Factorisations of distributive laws

Recently, Boehm and Stefan constructed duplicial (paracyclic) objects from distributive laws between (co)monads. Here we define the category of factorisations of a distributive law, show that it acts on this construction, and give some explicit examples.


Introduction
Distributive laws between monads were originally defined by Beck in [Bec69] and correspond to monad structures on the composite of the two monads. They have found many applications in mathematics as well as computer science; see e.g. [Bur09,Lod08,Tur96,VW06].
Recently, distributive laws have been used by Böhm and Ştefan [BŞ08] to construct new examples of duplicial objects [DK85], and hence cyclic homology theories. The paradigmatic example of such a theory is the cyclic homology HCpAq of an associative algebra A [Con85,Tsy83]. It was observed by Kustermans, Murphy, and Tuset [KMT03] that the functor HC can be twisted by automorphisms of A. The aim of the present paper is to extend this procedure to any duplicial object defined by a distributive law.
Given a distributive law χ we define in Section 3.1 the category F pχq of its factorisations. The main technical results are the definition of a monoidal structure on F pχq (Lemma 3.2 and Proposition 3.3), a characterisation of the comonoids in F pχq (Proposition 3.5), and the definition of actions of F pχq on the category of admissible data (septuples in [BŞ08]) which turns the latter into an F pχq-bimodule category (Theorem 3.8 and Corollary 3.9).
The remainder of the paper is devoted to examples. We begin by considering factorisations of distributive laws on Eilenberg-Moore categories, interpreting these as flat connections (Section 4.1). In particular, we present the twisting of cyclic homology in this framework (Section 4.2). We then describe examples arising from Hopf algebras (Section 4.3). The final examples are concerned with BD-laws, braidings (Section 4.4), and quantum doubles of Hopf algebras (Section 4.5).
Throughout this paper, A, B, C . . . are categories, A, B, C, . . . are functors, and greek letters are used to denote natural transformations. We use˝to denote composition of morphisms and vertical composition of natural transformations. The composition of functors and the horizontal composition of natural transformations will be denoted simply by concatenation. The identity morphism, functor and natural transformation is denoted by id. However, we denote the horizontal composition α id A β by αAβ.
Acknowledgements. UK is supported by the EPSRC First Grant "Hopf Algebroids and Operads" and the Polish Government Grant 2012/06/M/ST1/00169. PS is supported by an EPSRC Doctoral Training Award.

Preliminaries
In this section, we recall basic definitions and results that are needed later.
2.1. (Co)monads. Let A be a category.
Definition 2.1. A comonad on A is a triple C " pC, ∆, εq where C is an endofunctor on A, and ∆ : C ÝÑ CC and ε : C ÝÑ id A are natural transformations such that C∆˝∆ " ∆C˝∆, εC˝∆ " id C " Cε˝∆, that is, the two diagrams In other words, a comonad is a comonoid (or coalgebra) in the monoidal category rA, As of endofunctors on A (with composition as tensor product). Dually, a monad on a category C is a monoid (algebra) in rC, Cs.
2.2. Module categories. Next, we recall the notion of a module category (also known as an M-category) over a monoidal category pM, b, 1q. For the purpose of this paper, all monoidal categories and their module categories are strict, and by abuse of notation we will write M to refer to the whole triple pM, b, 1q.
Definition 2.2. A left module category for M is a pair pC, ✄q where C is a category and ✄ : MˆC ÝÑ C is a functor such that we have functorial identities for all objects X, Y in M and P in C. We call ✄ the left action of M on C.
Dually, one defines a right module category pD, ✁q. A bimodule category is a triple pC, ✄, ✁q where pC, ✄q and pC, ✁q are right respectively left module categories and the actions commute, i.e. for all objects X, Y in M and P in C we have again functorially in X, Y and P . We immediately have the following.
Lemma 2.3. Let pC, ✄q and pD, ✁q be left respectively right module categories. Then CˆD is a bimodule category with actions given by 2.3. Eilenberg-Moore categories. The comonads we are mostly interested in arise as restrictions of monads to their Eilenberg-Moore categories.
Definition 2.4. Let pC, ✄q be a left module category for a monoidal category M, and let B " pB, µ, ηq be a monoid in M. The Eilenberg-Moore category of B, denoted by C B , is the category whose objects are pairs pX, αq, where X is an object of C and α : B ✄ X ÝÑ X is a morphism in C such that the diagrams commute, and whose morphisms f : pX, αq ÝÑ pX 1 , α 1 q are morphisms f : X ÝÑ X 1 in C such that the diagram Now observe that the monoid B defines a comonadB " pB,∆,εq on A " C B whereB is defined on objects and morphisms bỹ and∆,ε are defined on objects pX, αq by In particular, every category C is in an obvious way a module category over rC, Cs. In this case, our definition of Eilenberg-Moore category of a monad B on C is the same as the usual definition [ML98,p. 139].
2.4. Distributive laws. Next we define distributive laws. Note that we consider them between (co)monads and arbitrary endofunctors as is common in the computer science literature, see e.g. [Tur96].
Definition 2.5. Let T " pT, ∆, εq be a comonad on A and let C be an endofunctor on A. A distributive law between the comonad T and the endofunctor C is a transformation χ : T C ÝÑ CT such that the two diagrams commute. We denote this by χ : T ÝÑ C. Analogously, we define a distributive law χ : T ÝÑ C between an endofunctor T and a comonad C. A comonad distributive law χ : T ÝÑ C is a transformation χ which is a distributive law between endofunctors and comonads in both ways.
Dually, we can define distributive laws involving monads; distributive laws from a monad to a comonad are usually called mixed distributive laws.
One application of distributive laws is to lift endofunctors to Eilenberg-Moore categories: let B be a monad on a category C and θ : B ÝÑ D be a distributive law. We define a functorD : C B ÝÑ C B as follows. On objects we definẽ DpX, αq " pDX, Dα˝θ X q and we defineDf " Df on morphisms. The distributive law θ lifts to give one θ :B ÝÑD whereB is the comonad described in Section 2.3. If D is part of a comonad D " pD, ∆, εq, and θ is a mixed distributive law B ÝÑ D, thenD is part of a comonadD " pD, ∆, εq and θ lifts to a comonad distributive law θ :B ÝÑD. See [Bec69,Bur73] for more details on distributive laws.
2.5. The categories of χ-coalgebras. Let T "`T, ∆ T , ε T˘a nd C "`C, ∆ C , ε Cb e comonads on A, and let χ : T ÝÑ C be a distributive law.
Definition 2.6. A right χ-coalgebra is a triple pM, X , ρq where X is a category, M : X ÝÑ A is a functor and ρ : T M ÝÑ CM is a natural transformation such that the diagrams A morphism of right χ-coalgebras between pM, X , ρq and pM 1 , X 1 , ρ 1 q is a pair pϕ, F q, where F : X ÝÑ X 1 is a functor and ϕ : M ÝÑ M 1 F is a natural transformation such that the diagram commutes. We define composition of morphisms by pϕ 1 , F 1 q˝pϕ, F q " pϕ 1 F˝ϕ, F 1 F q and we define identity morphisms by id pM,X ,ρq " pid M , id X q. We denote the category of right χ-coalgebras by Rpχq.
2.6. The construction of Böhm and Ştefan. Finally, we recall the construction of duplicial functors from a comonad distributive law χ : T ÝÑ C on a category A due to Böhm and Ştefan.
Definition 2.7. The category of admissible data over χ is the product category Spχq :" RpχqˆLpχq.
To every admissible datum pM, X , ρ, N, Y, λq there is an associated duplicial functor X ÝÑ Y defined by which is given objectwise by taking the bar resolution of M with respect to the comonad T, and then applying the functor N . If Y is an abelian category, we can apply the duplicial functor to an object X in X resulting in a duplicial object in Y of which we can take the cyclic homology.
This construction, which unifies and generalises the definition of the cyclic homology of an associative algebra as well as Hopf-cyclic homology, is detailed in [BŞ08] for the case that X " t˚u is the terminal category.

Theory
3.1. The category of factorisations F pχq. Throughout this section, let T " T, ∆ T , ε T˘a nd C "`C, ∆ C , ε C˘b e comonads on a category A, and let χ : T ÝÑ C be a distributive law. The main definition of the present paper is the following: Definition 3.1. A factorisation of χ is a triple pΣ, σ, γq where Σ is an endofunctor on A, and σ : T ÝÑ Σ and γ : Σ ÝÑ C are distributive laws satisfying the Yang-Baxter condition; that is, the hexagon commutes. A morphism α : pΣ, σ, γq ÝÑ pΣ 1 , σ 1 , γ 1 q of factorisations is a natural transformation α : Σ ÝÑ Σ 1 which is compatible with T and C in the sense that the diagrams commute. There are identity morphisms id pΣ,σ,γq " id Σ , and composition of morphisms is given by the vertical composite. This defines the category of factorisations which we denote by F pχq.
Similarly, we may also define factorisations of a monad or mixed distributive law.
Lemma 3.2. The assignment b is a well-defined functor.
Proof. Firstly, b is well-defined on objects if Σσ 1˝σ Σ 1 and γΣ 1˝Σ γ 1 satisfy the Yang-Baxter condition. Consider the following diagram The left square commutes by naturality of σ and the right square commutes by naturality of γ. The inner hexagons commute by the Yang-Baxter conditions. Therefore, the outer hexagon commutes, so the required condition is satisfied. Secondly, let α : pΣ, σ, γq ÝÑ pΓ, κ, νq and β : pΣ 1 , σ 1 , γ 1 q ÝÑ pΓ 1 , κ 1 , ν 1 q be morphisms in F pχq. Consider the diagram The bottom-left square commutes by naturality of α, the top-right square commutes by naturality of κ, and the two remaining inner squares commute since α and β are compatible with T . Therefore, the outer square commutes and α b β is compatible with T . A similar argument shows that α b β is compatible with C. It is clear that b respects composition of morphisms and identity morphisms. Therefore, b is well-defined on morphisms.
Let 1 denote the trivial factorisation pid A , id T , id C q.
Proposition 3.3. The triple pF pχq, b, 1q is a strict monoidal category.
Proposition 3.5. A factorisation pΣ, σ, γq is a comonoid in F pχq if and only if Σ is part of a comonad and σ, γ are distributive laws of comonads.
Dually, a factorisation pΣ, σ, γq is a monoid in F pχq if and only if Σ is part of a monad and σ, γ are mixed distributive laws between monads and comonads.
Corollary 3.6. Let χ : id A ÝÑ id A be the trivial distributive law given by the identity. Then pT, ∆, εq is a comonad on A if and only if pT, id T , id T q is a comonoid in F pχq, and pB, µ, ηq is a monad on A if and only if pB, id B , id B q is a monoid in F pχq.
Proposition 3.7. The assignment ✄ is a well-defined functor.
Proof. Consider the diagram The top-left and bottom rectangles commute by the distributive law axioms, the middle-left rectangle commutes because pM, X , ρq is a right χ-coalgebra, the topright diagram commutes by the Yang-Baxter condition, and the remaining squares commute by naturality of σ, γ. Therefore, the outer rectangle commutes.
Consider the triangle The middle triangle commutes because pM, X , ρq is a right χ-coalgebra, and the other two inner triangles commute by the distributive law axioms. Therefore, the outer triangle commutes. This shows that ✄ is well-defined on objects.
Let pϕ, F q : pM, X , ρq ÝÑ pM 1 , X 1 , ρ 1 q and α : pΣ, σ, γq ÝÑ pΣ 1 , σ 1 , γ 1 q be morphisms of right χ-coalgebras and factorisations, respectively. Consider the diagram The top-left square commutes since α is compatible with T , the top-right square commutes by naturality of σ, the bottom-left square commutes by naturality of α, and the bottom-right square commutes since pϕ, F q is a right χ-coalgebra morphism. Thus the outer square commutes, which shows that α✄pϕ, F q is a right χ-coalgebra morphism.
It is clear that ✄ respects identities and composition of morphisms (because the vertical and horizontal compositions of natural transformations are compatible with each other), so ✄ is well-defined on morphisms.
Theorem 3.8. The category Rpχq is a strict left module category for F pχq, with left action given by the functor ✄. Furthermore, the category Lpχq is a strict right module category for F pχq, with right action given by the functor ✁.
Corollary 3.9. The category Spχq is a strict bimodule category for F pχq.
Proof. This follows immediately by applying Lemma 2.3 to Theorem 3.8.

Flat connections.
Let B " pB, µ, ηq be a monad on a category C. The forgetful functor U : C B ÝÑ C has a left adjoint F defined by F pX, αq " pBX, µ X q, F pf q " Bf.
The unit of this adjunction is given by η and the counit isε pX,αq " α. LetB denote the functor F U and let∆ denote the natural transformation F η U . The adjunction gives rise to a comonadB " pB,∆,εq, which is the same as the comonad discussed in Section 2.3. Let Σ : C B ÝÑ C B be an endofunctor. For every object pX, αq in C B there are natural isomorphisms C B pBΣpX, αq, ΣBpX, αqq -CpU ΣpX, αq, U ΣBpX, αqq given by the adjunction, so there is a one-to-one correspondence between natural transformations σ :BΣ ÝÑ ΣB and natural transformations ∇ : U Σ ÝÑ U ΣB. In fact, σ is a distributive law if and only if the diagrams Definition 4.1. We say that the natural transformation σ is a connection ifε is compatible with σ, i.e. the second diagram above commutes for the corresponding natural transformation ∇. We say that a connection σ is flat if∆ is compatible with σ, i.e. σ is a distributive law, or equivalently, both diagrams above commute.
The terminology is motivated by the special case discussed in detail in the following section. 4.2. pA, Aq-bimodules. Let k be a commutative ring and let A be a unital associative algebra over k. Let C " A-Mod be the category of left A-modules. The functor B "´b k A : C ÝÑ C, together with the natural transformations defines a monad B on C which lifts to a comonadB on C B . The latter is isomorphic to the category of pA, Aq-bimodules (with symmetric action of k). The functor D " A b k´: C ÝÑ C, together with the natural transformations There is a mixed distributive law θ : B ÝÑ D given by rebracketing on components so this lifts to a comonad distributive law θ :B ÝÑD.
Let N be an pA, Aq-bimodule and Σ : C B ÝÑ C B be the functor defined by ΣpM q " M b A N . We have that ΣD "DΣ so the identity id ΣD : Σ ÝÑ D is a distributive law.
In this case, the components of a natural transformation ∇ : U Σ ÝÑ U ΣB are given by a left A-linear map The corresponding natural transformation σ :B ÝÑ Σ is given by The natural transformation ∇ defines a connection if and only if each ∇ M splits the quotient map M b k N ÝÑ M b A N . Taking M " A yields an A-linear splitting of the action Ab k N ÝÑ N , so N is k-relative projective. Conversely, given a splitting n Þ Ñ n p´1q b n p0q of the action, we obtain ∇ M as ∇ M pm b A nq " mn p´1q b n p0q .
Thus we have: Proposition 4.2. The functor Σ admits a connection σ if and only if N is krelative projective as a left A-module.
Composing ∇ A with the noncommutative De Rham differential d : A ÝÑ Ω 1 A,k , a Þ ÝÑ 1 b a´a b 1 gives the notion of connection in noncommutative geometry [Con94,III.3.5].
If N is not just k-relative projective but k-relative free, i.e. N -Ab k V as left Amodules, for some k-module V , then the assignment ∇ M pmb A pabvqq " mabp1bvq defines a flat connection. Thus we have: Proposition 4.3. The triple pΣ, σ, id ΣD q is a factorisation of θ.
In particular, let σ : A ÝÑ A be an algebra map and N " A σ , the pA, Aqbimodule which is A as a left A-module with right action of a P A given by right multiplication by σpaq. Then we have ΣpM q " M b A A σ -M σ . Since A σ is free as a left A-module we get a factorisation pΣ, σ, id ΣD q by Proposition 4.3, where σ :B ÝÑ Σ is the flat connection defined on components by Note that we use σ to denote both the algebra map and the flat connection.
From the general theory developed in Section 3 we obtain therefore an action of the group of endomorphisms of A on the category of admissible data for θ. In particular, we can act on the standard cyclic object associated to A [Con85,Tsy83], which corresponds to the following admissible datum.
Consider A as a functor A : t˚u ÝÑ C B from the one-morphism category to the category of pA, Aq-bimodules. SinceBA "DA " A b k A we have a natural transformation ρ " id Ab k A :BA ÝÑDA. The triple pA, t˚u, ρq is a right θ-coalgebra.
Considering pA, Aq-bimodules as either left or right A e :" A b k A op -modules, we view the zeroth Hochschild homology as a functor H "´b A e A : C B ÝÑ k-Mod. We define a natural transformation λ : HD ÝÑ HB by The pair pH, k-Mod, λq is a left θ-coalgebra, and the duplicial k-module associated to the admissible datum pA, t˚u, ρ, H, k-Mod, λq is indeed the cyclic object defining the cyclic homology HCpAq.
The cyclic homology of the duplicial object associated to the admissible datum pΣ, σ, id ΣD q ✄ pA, t˚u, ρ, H, k-Mod, λq " pA σ , t˚u, ρ˝σ A , H, k-Mod, λq is HC σ pAq, the σ-twisted cyclic homology of A. This was first considered in [KMT03] and is discussed in Section 5.2 of [KK11] in the context of Hopf algebroids. Thus the action of the category of factorisations generalises this twisting procedure. We consider three special cases of this construction. The distributive laws used therein are instances of one defined on the category of right U -modules, where U is a left Hopf algebroid, which is defined and discussed in [KKS].
Example 4.4. Suppose that σ : B ÝÑ B is a monad morphism which is compatible with θ; that is σ : B ÝÑ B is a natural transformation such that the three diagrams commute. The first two diagrams say that σ : B ÝÑ id C is a distributive law. The triple pid C , σ, id SD q is a factorisation of θ : B ÝÑ D, so we get a factorisation pΣ, σ, id ΣD q of θ :B ÝÑD. Explicitly, Σ : C B ÝÑ C B is given by ΣpX, αq " pX, α˝σ X q, Σpf q " f.
Observe that the composition of monad morphisms corresponds under this assignment to the monoidal structure in F pθq, so when viewing the monad morphisms as a monoidal category with composition as tensor product and the identity id B as unit object, we have: Proposition 4.5. The assignment σ Þ ÝÑ pΣ, σ, id ΣD q is a monoidal functor.
The factorisation given in Proposition 4.3 arises in this way.
Example 4.6. Let k be a commutative ring and let U be a Hopf algebra over k. We use Sweedler notation to denote the coproduct See [Swe69,Mon93] for more information about Hopf algebras.
Consider the category C " k-Mod. The functor B "´b k U : C ÝÑ C is part of a monad B where the multiplication is given by the multiplication of the algebra U and the unit is given by the unit of the algebra U . Dually, the functor D " U b k´: C ÝÑ C is part of a comonad, whose structure is given by the comultiplication and counit of the coalgebra U . There is a mixed distributive law θ : B ÝÑ D given by Let P be any right U -module. This defines a functor P b k´: C ÝÑ C. The maps define a distributive law σ : B ÝÑ P b k´a nd the maps define a distributive law γ : P b k´Ý Ñ D. The triple pP b´, σ, γq is a factorisation of θ : B ÝÑ D, and so this gives a factorisation of θ :B ÝÑD in the category C B -Mod -U .
Example 4.7. Let C " k-Mod where k is a commutative ring, and consider the functor B " U b k´: C ÝÑ C. Similarly to Example 4.7, this is simultaneously part of a monad B and a comonad D. There is a mixed distributive law θ : B ÝÑ D given by and a distributive law τ : B ÝÑ B given by If U is commutative (or even just if the antipode S maps into the centre of U ), then pB, τ, θq is a factorisation of θ : B ÝÑ D and so pB, τ, θq is a factorisation of θ :B ÝÑB in C B -U -Mod.

4.4.
Braided distributive laws. Let χ : T ÝÑ C be a comonad distributive law on a category A.
Definition 4.8. A distributive law τ : T ÝÑ T between the comonad T and the endofunctor T is braided with respect to χ if the hexagon commutes. Dually, we say that a distributive law ϕ : C ÝÑ C between the endofunctor C and the comonad C is braided with respect to χ if a similar hexagon commutes.
Clearly, τ is braided if and only if pT, τ, χq is a factorisation of χ, since the above hexagon is just the Yang-Baxter condition in that case. In the dual case, pC, χ, ϕq would be a factorisation of χ.
Example 4.9. In Example 4.7, the distributive law τ is braided with respect to θ.
Example 4.10. Let τ : T ÝÑ T be a BD-law. These are defined in [KLV04] and are exactly those distributive laws which are braided with respect to themselves. Thus pT, τ, τ q is a factorisation of τ .
Example 4.11. For this example we relax the assumption that monoidal categories are strict. Let A be a braided monoidal category with tensor product b, associator morphisms α and braiding morphisms b. Let U " pU, ∆ U , ε U q and V " pV, ∆ V , ε V q be comonoids in A. The comonoids U, V define two comonads U, V with endofunctors U b´, V b´respectively, and three distributive laws χ : U ÝÑ V, τ : U ÝÑ U and ϕ : V ÝÑ V defined by respectively. The distributive laws τ and ϕ are both braided with respect to χ so we get two factorisations pU b´, τ, χq and pV b´, χ, ϕq of χ. By Proposition 3.5 these are both comonoids in F pχq. This example comes from the dual of Example 1.11 in [BŞ09].
4.5. Quantum doubles. In our final example, we consider the distributive laws corresponding to quantum doubles: let B and C be two Hopf algebras over a commutative ring k and R P C b k B be an invertible 2-cycle, meaning that we have where R´1 refers to the multiplicative inverse in the tensor product algebra C b k B and subscripts denote components in C b k C b k B respectively C b k B b k B. We refer to [CP95] for more background information.
The coalgebras B and C define comonads T and C on A " k-Mod given by B b k´a nd C b k´w ith structure maps given by the coproducts and the counits. The 2-cycle R defines a distributive law χ : T ÝÑ C given by In this case, every pB, C op q-bimodule M , that is, a k-module M with two commuting left actions of B and C, gives rise to a factorisation of χ: let Σ : A ÝÑ A be the functor M b k´. We define distributive laws Then a straightforward computation shows that pΣ, σ, γq is a factorisation of χ.
When M " k with trivial actions given by the counits, we recover Example 4.11.