Theory and Applications of Categories, Vol. 30, No. 32, 2015, pp. 1067–1095.

CYCLIC HOMOLOGY ARISING FROM ADJUNCTIONS

NIELS KOWALZIG, ULRICH KRÄHMER, AND PAUL SLEVIN

Abstract. Given a monad and a comonad, one obtains a distributive law between them from lifts of one through an adjunction for the other. In particular, this yields for any bialgebroid the Yetter-Drinfel’d distributive law between the comonad given by a module coalgebra and the monad given by a comodule algebra. It is this self-dual setting that reproduces the cyclic homology of associative and of Hopf algebras in the monadic framework of Böhm and Ştefan. In fact, their approach generates two duplicial objects and morphisms between them which are mutual inverses if and only if the duplicial objects are cyclic. A 2-categorical perspective on the process of twisting coefficients is provided and the rôle of the two notions of bimonad studied in the literature is clarified.

1. Introduction

1.1. Background and aim. The Dold-Kan correspondence generalises chain complexes in abelian categories to general simplicial objects, and thus homological algebra to homotopical algebra. The classical homology theories defined by an augmented algebra (such as group, Lie algebra, Hochschild, de Rham and Poisson homology) become expressed as the homology of suitable comonads 𝕋, defined via simplicial objects C_{𝕋} N, M obtained from the bar construction (see, e.g., [Wei94]).

Connes’ cyclic homology created a new paradigm of homology theories defined in terms of mixed complexes [Kas87, DK85]. The homotopical counterparts are cyclic [Con83] or more generally duplicial objects [DK85, DK87], and Böhm and Ştefan [BŞ08] showed how C_{𝕋} N, M becomes duplicial in the presence of a second comonad 𝕊 compatible in a suitable sense with N, M and 𝕋.

The aim of the present article is to study how the cyclic homology of associative algebras and of Hopf algebras in the original sense of Connes and Moscovici [CM98] fits into this monadic formalism, extending the construction from [KK11], and to clarify the rôle of different notions of bimonad in this generalisation.

1.2. Distributive laws arising from adjunctions. Inspired by [MW14, AC12] we begin by describing the relation of distributive laws between (co)monads and of lifts of

N.K. acknowledges support by UniNA and Compagnia di San Paolo in the framework of the program STAR 2013, U.K. by the EPSRC grant EP/J012718/1 and the Polish Government Grant 2012/06/M/ST1/00169, and P.S. by an EPSRC DTA. We would like to thank Gabriella Böhm, Steve Lack, Tom Leinster, Danny Stevenson, and the referee for helpful suggestions and discussions.

Received by the editors 2015-06-15 and, in revised form, 2015-7-31. Transmitted by Ross Street. Published on 2015-08-04. 2010 Mathematics Subject Classification: 16T05, 18C15, 18C20, 19D55. Key words and phrases: cyclic homology, Hopf algebroids, Hopf monads, distributive laws. \begin{matrix}c \\ ◯ \end{matrix} Niels Kowalzig, Ulrich Krähmer, and Paul Slevin, 2015. Permission to copy for private use granted.

one of them through an adjunction for the other. In particular, we have:

1.3. Theorem. Let F U be an adjunction, 𝔹 : B, μ, η , B UF, and 𝕋 T, Δ, ε , T FU, be the associated (co)monads, and 𝕊 S, Δ^{S}, ε^{S} and ℂ C, Δ^{C}, ε^{C} be comonads with a lax isomorphism Ω: CU US,

[figure: B; U; ; S; B; U; ; A; F; C; A; F]

If Λ: FC SF corresponds under the adjunction to ΩF Cη: C USF, where η is the unit of B, then the following are (mixed) distributive laws:

\xymatrix{θ: BC UFC \ar[r]^{UΛ} & USF \ar[r]^{Ω^{1} F} & CUF CB,}
\xymatrix{χ: TS FUS \ar[r]^{FΩ^{1}} & FCU \ar[r]^{ΛU} & SFU ST.}

See Theorem 2.9 on p. 1072 for a more detailed statement. For Eilenberg-Moore adjunctions (B A^{𝔹}), such lifts 𝕊 of a given comonad ℂ correspond bijectively to mixed distributive laws between 𝔹 and ℂ (a dual statement holds for coKleisli adjunctions A B_{𝕋}), cf. Section 2.13.

Sections 2–4 contain various technical results that we would like to add to the theory developed in [BŞ08], while the final two Sections 5 and 6 discuss examples. In particular, we further develop the 2-categorical viewpoint of [BŞ12], interpreting the passage from mixed distributive laws between 𝔹, ℂ to distributive laws between 𝕋, 𝕊 in the case of an Eilenberg-Moore adjunction as the application of a 2-functor (Proposition 2.15). Furthermore, Section 2.17 describes how different lifts S, V of a given functor C are related to each other.

1.4. Coefficients. In Section 3, we discuss left and right χ-coalgebras N respectively M that serve as coefficients of cyclic homology.

The structure of right χ-coalgebras is easily described in terms of ℂ-coalgebra structures on UM (Proposition 3.5). In the example from [KK11] associated to a Hopf algebroid H, these are simply right H-modules and left H-comodules, see Section 5.14 below.

A special case is when the ℂ-coalgebra structure on UM arises from an 𝕊-coalgebra structure on M. In the Hopf algebroid case, these are given by Hopf modules. We show that in general, such coefficients are homologically trivial (Proposition 4.10) and can be also interpreted as 1-cells from the trivial distributive law (Propositions 3.10 and 3.11). One reason for discussing them is to point out that general χ-coalgebras can not be reinterpreted as 1-cells.

Similarly, 𝕋-opcoalgebras yield homologically trivial left χ-coalgebras. In the Hopf algebroid example, we present a construction of homologically non-trivial examples from Yetter-Drinfel’d modules, where the Yetter-Drinfel’d condition is necessary for the welldefinedness of the left χ-coalgebra structure; it does not arise as a condition for the resulting duplicial functor to be cyclic.

1.5. Duplicial objects. Section 4 recalls the construction of duplicial objects. We emphasize the self-duality of the situation by defining in fact two duplicial objects C_{𝕋} N, M and C_{𝕊}^{op} N, M , arising from bar resolutions using 𝕋 respectively 𝕊. There is a canonical pair of morphisms of duplicial objects between these which are mutual inverses if and only if the two objects are cyclic (Proposition 4.8).

Furthermore, we describe in Section 4.11 the process of twisting a pair of coefficients M, N by what we called a factorisation in [KS14]. This is motivated by the example of the twisted cyclic homology of an associative algebra [KMT03] and constitutes our main application of the 2-categorical language.

1.6. Hopf monads. One of our motivations in this project is to understand how various notions of bimonads studied in the literature lead to examples of the above theory that generalise known ones arising from bialgebras and bialgebroids.

All give rise to distributive laws, but it seems to us that opmodule adjunctions over opmonoidal adjunctions as studied recently by Aguiar and Chase [AC12] are the underpinning of the cyclic homology theories from noncommutative geometry: such adjunctions are associated to opmonoidal adjunctions

[figure: E; H; ; H; E] ,

so here H and E are monoidal categories, E is a strong monoidal functor and H is an opmonoidal functor, see Section 5.1. In the key example, H is the category H-Mod of modules over a bialgebroid H and E is the category of bimodules over the base algebra A of H. In the special case of the cyclic homology of an associative algebra A, we have H E and H E id, so this adjunction is irrelevant. Now the actual opmodule adjunctions defining cyclic homology are formed by an H-module category B and an E-module category A. In the example, one can pick any H-module coalgebra C and any H-comodule algebra B, take B to be the category B-Mod of B-modules, A be the category A-Mod of A-modules, and the pair of comonads 𝕊, ℂ is given by C _{A} . To obtain the cyclic homology of an associative algebra one takes B to be the category of A-bimodules (or rather right A^{e}-modules). Another very natural example is given by a quantum homogeneous space [MS99], where A k is commutative, H is a Hopf algebra, B is a left coideal subalgebra and C : A AB^{}where B^{}is the kernel of the counit of H restricted to B. So here the distributive law arises from the fact that B admits a C-Galois extension to a Hopf algebra H; following, e.g., [MM02] we call B, C a Doi-Koppinen datum.

Bimonads in the sense of Mesablishvili and Wisbauer also provide examples of the theory considered. There is no monoidal structure required on the categories involved, but instead we have B C, see Section 6. At the end of the paper we give an example of such a bimonad which is not related to bialgebroids and noncommutative geometry, but indicates potential applications of cyclic homology in computer science.

2. Distributive laws

2.1. Distributive laws. We assume the reader is familiar with 2-categorical constructions, see, e.g., [KS72, Lei04] for more background. Given a 2-category C, we denote by C_{}the 2-category obtained by reversing all 2-cells in C, and by C^{}we denote the 2category obtained by reversing all 1-cells in C. We write Cmd C to refer to the 2-category Mnd C_{}_{}of comonads in C, and End C for the 2-category of endo-1-cells in C, the 1-cells of which are of the same variance as those in Cmd C , so that there is a forgetful 2-functor Cmd C End C . See also [Str72] for more background on (co)monads.

2.2. Definition. Let 𝔹 B, μ^{B}, η^{B} and 𝔸 A, μ^{A}, η^{A} be monads on categories C respectively D, and let Σ: C D be a functor. A natural transformation σ: AΣ ΣB is called a lax morphism of monads if Σ, σ is a 1-cell in Mnd Cat , that is, if the two diagrams

\xymatrix{AAΣ \ar[d]_{μ^{A}Σ} \ar[r]^{Aσ} & AΣB \ar[r]^{σB} & ΣBB \ar[d]^{Σμ^{B}} & Σ \ar[r]^{η^{A}Σ} \ar[dr]_{Ση^{B}} & AΣ \ar[d]^{σ} \\ AΣ \ar[rr]_{σ} &  & ΣB &  & ΣB}

commute. We denote this by σ: 𝔸Σ Σ𝔹.

Analogously, one defines colax morphisms σ: Σ𝔸 𝔹Σ, where Σ: D C and 𝔸, 𝔹 are as before, and (co)lax morphism of comonads. Note that a lax morphism of (co)monads can be equivalently described as a (co)monad in End Cat .

2.3. Remark. Note that while a lax morphism of monads is a 1-cell in Mnd Cat , a lax morphism of comonads is a 1-cell in Cmd Cat_{}_{}. Dually, a colax morphism of monads is a 1-cell in Mnd Cat_{}_{}whilst a colax morphism of comonads is a 1-cell in Cmd Cat .

2.4. Definition. A distributive law χ: 𝔸𝔹 𝔹𝔸 between monads 𝔸, 𝔹 is a natural transformation χ: AB BA which is both a lax and a colax morphism of monads.

Analogously, one defines distributive laws between comonads and mixed distributive laws [Bur73] between monads and comonads.

2.5. The 2-categories Dist and Mix. Since this will simplify the presentation of some results, we turn comonad and mixed distributive laws into the 0-cells of 2-categories Dist respectively Mix. This closely follows Street [Str72], see also [KS14]:

2.6. Definition. We denote by Dist the 2-category Cmd Cmd Cat ^{}^{}of comonads (with colax morphisms as 1-cells) in the 2-category of comonads (with lax morphisms as 1-cells). Thus explicitly, 1. 0-cells are quadruples B, χ, 𝕋, 𝕊 , where χ: 𝕋𝕊 𝕊𝕋 is a comonad distributive law

on a category B,

2. 1-cells B, χ, 𝕋, 𝕊 D, τ, 𝔾, ℂ are triples Σ, σ, γ , where Σ: B D is a functor, σ: 𝔾Σ Σ𝕋 is a lax morphism of comonads, and γ: Σ𝕊 ℂΣ is a colax morphism of comonads satisfying the Yang-Baxter equation, i.e.,

\xymatrix{ & ΣTS \ar[r]^{Σχ} & ΣST \ar@{-->}[dr]^{γT} \\ GΣS \ar@{-->}[dr]_{Gγ} \ar@{-->}[ur]^{σS} &  &  & CΣT \\  & GCΣ \ar[r]_{τΣ} & CGΣ \ar@{-->}[ur]_{Cσ}}

commutes, and

3. 2-cells Σ, σ, γ Σ^{}, σ^{}, γ^{}are natural transformations α: Σ Σ^{}for which the

diagrams

\xymatrix{GΣ \ar[d]_{σ} \ar[r]^{Gα} & GΣ^{} \ar[d]^{σ^{}} & ΣS \ar[r]^{αS} \ar[d]_{γ} & Σ^{}S \ar[d]^{γ^{}} \\ ΣT \ar[r]_{αT} & Σ^{}T & CΣ \ar[r]_{Cα} & CΣ^{}}

commute.

In the sequel, we will denote 1-cells diagrammatically as:

[figure: B; 𝕋; 𝕊; Σ,σ,γ; D; 𝔾; ℂ; χ; τ]

Similarly, we define the 2-category Mix : Mnd Cmd Cat of mixed distributive laws.

2.7. Distributive laws arising from adjunctions. The topic of this paper is distributive laws that are compatible in a specific way with an adjunction for one of the involved comonads: let 𝔹 B, μ, η be a monad on a category A. Suppose

[figure: A; F; ; B; U]

is an adjunction for 𝔹, that is, B UF, and let 𝕋 : T, Δ, ε with T : FU be the induced comonad on B.

2.8. Definition. If S: B B and C: A A are endofunctors for which the diagram

\xymatrix{B \ar[d]_{U} \ar[r]^{S} & B \ar[d]^{U} \\ A \ar[r]_{C} & A}

commutes up to a natural isomorphism Ω: CU US, then we call C an extension of S and S a lift of C through the adjunction.

In general, any natural transformation Ω: CU US uniquely determines a mate Λ: FC SF that corresponds to

\xymatrix{C \ar[r]^{Cη} & CUF \ar[r]^{ΩF} & USF}

under the adjunction [Lei04]. The following theorem, which closely follows [Lei04, Lemmata 6.1.1 and 6.1.4], constructs a canonical pair of distributive laws from Λ:

2.9. Theorem. Suppose that S, C, and Ω are as in Definition 2.8. Then the natural transformations

\xymatrix{θ: BC UFC \ar[r]^{UΛ} & USF \ar[r]^{Ω^{1} F} & CUF CB}

and

\xymatrix{χ: TS FUS \ar[r]^{FΩ^{1}} & FCU \ar[r]^{ΛU} & SFU ST}

are lax endomorphisms of the monad 𝔹 respectively the comonad 𝕋. If C and S are parts of comonads ℂ C, Δ^{C}, ε^{C} respectively 𝕊 S, Δ^{S}, ε^{S} , and if Ω is a lax morphism of comonads, then θ is a mixed distributive law and χ is a comonad distributive law.

Proof. Both U, Ω^{1} and F, Λ are 1-cells in the 2-category End Cat of endofunctors in Cat. The unit and the counit of the adjunction F U are 2-cells by construction; hence F, Λ U, Ω^{1} is an adjunction defining a monad and a comonad whose underlying 1-cells are UF, θ and FU, χ respectively (which means that θ and χ are lax endomorphisms of 𝔹 respectively 𝕋). If C and S are parts of comonads ℂ respectively 𝕊 and Ω is a lax morphism of comonads, then the same argument applies with End Cat replaced by Cmd Cat , but in this case θ and χ are distributive laws.

2.10. Remark. Note that the lax morphisms θ, χ are unique such that the following diagrams commute respectively:

\xymatrix{UFCU \ar[rr]^{θU} \ar[d]_{UFΩ} &  & CUFU \ar[d]^{CUε} & US \ar[r]^{Ω^{1}} \ar[d]_{ηUS} & CU \ar[r]^{CηU} & CUFU \ar[d]^{ΩFU} \\ UFUS \ar[r]_{UεS} & US \ar[r]_{Ω^{1}} & CU & UFUS \ar[rr]_{Uχ} &  & USFU}

2.11. Definition. A comonad distributive law χ as in Theorem 2.9 is said to arise from the adjunction F U.

2.12. Example. A trivial example which will nevertheless play a rôle below is the case where C B, S T, and Ω id. In this case, χ and θ are given by

\xymatrix{TT FUFU \ar[r]^{εFU} & FU \ar[r]^{FηU} & FUFU TT,}
\xymatrix{BB UFUF \ar[r]^{UεF} & UF \ar[r]^{UFη} & UFUF BB.}

2.13. The Eilenberg-Moore and the coKleisli cases. Functors do not necessarily lift respectively extend through an adjunction (for example, the functor on Set which assigns the empty set to each set does not lift to k-Mod), and if they do, they may not do so uniquely. Theorem 2.9 says only that once a lift respectively extension is chosen, there is a unique compatible pair of lax endomorphisms θ and χ.

One extremal situation in which specifying a lax endomorphism θ: 𝔹C C𝔹 uniquely determines a lift S of C is when B is the Eilenberg-Moore category A^{𝔹}. In this case, we write B̃ : T, 𝔹̃ : 𝕋 and θ̃ : χ. The unique lift S is given on objects X, α by S X, α CX, Cα θX . Using Theorem 2.9 (with Ω id), one recovers θ, see, e.g., [App65, Joh75]. In the sequel, we will denote S by C^{θ}. If C is part of a comonad ℂ and Ω is a lax morphism, then we denote the corresponding comonad lift 𝕊 by ℂ^{θ}.

Dually, one can take A to be the coKleisli category B_{𝕋} in which case a lax endomorphism χ yields an extension C of a functor S. This means that every comonad distributive law and every mixed distributive law arises from an adjunction.

In the remainder of this section, we discuss the functoriality of the above constructions. This hinges on the following remark:

2.14. Lemma. The 2-category Cmd Cat admits Eilenberg-Moore constructions for monads.

In other words, the inclusion 2-functor Cmd Cat Mnd Cmd Cat has a right 2-adjoint, which is defined by

[figure: A; 𝔹; ℂ; Σ,σ,γ; ; A^{𝔹}, ℂ^{θ}; Σ^{σ},γ̃; D; 𝔸; 𝔻; D^{𝔸}, 𝔻^{ψ}; θ; ψ; Σ, σ, γ; α; ; Σ^{σ}, γ̃; α̃; Σ^{}, σ^{}, γ^{}; Σ^{σ^{}}, γ̃^{}]

Here, Σ^{σ} is the lifting of the functor Σ via the lax morphism of monads σ. The natural transformation γ: ΣC DΣ lifts to a natural transformation γ̃: Σ^{σ}C^{θ} D^{ψ}Σ^{σ} if and only if the Yang-Baxter equation is satisfied, and γ̃ is a colax morphism of comonads if and only if γ is. The natural transformation α lifts to a natural transformation α̃: Σ^{σ} Σ^{σ^{}} if and only if α: Σ, σ Σ^{}, σ^{}is a 2-cell in Mnd Cat , and α̃ becomes a 2-cell in Cmd Cat if and only if α: Σ, γ Σ^{}, γ^{}is a 2-cell in Cmd Cat .

Having fixed this notation, observe finally that a natural transformation σ: AΣ ΣB lifts to a natural transformation σ̃: ÃΣ^{σ} Σ^{σ}B̃ if and only if σ is compatible with the multiplication of the monads 𝔸 and 𝔹. In this situation, we have:

2.15. Proposition. The assignment

[figure: A; 𝔹; ℂ; Σ,σ,γ; ; A^{𝔹}; 𝔹̃; ℂ^{θ}; Σ^{σ},σ̃,γ̃; D; 𝔸; 𝔻; D^{𝔸}; 𝔸̃; 𝔻^{ψ}; θ; ψ; θ̃; ψ̃; Σ, σ, γ; α; ; Σ̃, σ̃, γ̃; α̃; Σ^{}, σ^{}, γ^{}; Σ^{σ^{}}, σ̃^{}, γ̃^{}]

is a 2-functor i: Mix Dist.

Proof. For any 2-category C which admits Eilenberg-Moore constructions for monads, there is a 2-functor \overline{Alg}_{C}: Mnd C Cmd C^{}^{}(see [Str72, p. 160]). By choosing C Cmd Cat in this construction, we get the above 2-functor.

Analogously, we obtain a 2-functor j: Dist Mix by taking extensions to coKleisli categories. It is those distributive laws in the image of the 2-functor i that are the main object of study in this paper.

2.16. Remark. Let F U be an adjunction and let 𝕊 be the lift of a comonad ℂ through the adjunction via Ω as in Section 2.7. Consider the comparison functor U^{Uε}: B A^{𝔹}. The colax morphism Ω^{1} : U𝕊 ℂU lifts to a colax morphism Ω̃^{1} : U^{Uε}𝕊 ℂ^{θ}U^{Uε}. This gives rise to a 1-cell U^{Uε}, id, Ω̃^{1} : χ θ̃. Thus the image under i of a general 1-cell θ ψ of mixed distributive laws can be composed with the 1-cell given by the comparison functor to give a new 1-cell χ ψ̃.

2.17. Generalising the Galois map. Theorem 2.9 yields comonad distributive laws from lifts through an adjunction, and different lifts produce different distributive laws. Here we describe how these are related in terms of suitable generalisations of the Galois map from the theory of Hopf algebras (see Section 6.2 below for the example motivating the terminology).

2.18. Definition. If S, V: B B are lifts of C: A A through F U with isomorphisms Ω: CU US and Φ: CU UV, we define a natural isomorphism

Γ^{S,V} : B F , S B F , V

of functors A^{op} B Set on components by the composition

\xymatrix{B FX, SY \ar[r] & A X, USY \ar[r] & A X, UVY \ar[r] & B FX, VY ,}

where the middle map is induced by Φ_{Y} Ω_{Y}^{1} : USY UVY and the outer ones are induced by the adjunction F U.

The following properties are easy consequences of the definition:

2.19. Proposition. Let S and V be two lifts of an endofunctor C through an adjunction F U. Then:

1. The inverse of Γ^{S,V} is given by Γ^{V,S}.

2. The natural transformation Γ^{S,V} maps a morphism f: FX SY to

\xymatrix{FX \ar[r]^{FηX} & FUFX \ar[r]^{FUf} & FUSY \ar[r]^{F Φ_{Y} Ω_{Y}^{1}} & FUVY \ar[r]^{εVY} & VY.}

3. If χ^{S} and χ^{V} denote the lax morphisms determined by the two lifts, then

Γ^{S,V} χ^{S} χ^{V}.

So, in the applications of Theorem 2.9, all distributive laws obtained from different lifts S, V of a given comonad through an adjunction are obtained from each other by application of Γ^{S,V}.

In particular, we can consider different lifts of B itself: recall the trivial Example 2.12 of Theorem 2.9, where C B and S T, and let V be any other lift of B through the adjunction. By taking X to be UY for an object Y of B, one obtains a natural transformation Γ^{T,V} : B T , T B T , V that we can evaluate on id: TY TY, which produces a natural transformation Γ^{T,V} id : T V.

Adapting [MW10, Definition 1.3], we define:

2.20. Definition. We say that F is V-Galois if

\xymatrix{Γ^{T,V} id : T FU \ar[r]^{FηU} & FUFU FUT \ar[r]^{FΦ} & FUV \ar[r]^{εV} & V}

is an isomorphism.

The following proposition provides the connection to Hopf algebra theory:

2.21. Proposition. If F is V-Galois and θ: BB BB is the lax morphism arising from the lift V of B, then the natural transformation

\xymatrix{β: BB \ar[r]^{BηB} & BBB \ar[r]^{θB} & BBB \ar[r]^{Bμ} & BB}

is an isomorphism.

Proof. If F is V-Galois, then UΓ^{T,V} id F is an isomorphism

\xymatrix{UTF UFUF \ar[r]^{UFηUF} & UFUFUF UFUTF \ar[r]^{UFΦF} & UFUVF \ar[r]^{UεVF} & UVF.}

Let now χ: TV VT be the lax morphism corresponding to θ as in Theorem 2.9. Inserting εV Vε χ and Uχ UFΦ ΦFU θU and B UF, the isomorphism becomes

\xymatrix{UTF BB \ar[r]^{BηB} & BBB \ar[r]^{θB} & BBB BUFUF \ar[r]^{ΦFUF} & UVFUF \ar[r]^{UVεF} & UVF}

Finally, we have by construction UεF μ, and using the naturality of Φ this gives UVεF ΦFUF ΦF BUεF. Composing the above isomorphism with Φ^{1} F gives β.

It is this associated map β that is used to distinguish Hopf algebras amongst bialgebras, see Section 6 below.

3. Coefficients

3.1. Coalgebras over comonads. Let 𝕋  T, Δ^{T}, ε^{T} and 𝕊  S, Δ^{S}, ε^{S} be

comonads on a category B, and let χ: 𝕋𝕊 𝕊𝕋 be a distributive law. First, we recall:

3.2. Definition. A 𝕋-coalgebra is a triple M, Y, ∇ , where M: Y B is a functor and ∇: M TM is a natural transformation such that the diagrams

[figure: M; ∇; ∇; TM; Δ^{T}M; TM; T∇; TTM; M; ∇; TM; ε^{T}M; M]

commute.

Dually, one defines 𝕋-opcoalgebras N, Z, ∇ where ∇: N NT, as well as algebras and opalgebras involving monads.

3.3. Coalgebras over distributive laws. We now discuss χ-coalgebras, which serve as coefficients in the homological constructions in the next section.

3.4. Definition. A right χ-coalgebra is a triple M, Y, ρ , where M: Y B is a functor and ρ: TM SM is a natural transformation such that the diagrams

\xymatrix{TM \ar[r]^{Δ^{T}M} \ar[d]_{ρ} & TTM \ar[r]^{Tρ} & TSM \ar[d]^{χM} &  & TM \ar[dl]_{ε^{T}M} \ar[d]^{ρ} \\ SM \ar[r]_{Δ^{S}M} & SSM & STM \ar[l]^{Sρ} & M & SM \ar[l]^{ε^{S}M}}

commute. Dually, we define left χ-coalgebras N, Z, λ .

The following characterises right χ-coalgebras in the setting of Theorem 2.9.

3.5. Proposition. In the situation of Theorem 2.9, let M: Y B be a functor.

1. Right χ-coalgebra structures ρ on M correspond to ℂ-coalgebra structures ∇ on the functor UM: Y A.

2. Let S and V be two lifts of the functor C through the adjunction, and let χ^{S} and χ^{V} denote the comonad distributive laws determined by the lifts S and V respectively. Then Γ^{S,V} maps right χ^{S}-coalgebra structures ρ^{S} on M bijectively to right χ^{V}-coalgebra structures ρ^{V} on M.

Proof. For part (1), right χ-coalgebra structures ρ: FUM SM are mapped under the adjunction to ∇: UM USM CUM. Part (2) follows immediately since Γ^{S,V} is the composition of the adjunction isomorphisms and Φ Ω^{1} .

3.6. Remark. The referee of this paper helpfully pointed out that given an adjunction V G for the comonad 𝕊 and an extension ℚ of the comonad 𝕋 through the adjunction, left χ-coalgebra structures on N: B Z correspond in complete analogy to ℚ-opcoalgebra structures on NV.

3.7. Twisting by 1-cells. Here we show how factorisations of distributive laws as considered in [KS14] can be used to obtain new χ-coalgebras from old ones. To this end, fix a 1-cell in the 2-category Dist:

[figure: B; 𝕋; 𝕊; Σ,σ,γ; D; 𝔾; ℂ; χ; τ]

3.8. Lemma. Let M, Y, ρ be a right χ-coalgebra. Then ΣM, Y, γM Σρ σM is a right τ-coalgebra.

Proof. This is proved for the case that χ τ in [KS14], but the same proof applies to this slightly more general situation.

Dually, left τ-coalgebras N, Z, λ define left χ-coalgebras NΣ, Z, Nσ λΣ Nγ . The following diagram illustrates the situation:

[figure: B; 𝕋; 𝕊; Σ,σ,γ; Y; M,ρ; Z; D; N,λ; 𝔾; ℂ; χ; τ]

The dotted arrows represent the induced χ-coalgebras from Lemma 3.8.

This will be applied in Section 4.11 below in the context of duplicial functors.

3.9. χ-coalgebras from coalgebras over comonads. In the remainder of this section, we discuss a class of coefficients that lead to contractible simplicial objects, see Proposition 4.10 below. In the Hopf algebroid setting, these are the Hopf (or entwined) modules as studied in [AC12, BM98].

Note first that 𝕋-coalgebras can be equivalently viewed as 1-cells from respectively to the trivial distributive law:

3.10. Proposition. Given an 𝕊-coalgebra M, Y, ∇^{S} and a 𝕋-opcoalgebra N, Z, ∇^{T} , there is a pair of 1-cells

[figure: Y; id; id; M, ε^{T}M, ∇^{S}; B; 𝕋; 𝕊; id; χ; B; 𝕋; 𝕊; N, ∇^{T}, Nε^{S}; Z; id; id; χ; id]

and all 1-cells id χ respectively χ id are of this form.

Furthermore, these 1-cells can also be viewed as χ-coalgebras:

3.11. Proposition. Let χ: 𝕋𝕊 𝕊𝕋 be a comonad distributive law. Then:

1. Any 𝕊-coalgebra M, Y, ∇^{S} defines a right χ-coalgebra M, Y, ε^{T}∇^{S} .

2. Any 𝕋-opcoalgebra N, Z, ∇^{T} defines a left χ-coalgebra N, Z, ∇^{T}ε^{S} .

Note, however, that there is no obvious way to associate a 1-cell in Dist to an arbitrary right or left χ-coalgebra.

3.12. Entwined algebras. Finally, we describe how χ-coalgebras as in Proposition 3.11 are in some sense lifts of entwined (also called mixed) algebras; throughout, θ: 𝔹ℂ ℂ𝔹 is a mixed distributive law between a monad 𝔹 and a comonad ℂ on a category A.

3.13. Definition. Let M: Y A be a functor with a 𝔹-algebra structure β: BM M and a ℂ-coalgebra structure ∇: M CM. We say that the quadruple M, Y, β, ∇ is an entwined algebra with respect to θ if the diagram

\xymatrix{BM \ar[d]_{B∇} \ar[r]^{β} & M \ar[r]^{∇} & CM \\ BCM \ar[rr]_{θM} &  & CBM \ar[u]_{Cβ}}  (3.1)

commutes.

Note that an entwined θ-algebra structure on a functor N: Y A is equivalent to an entwined Y, θ -algebra structure on the functor 1 Y, A which sends the unique object in 1 to N. Without loss of generality, we therefore assume Y 1 and thus consider entwined algebras as objects in A. With the obvious notion of morphism (given by natural transformations compatible with ∇ and β), entwined algebras form a category; this is evidently isomorphic to the category A^{𝔹} ^{ℂ^{θ}} of ℂ^{θ}-coalgebras. Dually we define an entwined opalgebra structure on a functor N: A Z for a distributive law ℂ𝔹 𝔹ℂ.

The following proposition explains the relation between entwined algebras and coalgebras for distributive laws χ arising from an adjunction:

3.14. Proposition. In the situation of Theorem 2.9, let M: Y B be a functor and let ∇: M SM be a natural transformation.

1. If ∇ is an 𝕊-coalgebra structure, then the structure morphisms

\xymatrix{BUM UFUM \ar[r]^{UεM} & UM} , \xymatrix{UM \ar[r]^{U∇} & USM \ar[r]^{Ω^{1}} & CUM}

turn UM into an entwined algebra with respect to θ.

2. If B A^{𝔹}, then the converse of (1) holds.

Proof. Throughout this proof, undecorated arrows denote forgetful functors. Let 𝟙 denote the identity monad id, id, id on B. The triangle identities of the adjunction F U imply that Uε: 𝔹U U𝟙 is a lax morphism of monads inducing a lifting

[figure: B; U^{Uε}; A^{𝔹}; B; U; A]

The universal property of θ, cf. Remark 2.10, is equivalent to the fact that Ω is a 2-cell in Mnd Cat :

[figure: B, 𝟙; S,id; U,Uε; Ω; A, 𝔹; C,θ; B, 𝟙; U,Uε; A, 𝔹]

Therefore, Ω lifts to a natural transformation Ω̃:

[figure: B; S; U^{Uε}; Ω̃; A^{𝔹}; C^{θ}; B; U^{Uε}; A^{𝔹}]

Since Ω: ℂU U𝕊 is a lax morphism of comonads, the lifting Ω̃: ℂ^{θ}U^{Uε} U^{Uε}𝕊 is a lax morphism of comonads, thus inducing a lifting

\xymatrix{B^{𝕊} \ar[d] \ar[r]^{U^{Uε} Ω̃^{1}} & A^{𝔹} ^{ℂ^{θ}} \ar[d] \\ B \ar[r]_{U^{Uε}} & A^{𝔹}}

The object map of the functor in the top row of this diagram is the construction in part (1). If B A^{𝔹}, then U^{Uε} id, Ω id and 𝕊 ℂ^{θ}, so the top functor in the diagram is just the identity, implying part (2).

Dually, entwined opalgebra structures on a 𝔹-opalgebra N, Z, ω are related to left χ-coalgebras if the codomain Z of N is a category with coequalisers. First, we define a functor N_{𝔹}: A^{𝔹} Z that takes a 𝔹-algebra morphism f: X, α Y, β to N_{𝔹} f defined using coequalisers:

\xymatrix{NBX \ar[d]_{NBf} \ar[r]^{ω_{X}} \ar[r]_{Nα} & NX \ar[d]_{Nf} \ar@{->>}[r]^{q_{X,α}} & N_{𝔹} X, α \ar@{.>}[d]^{N_{𝔹} f} \\ NBY \ar[r]^{ω_{Y}} \ar[r]_{Nβ} & NY \ar@{->>}[r]_{q_{Y,β}} & N_{𝔹} Y, β}

Thus N_{𝔹} generalises the functor _{B} N defined by a left module N over a ring B on the category of right B-modules.

Suppose that θ is invertible, and that N admits the structure of an entwined θ^{1} - opalgebra, with coalgebra structure ∇: N CN. There are two commutative diagrams:

\xymatrix{NBX \ar[d]_{∇_{BX}} \ar[r]^{ω_{X}} & NX \ar[dd]^{∇_{X}} & NBX \ar[d]_{∇_{BX}} \ar[rr]^{Nα} &  & NX \ar[dd]^{∇_{X}} \\ NCBX \ar[d]_{Nθ_{X}^{1}} &  & NCBX \ar[d]_{Nθ_{X}^{1}} \\ NBCX \ar[r]_{ω_{CX}} & NCX & NBCX \ar[r]_{Nθ_{X}} & NCBX \ar[r]_{NCα} & NCX}

Hence, using coequalisers, ∇ extends to a natural transformation ˜∇: N_{𝔹} N_{𝔹}C^{θ}, and in fact it gives N_{𝔹} the structure of a ℂ^{θ}-opcoalgebra. Since θ̃^{1} : ℂ^{θ}𝔹̃ 𝔹̃ℂ^{θ} is a comonad distributive law on A^{𝔹}, Proposition 3.11 gives examples of homologically trivial left θ̃^{1} - coalgebras of the form N_{𝔹}, Z, ˜∇ε .

4. Duplicial objects

4.1. The bar and opbar resolutions. Let 𝕋 T, Δ, ε be a comonad on a category B, and let M: Y B be a functor.

4.2. Definition. The bar resolution of M is the simplicial functor B 𝕋, M : Y B defined by

B 𝕋, M _{n} T^{n 1} M,  d_{i} T^{i}εT^{n i} M,  s_{j} T^{j}ΔT^{n j} M,

where the face and degeneracy maps above are given in degree n. The opbar resolution of M, denoted B^{op} 𝕋, M , is the simplicial functor obtained by taking the opsimplicial simplicial functor of B 𝕋, M . Explicitly:

B^{op} 𝕋, M _{n} T^{n 1} M,  d_{i} T^{n i} εT^{i}M,  s_{j} T^{n j} ΔT^{j}M.

Given any functor N: B Z, we compose it with the above simplicial functors to obtain new simplicial functors that we denote by

C_{𝕋} N, M : NB 𝕋, M , C_{𝕋}^{op} N, M : NB^{op} 𝕋, M .

4.3. Duplicial objects. Duplicial objects were defined by Dwyer and Kan [DK85] as a mild generalisation of Connes’ cyclic objects [Con83]:

4.4. Definition. A duplicial object is a simplicial object C, d_{i}, s_{j} together with additional morphisms t: C_{n} C_{n} satisfying

d_{i}t \begin{matrix}td_{i 1} , & 1 i n, \\ d_{n}, & i 0, \end{matrix} s_{j}t \begin{matrix}ts_{j 1} , & 1 j n, \\ t^{2}s_{n}, & j 0. \end{matrix}

A duplicial object is cyclic if T : t^{n 1} id.

Equivalently, a duplicial object is a simplicial object which has in each degree an extra degeneracy s_{1} : C_{n} C_{n 1} . This corresponds to t via

s_{1} : ts_{n}, t d_{n 1} s_{1} .

This turns a duplicial object also into a cosimplicial object, and hence a duplicial object C in an additive category carries a boundary and a coboundary map

b : _{i 0}^{n} 1 ^{i}d_{i}, s : _{j 1}^{n} 1 ^{j}s_{j}.

Dwyer and Kan called such chain and cochain complexes duchain complexes and showed that the normalised chain complex functor yields an equivalence between duplicial objects and duchain complexes in an abelian category, thus extending the classical Dold-Kan correspondence between simplicial objects and chain complexes.

If f_{n} ℤ x is given by 1 xf_{n} x 1 x ^{n 1} and B : sf_{n} bs , then one has

B^{2} 0, bB Bb id T,

and in this way cyclic objects give rise to mixed complexes C, b, B in the sense of [Kas87] that can be used to define cyclic homology.

4.5. The Böhm-Ştefan construction. Let B, χ, 𝕋, 𝕊 be a 0-cell in Dist, and let M, Y, ρ and N, Z, λ be right and left χ-coalgebras respectively. By abuse of notation, we let χ^{n} denote both natural transformations T^{n}S ST^{n} and TS^{n} S^{n}T obtained by repeated application of χ (up to horizontal composition of identities), where χ^{0} id. We furthermore define natural transformations

t_{n}^{𝕋}: C_{𝕋} N, M _{n} C_{𝕋} N, M _{n}, t_{n}^{𝕊}: C_{𝕊}^{op} N, M _{n} C_{𝕊}^{op} N, M _{n}

by the diagrams

\xymatrix{NT^{n}SM \ar[r]^{Nχ^{n}M} & NST^{n}M \ar[d]^{λT^{n}M} & NTS^{n}M \ar[r]^{Nχ^{n}M} & NS^{n}TM \ar[d]^{NS^{n}ρ} \\ NT^{n 1} M \ar[u]^{NT^{n}ρ} \ar@{.>}[r]_{t_{n}^{𝕋}} & NT^{n 1} M & NS^{n 1} M \ar[u]^{λS^{n}M} \ar@{.>}[r]_{t_{n}^{𝕊}} & NS^{n 1} M}

4.6. Theorem. The simplicial functors C_{𝕋} N, M and C_{𝕊}^{op} N, M become duplicial functors with duplicial operators given by t^{𝕋} respectively t^{𝕊}.

Proof. The first operator being duplicial is exactly the case considered in [BŞ08], and the second follows from a slight modification of their proof.

4.7. Cyclicity. For each n 0, we define a morphism R_{n}: NT^{n 1} M NS^{n 1} M in the following way. For each 0 i n, let r_{i,n} denote the morphism

\xymatrix{NS^{i}T^{n 1 i} M \ar[r]^{NS^{i}T^{n i} ρ} & NS^{i}T^{n i} SM \ar[r]^{NS^{i}χ^{n i} M} & NS^{i 1} T^{n i} M.}

Then set

R_{n} : r_{n,n} r_{0,n}.

Similarly, we can define a morphism L_{n}: NS^{n 1} M NT^{n 1} M whose definition involves the left χ-coalgebra structure λ on N.

4.8. Proposition. The above construction defines two morphisms

\xymatrix{C_{𝕋} N, M \ar[r]^{R} & C_{𝕊}^{op} N, M}, \xymatrix{C_{𝕊}^{op} N, M \ar[r]^{L} & C_{𝕋} N, M}

of duplicial functors. Furthermore, L R id if and only if C_{𝕋} N, M is cyclic, and R L id if and only if C_{𝕊}^{op} N, M is cyclic.

Proof. This is verified by straightforward computation. However, it is convenient to use a diagrammatic calculus as, e.g., in [BŞ08], in which natural transformations NVM NWM are visualised as string diagrams, where V and W are words in S, T. For example t^{𝕋} will be represented by the diagram

[figure: N; T; T; T; T; M; ; ; N; T; T; T; T; M]

Crossing of strings represents the distributive law χ and the bosonic propagators represent the χ-coalgebra structures λ: NS NT respectively ρ: TM SM.

For example, the identities Rd_{i} d_{i}R and Rs_{j} s_{j}R follow from the commutative diagrams in Definition 3.4, which are represented diagrammatically by

[figure: T; M; T; M; ; ; M; ; M]

respectively

[figure: T; M; ; S; S; M; T; M; S; S; M]

The relation Rt^{𝕋} t^{𝕊}R for n 2 becomes

[figure: N; T; T; T; M; ; N; S; S; S; M; N; T; T; T; M; N; S; S; S; M]

which reflects the naturality of λ, ρ, and χ. Similarly, L is a morphism of duplicial objects, and one has L R _{n} t_{n}^{𝕋} ^{n 1} and R L _{n} t_{n}^{ℂ} ^{n 1} .

4.9. Homologically trivial χ-coalgebras. As we had announced above, χ-coalge-bras as in Proposition 3.11 lead to contractible simplicial objects:

4.10. Proposition. Let χ: 𝕋𝕊 𝕊𝕋 be a comonad distributive law on a category B, and let M, Y, ρ and N, Z, λ be left and right χ-coalgebras respectively. Suppose also that Z is an abelian category. If either of N, Z, λ , M, Y, ρ arises as in Proposition 3.11, then the chain complexes associated to both C_{𝕋} N, M and C_{𝕊}^{op} N, M are contractible.

Proof. Assume there is a 𝕋-opcoalgebra structure ∇: N NT on N. The morphisms ∇T^{n}M: NT^{n 1} M NT^{n 2} M provide a contracting homotopy for the complex associated to C_{𝕋} N, M , and the morphisms

\xymatrix{NS^{n 1} M \ar[r]^{∇S^{n 1} M} & NTS^{n 1} M \ar[r]^{Nχ^{n 1} M} & NS^{n 1} TM \ar[r]^{NS^{n 1} ρ} & NS^{n 2} M}

provide a contracting homotopy for the complex associated to C_{𝕊}^{op} N, M . The other case is similar.

4.11. Twisting by 1-cells. Applying the twisting procedure described in Section 3.7, a 1-cell

[figure: B; 𝕋; 𝕊; Σ,σ,γ; D; 𝔾; ℂ; χ; τ]

in the 2-category Dist gives rise to morphisms between duplicial functors of the form considered above: Theorem 4.6 and Lemma 3.8 yield duplicial structures on the simplicial functors

C_{𝕋} NΣ, M , C_{𝕊}^{op} NΣ, M , C_{𝔾} N, ΣM , C_{ℂ}^{op} N, ΣM ,

and from Proposition 4.8 we obtain morphisms

\xymatrix{C_{𝕋} NΣ, M \ar[r]^{R^{χ}} & C_{𝕊}^{op} NΣ, M , & C_{𝕊}^{op} NΣ, M \ar[r]^{L^{χ}} & C_{𝕋} NΣ, M ,}
\xymatrix{C_{𝔾} N, ΣM \ar[r]^{R^{τ}} & C_{ℂ}^{op} N, ΣM , & C_{ℂ}^{op} N, ΣM \ar[r]^{L^{τ}} & C_{𝔾} N, ΣM}

of duplicial objects which determine the cyclicity of each functor.

Additionally, repeated application of σ: GΣ ΣT and γ: ΣS CΣ yields two duplicial morphisms

\xymatrix{C_{𝔾} N, ΣM \ar[r] & C_{𝕋} NΣ, M , & C_{𝕊}^{op} NΣ, M \ar[r] & C_{ℂ}^{op} N, ΣM .}

Note that for arbitrary functors M and N these are simplicial morphisms which become duplicial morphisms if M and N have coalgebra structures.

5. Hopf monads and Hopf algebroids

5.1. Opmodule adjunctions. One example of Theorem 2.9 is provided by an opmonoidal adjunction between monoidal categories:

5.2. Definition. An adjunction

[figure: E, _{E}, 1_{E}; H; ; H, _{H}, 1_{H}; E]

between monoidal categories is opmonoidal if both H and E are opmonoidal functors and the unit and the counit of the adjunction are opmonoidal natural transformations.

Thus by definition, there are natural transformations

Ξ: H X _{E} Y HX _{H} HY, Ψ: E K _{H} L EK _{E} EL

and morphisms Ξ_{0} : H 1_{E} 1_{H} and Ψ_{0} : E 1_{H} 1_{E}. It follows that Ψ and Ψ_{0} are in fact isomorphisms. Hence opmonoidal adjunctions are a special case of doctrinal adjunctions [Kel72]; some authors call opmonoidal adjunctions comonoidal adjunctions or bimonads. We refer, e.g., to [AC12, BLV11, McC02, MW14, Moe02] for more information.

It follows that

H 1_{E} _{H} EH 1_{E} _{E}

form a compatible pair of comonads as in Theorem 2.9 whose comonad structures are induced by the natural coalgebra (comonoid) structures on 1_{E}.

However, the examples we are more interested in are given by opmodule adjunctions (introduced under the name comodule adjunctions in [AC12, Definition 4.1.1]). First, we recall:

5.3. Definition. Let H: E, _{E}, 1_{E} H, _{H}, 1_{H} be an opmonoidal functor and let A, _{A} and B, _{B} be (left) E-module respectively H-module categories. An H-opmodule is a functor F: A B together with a natural transformation Θ: F Y _{A}Z HY _{B} FZ such that the diagrams

\xymatrix{F X _{A} Y _{A} Z \ar[d]_{Θ} \ar[r]^{Θ} & H X _{E} Y _{B} FZ \ar[d]^{Ξ _{B}id} \\ HX _{B} F Y _{A} Z \ar[r]_{id _{B}Θ} & HX _{B} HY _{B} FZ}

and

[figure: F 1_{E} _{A} Z; Θ; H 1_{E} _{B} FZ; Ξ_{0} _{B}id; FZ; 1_{H} _{B} FZ]

commute, where we have assumed for simplicity that all involved categories are strict.

5.4. Definition. Let H E be an opmonoidal adjunction between monoidal categories E, _{E}, 1_{E} respectively H, _{H}, 1_{H} . An opmodule adjunction over H E is an adjunction F U between an E-module category A, _{A} and an H-module category B, _{B} , together with natural transformations

Θ: F Y _{A} Z HY _{B} FZ, Ω: U L _{B} M EL _{A} UM

that turn F and U into H- respectively E-opmodules, such that

\xymatrix{Y _{A} Z \ar[d]_{η^{UF}} \ar[r]^{η^{EH} _{A}η^{UF}} & EHY _{A} UFZ \\ UF Y _{A} Z \ar[r]_{UΘ} & U HY _{B} FZ \ar[u]_{Ω}}

and

\xymatrix{HEL _{B} FUM \ar[r]^{ε^{HE} _{B}ε^{FU}} & L _{B} M \\ F EL _{A} UM \ar[u]^{Θ} & FU L _{B} M \ar[l]^{FΩ} \ar[u]_{ε^{FU}}}

commute.

It follows that Ω is an isomorphism (see [AC12, Proposition 4.1.2] and also [Kel72, Theorem 1.4] for a more general theorem in the setting of doctrinal adjunctions).

Now any coalgebra C in H defines a compatible pair of comonads

S C _{B} , C EC _{A}

on B respectively A. It is such an instance of Theorem 2.9 that provides the monadic generalisation of the setting from [KK11], see Section 5.14.

5.5. Bialgebroids and Hopf algebroids. Opmonoidal adjunctions can be seen as categorical generalisations of bialgebras and more generally (left) bialgebroids. We briefly recall the definitions but refer to [Böh09, DS97, KK11, Szl03] for further details and references.

5.6. Definition. If E is a k-algebra, then an E-ring is a k-algebra map η : E H.

In particular, when E A^{e} : A _{k} A^{op} is the enveloping algebra of a k-algebra A, then H carries two A-bimodule structures given by

a h b : η a _{k} b h, a h b : hη b _{k} a .

5.7. Definition. [Tak77] A bialgebroid is an A^{e}-ring η : A^{e} H for which _{}H_{}is a coalgebra in A^{e}-Mod, _{A}, A whose coproduct Δ: H H_{}_{A} _{}H satisfies

a Δ h Δ h a, Δ gh Δ g Δ h ,

and whose counit ε: H A defines a unital H-action on A given by h a : ε a h .

Finally, by a Hopf algebroid we mean left rather than full Hopf algebroid, so there is in general no antipode [KR13]:

5.8. Definition. [Sch00] A Hopf algebroid is a bialgebroid with bijective Galois map

β: _{}H _{A^{op}} H_{}H_{}_{A} _{}H, g _{A^{op}} h Δ g h.

As usual, we abbreviate

Δ h : h_{1} _{A} h_{2},  β^{1} h _{A} 1 : h_{}_{A^{op}} h_{}.  (5.1)

5.9. The opmonoidal adjunction. Every E-ring H defines a forgetful functor

E: H-Mod E-Mod

with left adjoint H H _{E} . In the sequel, we abbreviate H : H-Mod and E : E-Mod. If H is a bialgebroid, then H is monoidal with tensor product K _{H} L of two left Hmodules K and L given by the tensor product K _{A} L of the underlying A-bimodules whose H-module structure is given by

h k _{H} l : h_{1} k _{A} h_{2} l .

So by definition, we have E K _{H} L EK _{A} EL. The opmonoidal structure Ξ on H is defined by the map [BLV11, AC12]

H X _{A} Y H _{A^{e}} X _{A} Y  HX _{H} HY H _{A^{e}} X _{A} H _{A^{e}} Y ,
h _{A^{e}} x _{A} y  h_{1} _{A^{e}} x _{A} h_{2} _{A^{e}} y .

Schauenburg proved that this establishes a bijective correspondence between bialgebroid structures on H and monoidal structures on H-Mod [Sch98, Theorem 5.1]:

5.10. Theorem. The following data are equivalent for an A^{e}-ring η: A^{e} H:

1. A bialgebroid structure on H.

2. A monoidal structure , 1 on H-Mod such that the adjunction

[figure: A^{e}-Mod, _{A}, A; H-Mod, , 1]

induced by η is opmonoidal.

Consequently, we obtain an opmonoidal monad

EH _{}H_{}_{A^{e}}

on E A^{e}-Mod. This takes the unit object A to the cocentre H _{A^{e}} A of the A-bimodule H_{}, and the comonad H 1_{E} _{E} is given by

H _{A^{e}} A _{A} ,

where the A-bimodule structure on the cocentre is given by the actions , on H. The lift to H H-Mod takes a left H-module L to H _{A^{e}} A _{A} L with action

g h _{A^{e}} 1 _{A} l g_{1}h _{A^{e}} 1 _{A} g_{2}l,

and the distributive law resulting from Theorem 2.9 is given by

χ: g _{A^{e}} h _{A^{e}} 1 _{A} l g_{1}h _{A^{e}} 1 _{A} g_{2} _{A^{e}} l .

That is, it is the map induced by the Yetter-Drinfel’d braiding

H_{}_{A} _{}H H_{}_{A} _{}H, g _{A} h g_{1}h _{A} g_{2}.

For A k, that is, when H is a Hopf algebra, and also trivially when H A^{e}, the monad and the comonad on A^{e}-Mod coincide and are also a bimonad in the sense of Mesablishvili and Wisbauer, cf. Section 6. An example where the two are different is the Weyl algebra, or more generally, the universal enveloping algebra of a Lie-Rinehart algebra [Hue98]. In these examples, A is commutative but not central in H in general, so H_{}_{A^{e}} is different from H_{}_{A} .

5.11. Doi-Koppinen data. The instance of Theorem 2.9 that we are most interested in is an opmodule adjunction associated to the following structure:

5.12. Definition. A Doi-Koppinen datum is a triple H, C, B of an H-module coalgebra C and an H-comodule algebra B over a bialgebroid H.

This means that C is a coalgebra in the monoidal category H-Mod. Similarly, the category H-Comod of left H-comodules is also monoidal (see, e.g., [Böh09, Section 3.6]), and this defines the notion of a comodule algebra. Explicitly, B is an A-ring η_{B}: A B together with a coassociative coaction

δ: B H_{}_{A} B, b b_{1} _{A} b_{0},

which is counital and an algebra map,

η_{B} ε b_{1} b_{0} b, bd _{1} bd _{0} b_{1} d_{1} b_{0}d_{0}.

Similarly, as in the definition of a bialgebroid itself, for this condition to be well-defined one must also require

b_{1} _{A} b_{0}η_{B} a a b_{1} _{A} b_{0}.

The key example that reproduces [KK11] is the following:

5.13. The opmodule adjunction. For any Doi-Koppinen datum H, C, B , the Hcoaction δ on B turns the Eilenberg-Moore adjunction \xymatrix{A-Mod \ar[r] & B-Mod \ar[l]} for the monad B : B _{A} into an opmodule adjunction for the opmonoidal adjunction \xymatrix{E \ar[r] & H \ar[l]} defined in Section 5.9. The H-module category structure of B-Mod is given by the left B-action

b l _{A} m : b_{1} l _{A} b_{0}m,

where b B, l L (an H-module), and m M (a B-module).

Hence, as explained in Section 5.1, C defines a compatible pair of comonads C _{A} on B-Mod and A-Mod. The distributive law resulting from Theorem 2.9 generalises the Yetter-Drinfel’d braiding, as it is given for a B-module M by

χ: B _{A} C _{A} M  C _{A} B _{A} M ,
b _{A} c _{A} m  b_{1} c _{A} b_{0} _{A} m .

5.14. The main example. If H is a bialgebroid, then C : H is a module coalgebra with left action given by multiplication and coalgebra structure given by that of H. If H is a Hopf algebroid, then B : H^{op} is a comodule algebra with unit map η_{B} a : η 1 _{k} a and coaction

δ: H^{op} H_{}_{A} _{}H^{op}, b b_{}_{A} b_{}.

In the sequel we write B as _{A^{op}} H rather than H^{op} _{A} to work with H only. Then the distributive law becomes

χ: H _{A} M _{A^{op}} H  H _{A} M _{A^{op}} H ,
c _{A} m _{A^{op}} b  b_{}c _{A} m _{A^{op}} b_{},

for b, c H.

Proposition 3.5 completely characterises the right χ-coalgebras: in this example, they are given by right H-modules and left H-comodules M with right χ-coalgebra structure

ρ : m _{A^{op}} h h_{}m_{1} _{A} m_{0}h_{}.

In general, the characterisation of left χ-coalgebras mentioned in Remark 3.6 does not provide us with such an explicit description. Note, however, that one obtains left χcoalgebras from (left-left) Yetter-Drinfel’d modules:

5.15. Definition. A Yetter-Drinfel’d module over H is a left H-comodule and left Hmodule N such that for all h H, n N, one has

hn _{1} _{A} hn _{0} h_{1} n_{1} h_{}_{A} h_{2} n_{0}.

Indeed, each such Yetter-Drinfel’d module defines a left χ-coalgebra

N : _{H} N: H^{op}-Mod k-Mod

whose χ-coalgebra structure is given by

λ : h _{A} x _{H} n xn_{1} h_{}_{A^{op}} h_{}n_{1} _{H} n_{0}.

The resulting duplicial object C_{𝕋} N, M is the one studied in [KK11, Kow13]. Identifying _{A^{op}} H _{H} N _{A^{op}} N, the χ-coalgebra structure becomes

λ : h _{A} x _{H} n xn_{1} h_{}_{A^{op}} h_{}n_{1} n_{0}.

Using this identification, we give explicit expressions of the operators L_{n} and R_{n} as well as t_{n}^{𝕋} that appeared in Sections 4.5 and 4.7: first of all, observe that the right H-module structure on SM : H_{}_{A} M is given by

h _{A} m g : g_{}h _{A} mg_{},

whereas the right H-module structure on TM : M _{A^{op}} H_{}is given by

m _{A^{op}} h g : m _{A^{op}} hg.

The cyclic operator from Section 4.5 then results as

t_{n}^{𝕋} m _{A^{op}} h^{1} _{A^{op}} _{A^{op}} h^{n} _{A^{op}} n
m_{0}h_{}^{1} _{A^{op}} h_{}^{2} _{A^{op}} _{A^{op}} h_{}^{n}
A^{op} n_{1} h_{}^{n} h_{}^{1} m_{1} _{}_{A^{op}} n_{1} h_{}^{n} h_{}^{1} m_{1} _{}n_{0},

and for the operators L and R from Section 4.7 one obtains with the help of the properties [Sch00, Prop. 3.7] of the translation map (5.1):

\begin{matrix}L_{n} : h^{1} & _{A}  _{A} h^{n 1} _{A} m _{H} n \\ mn_{1} h_{}^{1} _{A^{op}} h_{}^{1} h_{}^{2} _{A^{op}}  _{A^{op}} h_{}^{n 1} n_{1}  _{H} n_{0}, \end{matrix}

along with

R_{n} : m _{A^{op}} h^{1} _{A^{op}}  _{A^{op}} h^{n} _{A^{op}} 1 _{H} n
m_{n 1} _{A} m_{n} h_{1}^{1} _{A} m_{n 1} h_{2}^{1}h_{1}^{2} _{A}
A m_{1} h_{n}^{1}h_{n 1}^{2} h_{1}^{n} _{A} m_{0} _{H} h_{n 1}^{1} h_{n}^{2} h_{2}^{n}n.

Compare these maps with those obtained in [KK11, Lemma 4.10]. Hence, one has:

L_{n} R_{n} m _{A^{op}} h^{1} _{A^{op}}  _{A^{op}} h^{n} _{A^{op}} 1 _{H} n
m_{0} h_{n 1}^{1} h_{n}^{2} h_{2}^{n}n _{1} m_{n 1} _{A^{op}} m_{n 1} m_{n} h_{1}^{1}
A^{op} h_{1}^{1} m_{n} m_{n 1} h_{2}^{1} h_{1}^{2} _{A^{op}}
A^{op} h_{1}^{n} h_{n}^{1} m_{1} h_{n 1}^{1} h_{2}^{n}n _{1} h_{n 1}^{1} h_{2}^{n}n _{0}
m_{0} h_{2}^{1} h_{2}^{n}n _{1} m_{1} _{}_{A^{op}} h_{1}^{1} _{A^{op}}
A^{op} h_{1}^{n} _{A^{op}} h_{1}^{n} h_{1}^{1} h_{2}^{1} h_{2}^{n}n _{1} m_{1} _{}h_{2}^{1} h_{2}^{n}n _{0}.

Finally, if M _{A^{op}} N is a stable anti Yetter-Drinfel’d module [BŞ08], that is, if

m_{0} n_{1} m_{1} _{}_{A^{op}} n_{1} m_{1} _{}n_{0} m _{A^{op}} n

holds for all n N, m M, we conclude by

L_{n} R_{n} m _{A^{op}} h^{1} _{A^{op}}  _{A^{op}} h^{n} _{A^{op}} n
m _{A^{op}} h_{1}^{1} _{A^{op}} _{A^{op}} h_{1}^{n} _{A^{op}} h_{1}^{n} h_{1}^{1} h_{2}^{1} h_{2}^{n}n
m _{A^{op}} h^{1} _{A^{op}} _{A^{op}} h^{n} _{A^{op}} n.

Observe that in [Kow13] this cyclicity condition was obtained for a different complex which, however, computes the same homology.

5.16. The antipode as a 1-cell. If A k, then the four actions , , , coincide and H is a Hopf algebra with antipode S: H H given by S h ε h_{}h_{}. The aim of this brief section is to remark that this defines a 1-cell that connects the two instances of Theorem 2.9 provided by the opmonoidal adjunction and the opmodule adjunction considered above.

Indeed, in this case we have A^{e}-Mod A-Mod k-Mod, but H^{op}-Mod H-Mod unless H is commutative. However, S defines a lax morphism σ: _{k}H id H _{k} id, given in components by

σ_{X}: X _{k} H H _{k} X, x _{k} h S h _{k} x.

The fact that this is a lax morphism is equivalent to the fact that S is an algebra antihomomorphism. Also, the lifted comonads agree and are given by H _{k} with comonad structure given by the coalgebra structure of H; clearly, γ id: idH _{k} H _{k} id is a colax morphism. Furthermore, the Yang-Baxter condition is satisfied, so we have that id, σ, γ is a 1-cell in the 2-category of mixed distributive laws. If we apply the 2-functor i to this, we get a 1-cell Σ, σ̃, γ̃ between a comonad distributive law on the category of left H-modules and one on the category of right H-modules. The identity lifts to the functor Σ: H-Mod Mod-H which sends a left H-module X to the right H-module with right action given by

x ⪦ h : S h x.

6. Hopf monads à la Mesablishvili-Wisbauer

6.1. Bimonads. A bimonad in the sense of [MW11] is a sextuple A, μ, η, Δ^{A}, ε^{A}, θ , where A: C C is a functor, A, μ, η is a monad, A, Δ^{A}, ε^{A} is a comonad and θ: AA AA is a mixed distributive law satisfying a list of compatibility conditions.

In particular, μ and Δ^{A} are required to be compatible in the sense that there is a commutative diagram

\xymatrix{AA \ar[d]_{AΔ^{A}} \ar[r]^{μ} & A \ar[r]^{Δ^{A}} & AA \\ AAA \ar[rr]_{θA} &  & AAA \ar[u]_{Aμ}}  (6.1)

The other defining conditions rule the compatibility between the unit and the counit with each other and with μ respectively Δ^{A}, see [MW11] for the details.

It follows immediately that we also obtain an instance of Theorem 2.9 in this situation: if we take A C^{𝔹} to be the Eilenberg-Moore category of the monad 𝔹 A, μ, η as in Section 2.13, then the mixed distributive law θ defines a lift 𝕍 V, Δ^{V}, ε^{V} of the comonad ℂ A, Δ^{A}, ε^{A} to A.

Note that in general, neither A nor C need to be monoidal, so B is in general not an opmonoidal monad. Conversely, recall that for the examples of Theorem 2.9 obtained from opmonoidal monads, B need not equal C.

6.2. Examples from bialgebras. In the main example of bimonads in the above sense, we in fact do have B C and we are in the situation of Section 5.9 for a bialgebra H over A k. The commutativity of (6.1) amounts to the fact that the coproduct is an algebra map.

This setting provides an instance of Proposition 2.19 since there are two lifts of B C from A k-Mod to B H-Mod: the canonical lift S T FU which takes a left H-module L to the H-module H _{k} L with H-module structure given by multiplication in the first tensor component, and the lift V which takes L to H _{k} L with H-action given by the codiagonal action g h _{k} y g_{1}h _{k} g_{2}y, that is, the one defining the monoidal structure on B.

In this example, the map β from Proposition 2.21 is given by

H _{k} L H _{k} L, g _{k} y g_{1} _{k} g_{2}y

which for L H is the Galois map from Definition 5.8. This is bijective for all L if and only if it is so for L H, which is also equivalent to H being a Hopf algebra. However, this Galois map should not just be viewed as a k-linear map, but as a natural H-module morphism between the two H-modules T L and V L , and this is the natural transformation Γ^{T,V} id from Section 2.17.

As shown in [LMW15, Theorem 5.8(c)], this characterisation of Hopf algebras in terms of the bijectivity of the Galois map extends straightforwardly to Hopf monads.

6.3. An example not from bialgebras. Another example of a bimonad is the nonempty list monad 𝕃^{}on Set, which assigns to a set X the set L^{}X of all nonempty lists of elements in X, denoted x_{1}, …, x_{n} . The monad multiplication is given by concatenation of lists and the unit maps x to x . The comonad comultiplication is given by Δ x_{1}, …, x_{n} x_{1}, …, x_{n} , …, x_{n} , the counit is ε x_{1}, …, x_{n} x_{1}, and the mixed distributive law

θ: 𝕃^{}𝕃^{}𝕃^{}𝕃^{}

is defined as follows: given a list

x_{1,1}, …, x_{1,n_{1}} , …, x_{m,1}, …, x_{m,n_{m}}

in L^{}L^{}X, its image under θ_{X} is the list with

_{i 1}^{m} n_{i} m i 1

terms, given by the lexicographic order, that is

x_{1,1}, x_{2,1}, x_{3,1} …, x_{m,1} , …, x_{1,n_{1}}, x_{2,1}, x_{3,1}, …, x_{m,1},
x_{2,1}, x_{3,1} …, x_{m,1} , …, x_{2,n_{2}}, x_{3,1}, … x_{m,1},
…,
x_{m,1} , x_{m,2} , …, x_{m,n_{m}} .

One verifies straightforwardly:

6.4. Proposition. 𝕃^{}becomes a bimonad on Set whose Eilenberg-Moore category is Set^{𝕃^{}}SemiGp, the category of (nonunital) semigroups.

The second lift V of the comonad 𝕃^{}that one obtains from the bimonad structure on SemiGp is as follows. Given a semigroup X, we have VX L^{}X as sets, but the binary operation is given by

VX VX VX
x_{1}, …, x_{m} y_{1}, …, y_{n} : x_{1}y_{1}, …, x_{m}y_{1}, y_{1}, …, y_{n} .

CYCLIC HOMOLOGY ARISING FROM ADJUNCTIONS  1093

Following Proposition 3.5, given a semigroup X, the unit turns the underlying set of X into an 𝕃^{}-coalgebra and hence we get a right χ-coalgebra structure on X. Explicitly, ρ_{X}: TX VX is given by

ρ x_{1}, …, x_{n} x_{1} x_{n}, x_{2} x_{n}, …, x_{n} .

The image of ρ is known as the left machine expansion of X [BR84]. 6.5. Proposition. The only θ-entwined algebra is the trivial semigroup . Proof. An 𝕃^{}-coalgebra structure β: T 𝕃^{}T is equivalent to T being a forest of at most countable height (rooted) trees, where each level may have arbitrary cardinality. The structure map β sends x to the finite list of predecessors of x. A θ-entwined algebra is therefore such a forest, which also has the structure of a semigroup such that for all x, y T with β y y, y_{1}, …, y_{n} we have

β xy xy, xy_{1}, …, xy_{n}, y, y_{1}, …, y_{n} .

Let T be a θ-entwined algebra. If T is non-empty, then there must be a root. We can multiply this root with itself to generate branches of arbitrary height. Suppose that we have a branch of height two; that is to say, an element y T with β y y, x (so, in particular, x y). Then β xy xy, y , but β xx xx, xy, x, y . This is impossible since x and y cannot both be the predecessor of xy.

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Università di Napoli Federico II, Dipartimento di Matematica e Applicazioni P.le Tecchio 80, 80125 Napoli, Italia

University of Glasgow, School of Mathematics and Statistics 15 University Gardens, G12 8QW Glasgow, UK

University of Glasgow, School of Mathematics and Statistics 15 University Gardens, G12 8QW Glasgow, UK

Email: niels.kowalzig@unina.it
      ulrich.kraehmer@glasgow.ac.uk
      p.slevin.1@research.gla.ac.uk
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