → (cid:104)− , F C,cog ( V ) (cid:105) R (cid:48) ◦ F E,cog ◦ L O is also an isomorphism. This proves the result for Z = L O ( K ) . In the general case, let us consider the following diagram F D,cog ( (cid:104) L O O ( Z ) , V (cid:105) R ) (cid:104) F E,cog L O O ( Z ) , F C,cog ( V ) (cid:105) R (cid:48) F D,cog ( (cid:104) L O ( Z ) , V (cid:105) R ) (cid:104) F E,cog L O ( Z ) , F C,cog ( V ) (cid:105) R (cid:48) F D,cog ( (cid:104) Z, V (cid:105) R ) (cid:104) F E,cog ( Z ) , F C,cog ( V ) (cid:105) R (cid:48) . The left vertical part is limiting as well as the left vertical part (by hypothesis). Moreover, the twofirst horizontal arrows are isomorphisms. Hence, the bottom horizontal map is also an isomorphism.This proves the result. (cid:3)
Remark 11. The third condition of Proposition 23 is true if in particular, the categories D and D (cid:48) have coreflexive equalisers that are preserved by F D , R and R (cid:48) .Remark 12. One has the same result for right adjoints of the functors − (cid:2) Y and − (cid:2) cog W . Itsuffices to apply the result to the opposite pairing Y (cid:2) op X = X (cid:2) Y. The example of chain complexes.
Let Ch be the category of chain complexes of modules overa ring (cid:75) and let Mod gr (cid:75) be the category of (cid:90) -graded (cid:75) -modules. Let us denote by U d the forgetfulfunctor from chain complexes to graded modules. Let us notice that the categories Ch and Mod gr (cid:75) are complete and cocomplete and that U d preserves limits and colimits. Definition 55.
We call a monad M (resp. a comonad Q ) on chain complexes "computed on gradedmodules" if there exists a monad M gr (resp. a comonad Q gr ) on graded modules so that we haveequalities U d ◦ M = M gr ◦ U d U d ◦ Q = Q gr ◦ U d that are consistent with respect to the structural morphism of monads and comonads. This determinesuniquely M gr and Q gr .The following diagrams of categories are commutative Ch × Ch ChMod gr (cid:75) × Mod gr (cid:75) Mod gr (cid:75) ; ⊗⊗ Ch op × Ch Ch ( Mod gr (cid:75) ) op × Mod gr (cid:75) Mod gr (cid:75) . [ − , − ][ − , − ] Moreover, the functors X ⊗ − , − ⊗ X , have the same right adjoint given by [ X, − ] and the functor [ − , X ] from Ch op to Ch has a left adjoint given actually by the same formula [ − , X ] .Let us consider three comonads on chain complexes Q, R, O and two monads
M, N . We supposethat these monads and comonads are computed on graded modules, that the comonads preservecoreflexive equalisers and that the monads preserve reflexive coequalisers.3.4.1.
Coalgebras in chain complexes.
Let us consider a bifunctor − ⊗ − : Cog Ch ( Q ) × Cog Ch ( R ) → Cog Ch ( O ) that lifts the tensor product of chain complexes. Since these comonads are computed at the level ofgraded modules, the bifunctor between categories of coalgebras described above lifts another bifunctor − ⊗ − : Cog
Mod gr (cid:75) ( Q gr ) × Cog
Mod gr (cid:75) ( R gr ) → Cog
Mod gr (cid:75) ( O gr ) , APPING COALGEBRAS I COMONADS 27 that lifts itself the tensor product of graded modules. By the adjoint lifting theorem as used in thissection, we obtain four bifunctors (cid:104)− , −(cid:105) : Cog Ch ( O ) × Cog Ch ( Q ) op → Cog Ch ( R ) ; {− , −} : Cog Ch ( R ) op × Cog Ch ( O ) → Cog Ch ( Q ) ; (cid:104)− , −(cid:105) gr : Cog
Mod gr (cid:75) ( O gr ) × Cog
Mod gr (cid:75) ( Q gr ) op → Cog
Mod gr (cid:75) ( R gr ) ; {− , −} gr : Cog
Mod gr (cid:75) ( R gr ) op × Cog
Mod gr (cid:75) ( R gr ) → Cog
Mod gr (cid:75) ( Q ) gr ; together with natural isomorphisms hom ( W, (cid:104) Z, V (cid:105) ) (cid:39) hom ( V ⊗ W, Z ) (cid:39) hom ( V, { W, Z } ) ;hom ( W (cid:48) , (cid:104) Z (cid:48) , V (cid:48) (cid:105) gr ) (cid:39) hom ( V (cid:48) ⊗ W (cid:48) , Z (cid:48) ) (cid:39) hom ( V (cid:48) , { W (cid:48) , Z (cid:48) } gr ) ; for any Q, R, O, Q gr , R gr , O gr -coalgebras respectively V, W, Z, V (cid:48) , W (cid:48) , Z (cid:48) . From Proposition 23, weget that the natural morphisms U d,cog ( (cid:104) Z, V (cid:105) ) → (cid:104) U d,cog ( Z ) , U d,cog ( V ) (cid:105) gr U d,cog ( { W, Z } ) → { U d,cog ( W ) , U d,cog ( Z ) } gr are isomorphisms.3.4.2. Algebras in chain complexes.
Let us consider a bifunctor [ − , − ] : Cog Ch ( Q ) op × Alg Ch ( M ) → Alg Ch ( N ) that lifts the internal hom of chain complexes. Since M, N, Q are computed at the level of gradedmodules, the bifunctor between categories of algebras and coalgebras described above lifts anotherbifunctor [ − , − ] : Cog
Mod gr (cid:75) ( Q gr ) op × Alg
Mod gr (cid:75) ( M gr ) → Cog
Mod gr (cid:75) ( N gr ) , that lifts itself the internal hom of graded modules. Thus, from the adjoint lifting theorem, we obtainfour bifunctors − (cid:2) − : Cog Ch ( Q ) × Cog Ch ( N ) → Alg Ch ( M ) ; {− , −} : Alg Ch ( N ) op × Alg Ch ( M ) → Cog Ch ( Q ) ; − (cid:2) gr − :: Cog
Mod gr (cid:75) ( Q gr ) × Alg
Mod gr (cid:75) ( N gr ) → Cog
Mod gr (cid:75) ( M gr ) ; {− , −} gr : Alg
Mod gr (cid:75) ( N gr ) op × Alg
Mod gr (cid:75) ( M gr ) → Cog
Mod gr (cid:75) ( Q gr ) ; together with natural isomorphisms hom ( V (cid:2) B, A ) (cid:39) hom ( B, [ V, A ]) (cid:39) hom ( V, { B, A } ) ;hom ( V (cid:48) (cid:2) gr B (cid:48) , A (cid:48) ) (cid:39) hom ( B (cid:48) , [ V (cid:48) , A (cid:48) ]) (cid:39) hom ( V (cid:48) , { B (cid:48) , A (cid:48) } gr ) ; for any Q, Q gr -coalgebras V, V (cid:48) and any
M, N, M gr , N gr -algebras A, B, A (cid:48) , B (cid:48) . From Proposition 23,we get that the natural morphisms U d,cog ( V ) (cid:2) gr U d,alg ( B ) → U d,alg ( V (cid:2) B ) U d,cog ( { B, A } ) → { U d,alg ( B ) , U d,alg ( A ) } gr are isomorphisms.Appendix A. Category enriched tensored and cotensored over a monoidal categoryIn this appendix, we recall some notions related to categories enriched, tensored and cotensoredover a monoidal category. A.1.
Category tensored over a monoidal category.Definition 56.
A category C is said to be tensored over a monoidal category ( E , ⊗ , (cid:49) ) if it is equippedwith the structure of a lax E -module, that is there exists an oplax monoidal functor E → Fun ( C , C ) . Equivalently, there exists a bifunctor − (cid:2) − : E × C → C together with natural transformations (cid:49) (cid:2) X → X , (V (cid:48) ⊗ V) (cid:2) X → V (cid:48) (cid:2) (V (cid:2) X ) , that satisfy coherences.Let E and F be two monoidal categories and let G : E → F be a lax monoidal functor. Definition 57.
Let C and D be respectively a lax E -module a lax F -module. A G -strength on F is theadditional structure of a lax morphism of lax modules that is a natural morphism G (V) (cid:2) F ( X ) → F (V (cid:2) X ) for any V ∈ E and any X ∈ C , so that the following diagrams commute ( G (V) ⊗ G (V (cid:48) )) (cid:2) F ( X ) G (V) (cid:2) ( G (V (cid:48) ) (cid:2) F ( X )) G (V) (cid:2) F (V (cid:48) (cid:2) X ) G (V ⊗ V (cid:48) ) (cid:2) F ( X ) F ((V ⊗ V (cid:48) ) (cid:2) X ) F (V (cid:2) (V (cid:48) (cid:2) X )); (cid:49) (cid:2) F ( X ) G ( (cid:49) ) (cid:2) F ( X ) F ( X ) F ( (cid:49) (cid:2) X ) . In the case where all the structural morphisms G (V) (cid:2) F ( X ) → F (V (cid:2) X ) are isomorphisms, one talksabout G -tensorial strength isomorphism. In the case where G is the identity functor of E , one talksabout a E -tensorial strength for F .Let us consider two lax monoidal functors G, G (cid:48) : E → F between monoidal categories, a monoidalnatural transformation G → G (cid:48) , two categories C and D tensored over respectively E and F and twofunctors F, F (cid:48) : C → D . Let us suppose that F is equipped with a G -strength and that F (cid:48) is equippedwith a G (cid:48) -strength. Definition 58.
A strong natural transformation from F to F (cid:48) , with respect to their strength andwith respect to the monoidal natural transfomation G → G (cid:48) is a natural transformation so that thefollowing diagram commutes G (V) (cid:2) F ( X ) F (V (cid:2) X ) G (cid:48) (V) (cid:2) F (cid:48) ( X ) F (cid:48) (V (cid:2) X ) for any X ∈ C , V ∈ E .A.2. Tensorisation, cotensorisation and enrichment.Definition 59.
For any monoidal category ( E , ⊗ , (cid:49) ) , we denote by E tr = ( E , ⊗ , (cid:49) ) the transposedmonoidal category of E , that is the same category with the opposite monoidal structure X ⊗ tr Y := Y ⊗ X. Definition 60 (Category cotensored over a monoidal category) . Let ( E , ⊗ , (cid:49) ) be a monoidal categoryand let C be a category. We say that C is cotensored over E if there exists a bifunctor (cid:104)− , −(cid:105) : C × E op → C APPING COALGEBRAS I COMONADS 29 together with natural transformations X → (cid:104) X, (cid:49) (cid:105) , (cid:104)(cid:104) X, V (cid:105) , V (cid:48) (cid:105) → (cid:104) X, V ⊗ V (cid:48) (cid:105) , that makes the category C tensored over E optr = ( E op , ⊗ tr , (cid:49) ) . Definition 61.
A category enriched over a monoidal category E , (or E -category) C is the data of aset (possibly large) called the set of objects or the set of colours, an object { X, Y } ∈ E for any twocolours X, Y and morphisms γ : { Y, X } ⊗ {
X, Y } → {
X, Z } ; η X : (cid:49) → { X, X } for any three colours X, Y, Z that define a unital associative composition. Moreover, a morphism of E -categories from C to D is the data of a function on colours φ and morphisms { X, Y } → { φ ( X ) , φ ( Y ) } for any two colours X, Y of C that commute with units and compositions. This defines the category Cat E of E -categories. Definition 62.
A category tensored-cotensored-enriched (TCE for short) over E is the data of acategory C equipped with three bifunctors: − (cid:2) − : E × C → C {− , −} : C op × C → E (cid:104)− , −(cid:105) : C × E op → C , with natural isomorphisms, hom C (V (cid:2) X, Y ) (cid:39) hom E (V , { X, Y } ) (cid:39) hom C ( X, (cid:104) Y, V (cid:105) ) , and with (cid:46) a structure of a tensorisation on − (cid:2) − , (cid:46) or equivalently a structure of a cotensorisation on (cid:104)− , −(cid:105) , (cid:46) or equivalently a structure of an enrichment on {− , −} .Given a tensorisation − (cid:2) − , the composition and the unit on {− , −} are the adjoints maps of themorphisms ( { Y, Z } ⊗ {
X, Y } ) (cid:2) X → { Y, Z } (cid:2) ( { X, Y } (cid:2) X ) → { Y, Z } (cid:2) Y → Z , (cid:49) (cid:2) X → X. Given a cotensorisation (cid:104)− , −(cid:105) , the composition and the unit on {− , −} are the adjoints maps of themorphisms of the map X → (cid:104) Y, { X, Y }(cid:105) → (cid:104)(cid:104) Z, { Y, Z }(cid:105) , { X, Y }(cid:105) → (cid:104) Y, { Y, Z } ⊗ {
X, Y }(cid:105) ,X → (cid:104) X, (cid:49) (cid:105) . Conversely, given an enrichment {− , −} , the structural tensorisation morphisms for − (cid:2) − are adjointsto the maps V ⊗ V (cid:48) → { V (cid:48) (cid:2) X, V (cid:2) (V (cid:48) (cid:2) X ) } ⊗ { X, V (cid:48) (cid:2) X } → { X, V (cid:2) (V (cid:48) (cid:2) X ) } , (cid:49) → { X, X } , and the structural cotensorisation morphisms for (cid:104)− , −(cid:105) are adjoints to the maps V ⊗ V (cid:48) → {(cid:104) X, V (cid:105) , X } ⊗ {(cid:104)(cid:104) X, V (cid:105) , V (cid:48) (cid:105) , (cid:104) X, V (cid:105)} → {(cid:104)(cid:104) X, V (cid:105) , V (cid:48) (cid:105) , X } , (cid:49) → { X, X } . Remark 13. Given an adjunction between monoidal categories
E F LR where L is monoidal (and then R is lax monoidal), then any category C TCE over F is also TCE over E . Definition 63.
A biclosed monoidal category is a monoidal category ( E , ⊗ , (cid:49) ) TCE over itself in away so that the tensorisation is the tensor product of E . If it is symmetric, one says simply that it isclosed.Notation. When dealing with a biclosed (or a closed) monoidal category, the enrichment will usuallybe denoted [ − , − ] .A.3. Strength in the context of categories enriched-tensored and cotensored.
Let G : E → F be lax monoidal functor between monoidal categories. Moreover, let C and D be two categories TCErespectively over E and F and let F : C → D be a functor.A G -strength on F is equivalent to the data of natural morphisms F ( (cid:104) X, V (cid:105) ) → (cid:104) F ( X ) , G (V) (cid:105) forany V ∈ E and any X ∈ C , so that the following diagrams commute F ( (cid:104)(cid:104) X, V (cid:105) , W (cid:105) ) (cid:104) F ( (cid:104) X, V (cid:105) ) , G (W) (cid:105) (cid:104)(cid:104) F ( X ) , G (V) (cid:105) , G (W) (cid:105) F ( (cid:104) X, V ⊗ W (cid:105) ) (cid:104) F ( X ) , G (V ⊗ W) (cid:105) (cid:104) F ( X ) , G (V) ⊗ G (W) (cid:105) ; F ( (cid:104) X, (cid:49) (cid:105) ) (cid:104) F ( X ) , G ( (cid:49) ) (cid:105) F ( X ) (cid:104) F ( X ) , (cid:49) (cid:105) . It is also equivalent to the data of natural morphisms G { X, Y } → {
F X, F Y } for any X, Y ∈ C , sothat the following diagrams commute G ( { Y, Z } ) ⊗ G ( { X, Y } ) { F Y, F Z } ⊗ {
F X, F Y } G ( { Y, Z } ⊗ {
X, Y } ) { X, Z } G ( { F X, F Z } ); (cid:49) G ( (cid:49) ) { F X, F X } G ( { X, X } ) . Moreover, let us consider another lax monoidal functor G (cid:48) : E → F a monoidal natural transfor-mation G → G (cid:48) , and another functor F (cid:48) : C → D equipped with a G (cid:48) -strength. Then, a naturaltransformation F → F (cid:48) is strong if and only if the following diagram commutes F ( (cid:104) X, V (cid:105) ) (cid:104) F ( X ) , G (V) (cid:105) F (cid:48) ( (cid:104) X, V (cid:105) ) (cid:104) F (cid:48) ( X ) , G (cid:48) (V) (cid:105) (cid:104) F (cid:48) ( X ) , G (V) (cid:105) . for any X ∈ C , V ∈ E , if and only if the following diagram commutes G ( { X, Y } ) { F ( X ) , F ( Y ) } G (cid:48) ( { X, Y } ) { F (cid:48) ( X ) , F (cid:48) ( Y ) } { F ( X ) , F (cid:48) ( Y ) } . for any X, Y ∈ C . APPING COALGEBRAS I COMONADS 31
A.4.
Strong adjunctions.
Let us consider an adjunction
C D . LR We suppose that C and D are TCE over a monoidal category E and F . Definition 64.
The adjunction L (cid:97) R is said to be strong if L and R are equipped with strengthswith respect to E so that the natural transformations Id → RL and LR → Id are strong.Equivalently, by doctrinal adjunction, L (cid:97) R is strong if L is equipped with a strength so that thenatural morphism V (cid:2) L ( X ) → L ( V (cid:2) X ) is an isomorphism for any V, X ∈ E × C . Then, the strength on R is given by the formula V (cid:2) R ( Y ) → RL ( V (cid:2) R ( Y )) (cid:39) R ( V (cid:2) LR ( Y )) → R ( V (cid:2) Y ) . One has also an adjunction relating the opposite categories D op C op . R op L op The cotensorisation of C and D over E are actually tensorisation of C op and D op over E tr . Proposition 24.
The structure of a strong adjunction on L (cid:97) R with respect the tensorisation of C and D over E is equivalent to the structure on a strong adjunction on R op (cid:97) L op with respect thetensorisation of D op and C op over E tr .Proof. In Subsection A.3, we saw that we have a one to one correspondence between the strengthson L (resp. R ) with respect to E and the strengths on L op (resp. R op ) with respect to E tr . Then,given a pair of strength on L and R , the natural transformations Id → RL and LR → Id are strong ifand only if the natural transformations Id → L op R op and R op L op → Id are strong with respect to theinduced strength on L op and R op . (cid:3) Proposition 25.
Let us suppose that the functors L and R are equipped with strengths. Then, thefollowing assertions are equivalent. (1) the natural transformations Id → RL and LR → Id are strong (hence, the adjunction L (cid:97) R is strong); (2) the natural maps { LX, Y } → {
RLX, RY } → {
X, RY } ; { X, RY } → {
LX, LRY } → {
LX, Y } ; are isomorphisms inverse to each other for any X, Y ∈ C × D .Proof. We know from Subsection A.3 that the fact that the natural transformations Id → RL and LR → Id are strong is equivalent to the commutation of the following two diagrams { X, Y } {
RX, RY }{ LRX, Y } {
LRX, LRY } { X (cid:48) , Y (cid:48) } { LX (cid:48) , LY (cid:48) }{ X (cid:48) , RLY (cid:48) } { RLX (cid:48) , RLY (cid:48) } for any X, X (cid:48) , Y, Y (cid:48) .Let us suppose (1). Using the commutation of the diagram just above, it is straightforward toprove that the composite maps { LX, Y } → {
X, RY } → {
LX, Y } , { X, RY } → {
LX, Y } → {
X, RY } , are identities. Conversely, let us suppose (2). By naturality, the two following diagrams are commutative { X, Y } {
RX, RY }{ LRX, Y } {
RLRX, RY } { X (cid:48) , Y (cid:48) } { LX (cid:48) , LY (cid:48) }{ X (cid:48) , RLY (cid:48) } { LX (cid:48) , LRLY (cid:48) } for any X, X (cid:48) , Y, Y (cid:48) . Then, the commutation of the two previous squares follows from the fact thatthe maps { LRX, Y } → {
RX, RY }{ X (cid:48) , RLY (cid:48) } → { LX (cid:48) , LY (cid:48) } are respective inverses of the maps that appear in these previous diagrams. (cid:3) A.5.
Homotopical enrichment.Definition 65 (Homotopical enrichment) . Let M be a model category and let E be a monoidal modelcategory. We say that M is homotopically TCE over E if it TCE over E and if for any cofibration f : X → X (cid:48) in M and any fibration g : Y → Y (cid:48) in M , the morphism in E : { X (cid:48) , Y } → { X (cid:48) , Y } × { X,Y (cid:48) } { X, Y } is a fibration. Moreover, we require this morphism to be a weak equivalence whenever f or g is aweak equivalence.This is equivalent to the fact that for any cofibration f : X → Y in M and any cofibration g : V → W in E , the morphism V (cid:2) Y (cid:97) V (cid:2) X W (cid:2) X → W (cid:2) Y is a cofibration and it is acyclic whenever f or g is acyclic. This is also equivalent to the fact that forany fibration f : X → Y in M and any cofibration g : V → W in E , the morphism (cid:104) X, W (cid:105) → (cid:104) X, V (cid:105) × (cid:104) Y, V (cid:105) (cid:104) Y, W (cid:105) is a fibration and it is acyclic whenever f or g is acyclic.Appendix B. Monads, comonads, limits and colimits Proposition 26.
Let M be a monad on a category C . Then, the functor U M preserves and createslimits.Proof. The functor U M preserves limits since it is right adjoint. Moreover, it reflects limits since it isconservative. Finally, for any diagram D : I → Alg C ( M ) , if U M ◦ D has a limit, then this limit has thestructure of a M -algebra given by M (lim ←− U M D ) → lim ←− MU M D = lim ←− U M T M U M D → lim ←− U M D. This M -algebra is the limit of the diagram D . (cid:3) Corollary 8.
Let Q be a comonad on a category C . Then, the functor U Q preserves and createscolimits. Proposition 27.
Let M be a monad on a category C . Let us suppose C has all reflexive coequalisersand that M preserves these reflexive coequalisers. Then the category of M -algebras has all reflexivecoequalisers and these are preserved by the functor U M .Proof. Let I be the category generated by (cid:46) two objects and ; (cid:46) three nontrivial morphisms f , g : 0 → and h : 1 → APPING COALGEBRAS I COMONADS 33 with the relation f s = gs = Id . For any diagram D : I → Alg C ( M ) , let A be the colimit of thefunctor U M ◦ D . Then, A has the structure of a M algebra as follows M (A) = M (colim U M ◦ D ) (cid:39) colim( MU M ◦ D ) = colim( U M T M U M ◦ D ) → colim U M ◦ D = A . A straightforward check shows that this defines the structure of a M -algebra on A which gives thecolimit of the diagram D . It is then clear that U M preserves reflexive coequalisers. (cid:3) Proposition 28.
In the context of Proposition 27, let us suppose that C is cocomplete. Then thecategory of M -algebras is cocomplete.Proof. Given a family of M -algebras (A i ) i ∈ J , their coproduct is given by the reflexive coequaliser ofthe following diagram T M ( (cid:96) i M (A i )) T M ( (cid:96) i A i ) . It is enough to have all reflexive coequalisers and all coproducts to be cocomplete. (cid:3)
Corollary 9.
Let Q be a comonad on a category C . Let us suppose C has all coreflexive equalisers andthat Q preserves these coreflexive equalisers. Then the category of Q -coalgebras has all coreflexiveequalisers and these are preserved by the functor U Q . Corollary 10.
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