Oplax Hopf Algebras

OPLAX HOPF ALGEBRAS

arXiv:1710.01465v1 [math.CT] 4 Oct 2017

MITCHELL BUCKLEY, TIMMY FIEREMANS, CHRISTINA VASILAKOPOULOU, AND JOOST VERCRUYSSE Abstract. We introduce the notion of (op)lax Hopf monoids in a monoidal 2-category and show that Hopf V-categories introduced in [BCV16] are a particular type of oplax Hopf monoids in the 2-category Span|V described in [Böh16]. We also introduce Frobenius V-categories as the Frobenius objects in this category.

1. Introduction Over the last decades, a growing number of variations on the notion of Hopf algebra have surfaced, and this has lead to increasing investigations on the intriguing questions on how these various notions are related and can be unified in a single (categorical) framework. Weak Hopf algebras [BNS99] were introduced as a tool to describe extensions of Von Neumann algebras as crossed products by an action of a weak Hopf C ∗ -algebra. A weak Hopf algebra is an algebra that possesses at the same time a coalgebra structure, but their compatibility as well as the properties of the antipode are weakened compared to usual Hopf algebras. Weak Hopf algebras were one of the inspiring examples to define Hopf algebroids [BS04], which in turn lead to Hopf monads [BLV11]. Just as Hopf monads live in the monoidal 2-category of categories, functors and natural transformations, Hopf-type objects can be defined in any monoidal bicategory, and they have been studied in this way, e.g. [BL16], [DS97], [CLS10b] which also provide a setting to describe other generalized Hopf notions such as quasi Hopf algebras. In a different direction, multiplier Hopf algebras [Van94] provide a generalization of Hopf algebras to the non-unital setting. Recently, the theory of weak and multiplier Hopf algebras have been merged [VW15], [BGL15] and brought to the categorical setting [BL17]. In this paper we aim to settle a framework for an intermediate notion, called Hopf category [BCV16]. Just as a category can be viewed as a ‘monoid with many objects’, Hopf V-categories are a many-object generalization (or a categorification) of Hopf algebras. They are defined as categories enriched over comonoids (in a braided monoidal category V), that admit a suitable notion of antipode. Moreover, a Hopf category can be ‘packed’ (by taking the coproduct of it’s Hom-objects) and this leads to examples of weak (multiplier) Hopf algebras. In [Böh16], Böhm constructed a monoidal bicategory Span|K associated to any monoidal bicategory K. It has been shown that Hopf V-categories fit in the framework of [BL16], namely they can be viewed as a particular type of opmonoidal monads in Span|K, by considering the braided monoidal category V as a 1-object monoidal bicategory K. Similar to Hopf algebroids, the underlying monoid and comonoid structures of an opmonoidal monad make use of different tensor products (that together fit into a duoidal category structure). Recall however that for weak Hopf algebras, the underlying algebra and coalgebra structures were described over the same monoidal category, only their 1

2

BUCKLEY, FIEREMANS, VASILAKOPOULOU, AND VERCRUYSSE

compatibility was weakened. In the present paper, we provide an alternative description of Hopf categories in the setting of monoidal bicategories, which restores the symmetry between monoid and comonoid structures, as they are again defined over the same monoidal product. In order to obtain this new description, we introduce the notion of an oplax Hopf monoid in a monoidal 2-category or more general a monoidal bicategory. Similarly to weak Hopf algebras, an (op)lax Hopf algebra has a usual algebra and coalgebra structure, but their compatibility conditions are relaxed, as they are given by appropriate 2-cells rather than just identities. We give some immediate properties of oplax bimonoids and oplax Hopf monoids: In spirit of Tannaka-Krein duality, we proof that the category of lax modules over a monoid is monoidal if and only if this monoid is a lax bimonoid. We also discuss the relation between the existence of an antipode and the bijectivity of fusion morphism for a lax bimonoid. Other results such as fundamental theorem are postponed to a later paper. Next, following [Böh16], we construct in detail a suitable monoidal 2-category Span|V, where now the monoidal category V is considered as a monoidal 2-category with trivial 2-cells. In our approach, Hopf V-categories arise as a well-described class of oplax Hopf algebras inside Span|V. It follows then that the two descriptions of Hopf categories, in this work and [Böh16], are related via a dimension shift in the following sense: a Hopf category in [Böh16] is a 1-cell in a certain monoidal bicategory, whereas in our setting it is a 0-cell in a (different) monoidal bicategory. Moreover, the notion of a Frobenius monoid in (our) Span|V leads to the notion of Frobenius V-category. Although both Hopf and Frobenius categories are viewed as monoids and comonoids in the same monoidal (2-)category (albeit with different compatibility conditions), by a closer look, for a Hopf category the algebra and coalgebra structures are of different type, whereas for the Frobenius category, both algebra and coalgebra structure are of the same type. In a forthcoming paper [BFVV], we will show that a Hopf V-category that is locally rigid admits the structure of a Frobenius V-category, providing a generalization of the well-known Larson Sweedler theorem. Our paper is structured as follows. In Section 2, we recall some known results and fix notation, especially for monoidal bicategories and spans, that will used throughout the paper. In Section 4 we introduce oplax bialgebras and oplax Hopf monoids, and prove some first results for them. Section 4 consists of some elementary observations about groupoids viewed as objects in Span where they are shown to give examples of oplax Hopf monoids and Frobenius algebras. In Section 5 we recall the notion of Hopf category and introduce Frobenius V-categories. In Section 6, we describe extensively the category Span|V and show how the forgetful functor to Span is a 2-opfibration. In Section 7 we give our main theorem showing that Hopf categories are oplax Hopf monoids. 2. Preliminaries In this section, we provide some background material for what follows and fix the notation and terminology used. Monoidal, closed and rigid categories. We assume familiarity with (braided) monoidal categories (V, ⊗, I, α, λ, σ); see [Mac98] and [JS93].

OPLAX HOPF ALGEBRAS

3

An object A in a monoidal category V is left rigid or dualizable if it has a left dual A∗ , with evaluation and coevaluation maps coev : I → A ⊗ A∗ , ev : A∗ ⊗ A → I satisfying the usual triangle equalities. A left rigid or autonomous monoidal category is one where all objects are left rigid. A left rigid monoidal category is always left monoidal closed by setting [A, B] = B ⊗ A∗ . Dually, the category is right rigid when all objects have right duals; when the category is braided right duals are the same as left duals and we say that the category is simply rigid. As an example, for a commutative ring k, the rigid objects in Modk are the finitely generated and projective k-modules; restricting to these objects gives a rigid monoidal category Modfk . 2.1. Bimonoids, Hopf monoids, and Frobenius monoids. A monoid in a monoidal category V is an object A equipped with multiplication m : A ⊗ A → A and unit j : I → A that are associative and unitary. A monoid morphism is a map between the underlying objects which commutes with multiplications and units. Comonoids and their morphisms are defined dually, with comultiplication d : A → A ⊗ A and counit ǫ : A → I, and comonoid morphisms respecting the structure. Monoids and monoid morphisms in V form a category Mon(V) and similarly for comonoids Comon(V) – which is exactly Mon(V op )op . We also have categories of A-modules ModA for any monoid A, of objects M equipped with A-actions and morphisms that preserve them; dually, there are categories of comodules ComodC for any comonoid C. When V is braided, both categories of monoids and comonoids inherit a monoidal structure from V. A bimonoid (sometimes called bialgebra) is an object with a monoid structure and a comonoid structure that are compatible, in the sense that a bimonoid is a monoid in Comon(V) or equivalently a comonoid in Mon(V); bimonoids form a category Bimon(V) whose morphisms are maps that are both monoid and comonoid morphisms. A Hopf monoid is a bimonoid H equipped with an antipode s : H → H satisfying m ◦ (s ⊗ 1H ) ◦ d = j ◦ ǫ = m ◦ (s ⊗ 1H ) ◦ d. Since H has both a monoid and comonoid structure, we can define the convolution product f ∗ g of any two maps f, g ∈ V(H, H) as the composite m ◦ (f ⊗ g) ◦ d; this is an associative binary operation on V(H, H) with 1∗ := j ◦ ǫ as a unit. Using this terminology, the defining property of an antipode is that it is inverse to 1H under the convolution product. Hence, the antipode is unique when it exists, and bimonoid morphisms between Hopf monoids can be seen to automatically commute with the antipode. Therefore Hopf monoids form a full subcategory Hopf(V) of Bimon(V). There are further equivalent definitions of Hopf monoids. For example, the existence of an antipode on a bimonoid H is equivalent with stating that either of the canonical maps (1H ⊗ m) ◦ (d ⊗ 1H ) or (m ⊗ 1H ) ◦ (1H ⊗ d) are isomorphisms; these are sometimes referred to as fusion morphisms, see [Str98]. Alternatively, a monoid H is a bimonoid (resp. Hopf monoid) if and only if its category of modules ModH is monoidal (resp. monoidal closed) and the forgetful functor to V is strict monoidal (resp. monoidal and closed). A Frobenius monoid M is an object with a monoid and comonoid structure such that the usual Frobenius laws hold: (1M ⊗ m) ◦ (d ⊗ 1M ) = d ◦ m = (m ⊗ 1M ) ◦ (1M ⊗ d). Equivalently, a monoid M is Frobenius if and only if it is rigid, and M ∼ = M ∗ as M-modules: M with its regular M-action, and M ∗ with 1⊗coev

1⊗m⊗1

ev⊗1

M ∗ ⊗ M −−−−→ M ∗ ⊗ M ⊗ M ⊗ M ∗ −−−−→ M ∗ ⊗ M ⊗ M ∗ −−−→ M ∗

4


Another equivalent characterisation states that a monoid M is Frobenius if and only if the forgetful functor U : ModM → V has − ⊗ M as a right adjoint (note that for any monoid, it is always the case that (− ⊗ M) ⊣ U). A morphism between Frobenius monoids is just a morphism that is at the same time a monoid and comonoid morphism. For more details about Frobenius algebras, see [Str04b] and [CMZ02]. 2.2. Enriched categories. Recall that for a monoidal category V, a V-graph over a set X is a family of objects {Ax,y }x,y∈X in V; we give a preference to the notation Ax,y rather than the more common A(x, y). V-graph morphisms are functions between the sets of objects, along with arrows Fxy : Ax,y → Bf x,f y in V. A V-enriched category is a V-graph {Ax,y }X together with composition laws mxyz : Ax,y ⊗Ay,z → Ax,z and identities jx : I → Ax,x that satisfy the usual associativity and unity conditions; a V-functor is a V-graph morphism that respects the structure. We obtain categories V-Grph and V-Cat. If V is braided monoidal, then every V-category A has an opposite V-category Aop with the same objects and hom-objects Aop x,y := Ay,x . A standard reference for the theory of enriched categories is [Kel05]. One of our main working examples is Modk -categories, i.e. k-linear categories for k a commutative ring. This is particularly instructive for the lifting of classical results on Hopf and Frobenius algebras to their many-object, V-enriched generalizations. If A is a V-category, recall that a (right) A-module [Law73] is a V-graph {Nx,y }X ∈ V over the same set of objects as A, equipped with actions µxyz : Nx,y ⊗ Ay,z → Nx,z satisfying the usual conditions Nx,y ⊗ Ay,z ⊗ Az,w

µxyz ⊗1

Nx,z ⊗ Az,w

Nx,y ⊗ Ay,w

and

µxzw

1⊗myzw µxyw

Nx,y

1⊗jy

1

Nx,w

Nx,y ⊗ Ay,y µxyy

(1)

Nx,y

Morphisms are identity-on-objects V-graph maps that preserve the actions, namely ϕxy : Nx,y → Px,y with Nx,y ⊗ Ay,z

µxyz

ϕxz

ϕxy ⊗Ay,z

Px,y ⊗ Ay,z

Nx,z

µxyz

(2)

Px,z

These form a category of right A-modules which we call V-ModA . 2.3. Monoidal 2-categories. Recall that a monoidal 2-category K is a 2-category equipped with a pseudofunctor ⊗ : K × K → K and a unit object I : 1 → K which are associative and unital up to coherent equivalence; for an explicit description see [Car95; GPS95]. A monoidal 2-category is braided when it comes equipped pseudonatural equivalences σA,B : A⊗B → B⊗A and certain coherent invertible modifications; it is symmetric when σA,B is the equivalence inverse of σB,A . While our results could be expressed more generally in terms of a monoidal bicategory, we restrict ourselves to the more succinct 2-categorical context for simplicity and in light of the examples. Background for this, including coherence for symmetric monoidal bicategories, can be found for example in [GO13]. As a primary example, there is the symmetric monoidal 2-category (Cat, ×, 1) of categories, functors and natural transformations with the cartesian product of categories and the unit category. There is also (Bim, ⊗, k), the monoidal bicategory of k-algebras,

OPLAX HOPF ALGEBRAS

5

bimodules and bimodule maps, equipped with tensor product over the commutative base ring k, which is also the monoidal unit. Another example, central to what follows, is the monoidal 2-category of spans. 2.4. Spans. Recall that in any category C, a span from an object X to an object Y is a pair of maps with common domain as in the following diagram: S g

f

(3)

X

Y.

The map along the base marked with a bar is notation that indicates that the span is considered as a map from X to Y .If the category C has pullbacks, spans can be composed: given X o f S g / Y and Y o h T k / Z , the composite span is defined as S ×Y T πT

p

πS

S

.

T

f

g

h

X

(4)

k

Y

Z

The identity 1-cell is the identity span X o id X id / X . A 2-cell between spans X o f S g / Y and X o h T k / Y is a map u : S → T in C such that the following diagrams commutes S u

f

g

(5)

T h

k

X Y Vertically, these compose in a straightforward way. Horizontally, if u is a map between spans from X to Y and v is a map between spans from Y to Z then their horizontal composite is the unique morphism w that follows from the universal property of the pullback, as in the following diagram S ×Y Q S u g

f

Q

∃!w

T ×Y P m

T h

X

(6)

v n

P r

k

ℓ

Y

Z,

which forms a 2-cell X ⇓u∗v Z between the composite spans. Note that because pullbacks are only unique up to isomorphism, horizontal composition of 1-cells is associative and unitary only up to isomorphism; therefore the above

6


data forms a bicategory. Let Span1 (C) denote the 1-category of objects in C and isomorphism classes of spans (which makes composition strictly associative and unitary) and Span2 (C) the 2-category of objects in C, isomorphism classes of spans and span morphisms. From this point on, we will restrict ourselves to C = Set. In this case, as in any category with products, spans can be expressed as (f, g) : S → X × Y . Whenever we refer to Span below, it should be clear from the context whether we consider the ordinary category or the 2-category. It is not hard to verify that (Span, ×, 1) with the singleton set is a symmetric monoidal Y ×Y given by the span X × Y o id X × Y sw / Y × X (2-)category, with σ : X ×Y where sw is the switch map (x, y) 7→ (y, x) in Set. Notice that this is not a cartesian monoidal structure in Span: the categorical product in this category is given by the disjoint union. There is a faithful monoidal functor Set → Span

(7)

which acts as the identity on objects, and maps a function f : X → Y to the span X o id X f / Y . In a similar way there is an embedding from Setop to Span. 2.5. Pseudomonoids. If (K, ⊗, I) is a monoidal 2-category, recall [DS97, §3] that a pseudomonoid (or monoidale) A in K is a sixtuple (A, m, j, α, λ, ρ) with multiplication m : A ⊗ A → A, unit j : A → I, and invertible 2-cells A⊗A⊗A

A⊗m

A⊗A

A⊗j

A⊗I

m

α

∼ =

A⊗A

A

m

= m ∼

∼ =

∼

I ⊗A

ρ

λ

m⊗A

j⊗A

A⊗A

(8)

∼

A

satisfying appropriate coherence conditions. In particular, a strict monoid in K is a pseudomonoid where α, λ, ρ are identity 2-cells. For example, a pseudomonoid in (Cat, ×, I) is a monoidal category and a strict monoid is a strict monoidal category. Dually, we have the notions of a pseudocomonoid and strict comonoid in a monoidal 2-category. Clearly, a strict (co)monoid in a monoidal 2-category is an ordinary (co)monoid in the underlying monoidal 1-category. Definition 2.5.1. An oplax (or opmonoidal ) morphism between pseudomonoids A, B in a monoidal 2-category K is a 1-cell f : A → B equipped with 2-cells m

A⊗A

A

j

I

A

φ0 φ

f ⊗f

B⊗B

f

m

(9)

f

j

B

B

such that the following conditions hold, for α, λ, ρ as in (8): m

A⊗A A⊗m

A⊗A⊗A f ⊗f ⊗f

α

m⊗A

∼ = A⊗A ⇓ φ⊗1f

B⊗B⊗B

m

m

A⊗A

A f

A⊗m f ⊗f

B = A⊗A⊗A

⇓φ f ⊗f

B⊗m m

m⊗B

⇓ 1f ⊗φ

B⊗B

f ⊗f ⊗f

B⊗B⊗B

m⊗B

A ⇓φ

f

B⊗B

m

α

∼ = B⊗B

m

B

OPLAX HOPF ALGEBRAS

7

idA

idA ρ∼ =

λ∼ = m

A⊗A A⊗j 1f ⊗φ0

A∼ = A⊗I

f

= A

⇓ 1f

B

m

m

A⊗A

f

⇓φ B⊗B

f ⊗f f ⊗j

A

j⊗A

B =

φ0 ⊗1f

A∼ =I ⊗A

f

⇓φ B⊗B

f ⊗f j⊗f

λ∼ = f

A f

m

B

ρ∼ = f

where the associativity and unity constraints of the monoidal 2-category are suppressed. A lax morphism of pseudomonoids is defined similarly, with the 2-cells φ, φ0 pointing in the opposite direction and the conditions adjusted accordingly. For example, (op)lax morphisms between pseudomonoids in the monoidal 2-category Cat are precisely (op)lax monoidal functors; the usual structure maps are given by taking components of the natural transformations φ and φ0 , and the conditions they satisfy arise from the above pasted composites. We obtain a category PsMonopl (K) of pseudomonoids and oplax morphisms in any monoidal 2-category K, as well as PsMonlax (K) which was called Mon(K) in [CLS10a]. In particular, we can restrict to the full subcategory Monopl (K) of strict monoids and oplax 1-cells. Dually, we can talk of oplax morphisms between pseudocomonoids in any monoidal 2-category, and form the appropriate category. If Kop is the monoidal 2-category with reversed 1-cells, it is the case that PsMonopl (Kop ) ∼ = PsComonopl (K)op since the 2-cells do not reverse. This isomorphism effectively defines what we mean by oplax morphism of pseudocomonoids. In fact, PsMonopl (K) is a 2-category: if (f, φ, φ0 ) and (g, ψ, ψ0) are two oplax morphisms between pseudomonoids A and B, a 2-cell between them is a 2-cell α : f ⇒ g in K which is compatible with multiplications and units, in the sense that A

m f ⊗f

A⊗A

f

⇓φ

B

=

f

A

m

⇓α A⊗A

ψ

⇓

B

g

(10)

⇓ α⊗α B⊗B

g⊗g j

I

A ⇓ φ0

m

g⊗g

f

j

f

A

= B

m

B⊗B ⇓α

I

ψ0

j

⇓

g

B.

j

For K = Cat, these are monoidal natural transformations between oplax monoidal functors. In [McC00; GPS95] lax morphisms are called monoidal and 2-cells monoidal transformations. Furthermore, when K is braided monoidal 2-category with braiding σ, the 2-category PsMonopl (K) obtains a monoidal structure itself: if (A, mA , jA ) and (B, mB , jB ) are two pseudomonoids, the multiplication and unit on their tensor product are defined as A⊗σ⊗B

m ⊗m

B (A ⊗ B) ⊗ (A ⊗ B) −−−−→ (A ⊗ A) ⊗ (B ⊗ B) −−A−−−→ A⊗B

jA ⊗jB I∼ = I ⊗ I −−−−→ A ⊗ B

(11)

8


and the coherence data can be readily constructed also. 3. Oplax bimonoids and oplax Hopf monoids In this section, we define lax variations of usual bimonoids and Hopf monoids in a braided monoidal 2-category and give some first properties. Examples and applications will be provided in the following sections. 3.1. Oplax bimonoids. We already observed that in a braided monoidal 2-category K, the 2-category PsMonopl (K) of pseudomonoids with oplax morphisms and 2-cells inherits a monoidal structure. Hence, one can then consider pseudocomonoids in that category; if we restrict to strict monoids, one still has a monoidal 2-category Monopl (K) and by considering a strict comonoid in that category, we obtain the following definition. Definition 3.1.1. In a braided monoidal 2-category (K, ⊗, I, σ), an oplax bimonoid is a comonoid in Monopl (K). More explicitly, it is an object M endowed with a monoid structure (M, m, j) and a comonoid structure (M, δ, ε), and 2-cells δ⊗δ

M ⊗M

M ⊗M ⊗M ⊗M M ⊗σ⊗M

m

(12)

M ⊗M ⊗M ⊗M

θ⇑

m⊗m

M I j

∼

/

M

/

δ

M ⊗M

I ⊗I j⊗j

θ0 ⇑

M ⊗M

δ

m

ε⊗ε

/

I ⊗I ∼

χ⇑

M

M ⊗M

/

ε

I

I j

id

/

χ0 ⇑

M

ε

(13)

I

/

id

I

that express that δ and ε are oplax maps as in (9); this data is subject to a number of axioms which defer to Appendix A.1. Notice that while the individual monoid and comonoid structures are strict, the interaction between them is oplax. As a preliminary example, when K = (Cat, ×, 1) then an oplax bimonoid is simply a (strict) monoidal category because the tensor product is cartesian. Other trivial examples are given by bimonoids in a monoidal category V, that can be considered as a 2-category with trivial 2-cells. The appropriate notion of morphisms between oplax bimonoids can be obtained from the formally defined category Comonopl (Monopl (K)). Definition 3.1.2. An oplax bimonoid morphism between oplax bimonoids M and N with structure 2-cells (θ, θ0 , χ, χ0 ) and (ξ, ξ0 , ω, ω0) respectively, is a 1-cell f : M → N that is both an oplax monoid morphism and an oplax comonoid morphism (see Definition 2.5.1) with structure 2-cells m

M ⊗M

M

I

j

M

M

δ

M ⊗M

M

φ0 f ⊗f

N ⊗N

φ

f

m

N

j

f

N

f

N

ψ

δ

f ⊗f

N ⊗N

f

N

ψ0

(14)

ε

ε

I

OPLAX HOPF ALGEBRAS

9

satisfying four further axioms that express that ψ and ψ0 are monoidal 2-cells (10). The axioms are explicitly recorded in Appendix A.2. Remark 3.1.3. One might observe that oplax bimonoids have axioms that are very similar to those of (strict) duoidal (or 2-monoidal ) categories [AM10]. This can be explained as follows. An oplax bimonoid being an object in Comonopl (Monopl (K)) is in particular an object in PsComonopl (PsMonopl (K)). On the other hand, a duoidal category is exactly an object in PsMonopl (PsMonopl (Cat)) and a strict duoidal category is an object in Monopl (Monopl (Cat)). Hence these notions are ‘half-dual’ to each other. Furthermore, if (M, m, j) is a pseudomonoid in K such that m and j have left adjoints that we denote by m† and j † , then (M, m† , j † ) is a pseudocomonoid in K. In particular, if M is an oplax bimonoid in K whose comultiplication and counit have right adjoints, then this induces a duoidal structure on M [BL16]. We hope that future work will shed more light on this connection. We now state an appropriate module notion for pseudomonoids, which helps characterize oplax bimonoids among monoids based on their categories of modules. Definition 3.1.4. An oplax module over a pseudomonoid M is an object X ∈ K equipped with a 1-cell ρ : X ⊗ M → X and 2-cells X ⊗M ⊗M

M ⊗m

X ⊗M

X

X⊗j

X ⊗M ξ0

ρ⊗M

X ⊗M

ξ

ρ

(15)

ρ id

ρ

X

X

satisfying canonical compatibility axioms. When ξ and ξ0 are identities, we say that the module X is strict. One can analogously describe lax modules and pseudomodules for M, as well as the corresponding variations of comodules over a pseudocomonoid. The precise axioms can be obtained by observing that oplax modules are oplax algebras for the 2-monad (M ⊗ −) on K [Mar99, p.96]. Oplax modules form a 2-category that we denote as Modoplax and which has a forgetful 2-functor to K. We then obtain M the following theorem, for which we assume that the monoidal unit of K is a regular generator. Theorem 3.1.5. Let (M, m, j) be a (strict) monoid in a braided monoidal 2-category K. Then there is a bijective correspondence between oplax bimonoid structures on M and monoidal structures on Modoplax such that the forgetful functor to K is strict monoidal. M As this theorem is a direct generalization of the corresponding theorem for usual bimonoids, we only give a sketch of the constructions for the oplax case, leaving the details to the reader. Proof. Suppose first that (M, m, j, δ, ε) is a oplax bimonoid. If X and Y are oplax modules over M, then we define an action on X ⊗ Y by 1⊗1⊗δ

1⊗σ⊗1

ρ ⊗ρ

X Y ρ = X ⊗ Y ⊗ M −−−−→ X ⊗ Y ⊗ M ⊗ M −−−−→ .X ⊗ M ⊗ X ⊗ Y −− −−→ X ⊗Y

Then, even if X and Y are strict modules, the pair (X ⊗ Y, ρ) naturally becomes an oplax module, due to the structure 2-cells θ and θ0 from Definition 3.1.1. Similarly, by

10


defining an action ε : I ⊗ M ∼ = M → I on I, this becomes an oplax module due to χ and χ0 . Thanks to the strict coassociativity and counity of the comonoid structure of M, this then defines a monoidal structure on Modoplax whose forgetful functor to K is M strict monoidal. Conversely, suppose that M is a monoid in K such that Modoplax has a monoidal M structure with a strict monoidal functor to K. Since (M, m) is a (strict) M-module itself, M ⊗ M has an oplax M-module structure with action ρ. This allows to define a comultiplication on M as j⊗j⊗1

ρ

M −−−→ M ⊗ M ⊗ M − → M ⊗ M. Similarly, since I is also an oplax module, it comes with a coaction I ⊗ M ∼ =M →I which we choose as the counit. The monoidal associativity and unity constraints lead to the coassociativity and counity of M as a strict comonoid, and the structure 2-cells for both M ⊗ M and I as oplax modules lead to the four structure 2-cells for M as oplax bimonoid. 3.2. Oplax Hopf monoids. An oplax bimonoid is a variation on the notion of a bialgebra; our next aim is to describe the corresponding variation of a Hopf algebra. Let us start by describing the convolution monoidal structure [DS97, Prop.4] on a hom-category between a pseudomonoid and pseudocomonoid in a bicategory, that naturally generalizes the classical convolution for (co)monoids in monoidal categories. Lemma 3.2.1. Let (M, m, j) be a pseudomonoid and (C, δ, ε) a pseudocomonoid in K. Then the hom-category K(C, M) is a monoidal category with tensor product ∗ defined on 1-cells f, g : C → M by f ∗g :C

δ

/

f ⊗g

C⊗C

/

M ⊗M

m

/

M

and on two-cells α : f ⇒ f ′ , β : g ⇒ g ′ by C; ⊗ C

C

✇ δ ✇✇✇ ✇ ✇ ✇ ✇✇ ●● = ●● ●● ● δ ●●#

C⊗C

f ⊗g

⇓α⊗β

/

f ′ ⊗g ′

M ⊗M ■■ ■■ m ■■ ■■ ■$

M ✉: ✉✉ ✉ ✉ ✉✉ m ✉✉

/

=

.

M ⊗M

The monoidal unit is given by I ∗ = ε ◦ j. If moreover M and C are strict, then this monoidal structure is strict as well. If M is an oplax bimonoid in K, the above lemma assures that K(M, M) has a monoidal structure given by convolution. Notice that (K(M, M), ◦, 1M ) is also monoidal for any object M, where ◦ is the horizontal composition. In general, these structures are not compatible; however, if M is a so-called map monoidale, these form a duoidal structure on K(M, M). For a detailed discussion, we refer to [BL16, Section 3.3]. In the classical case, the antipode is a convolution inverse to the identity. In our relaxed notion, this condition will be weakened in an appropriate sense. Definition 3.2.2. Let V be a monoidal category. A generalized inverse for an object X in V is an object Y together with morphisms t1 : X ⊗ Y → I and t2 : Y ⊗ X → I

OPLAX HOPF ALGEBRAS

11

such that Y ⊗ t1 = t2 ⊗ Y and t1 ⊗ X = X ⊗ t2 , and these morphisms are moreover invertible. Definition 3.2.3. Suppose that (M, m, j, δ, ε) is an oplax bimonoid in a braided monoidal 2-category (K, ⊗, I, σ). An oplax antipode is defined to be a generalized inverse of 1M in the monoidal convolution category K(M, M). Explicitly, an antipode is a 1-cell s : M → M equipped with 2-cells τ1 : 1M ∗ s ⇒ I ∗ and τ2 : s ∗ 1M ⇒ I ∗ as in M ⊗M

M ⊗s

d

m

τ1 ǫ

M

M ⊗M j

I τ2

d

M ⊗M

s⊗M

M .

(16)

m

M ⊗M

such that s ∗ τ1 = τ2 ∗ s : s ∗ 1M ∗ s ⇒ s and τ1 ∗ 1M = 1M ∗ τ2 : 1M ∗ s ∗ 1M ⇒ 1M and these are invertible 2-cells. We refer to this situation by saying that s is a generalized convolution inverse for the identity on M. The above explicitly says that s ∗ M ∗ s ∼ = s and M ∗ s ∗ M ∼ = M via specific invertible 2-cells. Definition 3.2.4. An oplax Hopf monoid is an oplax bimonoid with an oplax antipode. A morphism of oplax Hopf monoids is an oplax bimonoid morphism between the underlying oplax bimonoids, which preserves the antipode. We obtain a category OplHopf(K) of oplax Hopf monoids and morphisms, with a faithful forgetful functor to Comonopl (Monopl (K)). Remark 3.2.5. The name ‘generalized convolution inverse’ is inspired by the notion of generalized inverse in semigroup theory. An element x in a semigroup S is called a generalized inverse for x′ ∈ S if and only if xx′ x = x and x′ xx′ = x′ . Since generalized inverses in arbitrary semigroups are not necessarily unique, at this moment the authors can not guarantee if an oplax antipode on an oplax bimonoid is unique if it exists. A possible solution might be to suppose the existence of an involution on the convolution category and to consider Moore-Penrose inverses instead of generalized. Although this might provide the wanted uniqueness of antipodes and such a structure is available in the example of Span we consider in the next section, we postpone this discussion to a future paper. The (possible) non-uniqueness of antipodes is also the reason why we define morphisms of oplax Hopf monoids as bimonoid morphisms that preserve the antipode. We know however that an invertible element in a semigroup is has a unique generalized inverse (which is in that case the usual inverse). Hence Hopf monoids can be considered as oplax Hopf monoids and have a unique antipode. The definition of an antipode in terms of the generalized inverse conditions the authors cannot currently deduce whether such a condition is too weak or too strong to capture the desired structure. In fact, in the examples we study in the next sections, the 2-cells s ∗ τ1 = τ2 ∗ s and τ1 ∗ 1M = 1M ∗ τ2 are all identities. Further work on this subject should confirm that the definition of antipode here is exactly the ‘correct’ notion of antipode, or present an improved definition.

12


Another consequence of the fact that generalized inverses are possibly not unique, is the fact that we have no immediate proof of the fact that the antipode is an anti-monoid, anti-comonoid morphism. The following result can be worked out, if we recall that a monoidal 2-functor between 2-categories preserve pseudomonoids and pseudocomonoids, see [DS97, Prop. 5]. Proposition 3.2.6. Let F : K → K′ be a braided monoidal 2-functor. If M is an oplax bimonoid (resp. Hopf monoid) in K, then F M is an oplax bimonoid (resp. Hopf ) monoid in K′ . We end this section by rephrasing the definition of oplax Hopf monoid in terms of fusion morphisms. Consider M ⊗ M as a (strict) left M-module and (strict) right M-comodule with actions given by m⊗1

ρ =M ⊗ M ⊗ M −−−−M → M ⊗M 1

⊗δ

M χ =M ⊗ M −− −→ M ⊗ M ⊗ M

Denote by M KM (M ⊗ M, M ⊗ M) the monoidal subcategory of the endo-hom category K(M ⊗ M, M ⊗ M) consisting of those 1-cells that are (strictly) left M-linear and right M-colinear, i.e. M-module and M-comodule maps. With this notation, we have the following result. Lemma 3.2.7. Then there is an isomorphism of monoidal categories (M KM (M ⊗ M, M ⊗ M), ◦, 1M ⊗M ) ∼ = (K(M, M), ∗, I ∗ ) Proof. Any f : M → M is mapped to 1

1

⊗δ

⊗f ⊗1

m⊗1

M M F (f ) = M ⊗ M −− −→ M ⊗ M ⊗ M −− −−−−M → M ⊗ M ⊗ M −−−−M → M ⊗ M.

One easily verifies that this is left M-linear and right M-colinear. Conversely, given any left M-module and right M-comodule g : M ⊗ M → M ⊗ M, define j⊗1M

1

g

⊗ε

M G(g) = M −−−→ M ⊗ M − → M ⊗ M −− −→ M

The bijection follows using the strict (co)unity conditions for M, and similarly for the morphisms. Furthermore, one easily verifies that F (f ∗ f ′ ) = F (f ) ◦ F (f ′ ), hence this is an isomorphism of monoidal categories. Under the above isomorphism of categories, the identity 1M ∈ K(M, M) corresponds to the so-called fusion 1-cell 1

⊗δ

m⊗1

M M ⊗ M −− −→ M ⊗ M ⊗ M −−−−M → M ⊗M

and following result is deduced. Corollary 3.2.8. An oplax bimonoid is an oplax Hopf monoid if and only if the fusion 1-cell has a generalized inverse in the monoidal category M KM (M ⊗ M, M ⊗ M). 4. The object X 2 in Span In this section, we describe various algebraic properties of X 2 as an object in Span. These are necessary for identifying Frobenius V-categories and Hopf V-categories as certain objects in Span|V of Section 7. Although the results of this section are also valid for arbitrary groupoids, we restrict ourselves to give explicit proofs only in the case X 2 and state the theorems in general case.

OPLAX HOPF ALGEBRAS

13

4.1. Trivial monoid and comonoid structures. Since every set X has a canonical comonoid structure in Set given by the diagonal map ∆ : X → X × X and unique map ! : X → 1, the monoidal embedding Set → Span as in (7) yields a comonoid structure on X in Span with (id, ∆) : X → X × X 2 as the comultiplication and (id, !) : X → X × 1 as the counit; we refer to this as the trivial comonoid structure on X in Span. Similarly, the contravariant embedding yields a monoid structure on X which is the reverse of the comonoid structure and is called the trivial monoid structure on X. For objects of the form X 2 in Span, the trivial comonoid structure is (X 2 , ζ, ν) via X2

X2 ∆X 2

id

X2

id

X4

ζ

(17)

!

X2

1

ν

Remark 4.1.1. Notice that while Comon(Set) ∼ = Set as is the case in any cartesian monoidal category, here Comon(Span) ≇ Span since the latter is no longer cartesian; hence our so-called ‘trivial’ comonoid structure above is not so in that usual way. 4.2. Groupoid monoid and comonoid structures. For any set X, one can form the codiscrete category whose objects are elements of X and where there is a single arrow (x, y) ∈ X 2 between any pair of objects x, y ∈ X. The identity arrows are the pairs (x, x) and composition is ((x, y), (y, z)) 7→ (x, z). This category is in fact a groupoid: inverses are computed by (x, y) 7→ (y, x). This groupoid gives rise to a monoid structure on X 2 in Span with the following multiplication and unit: X 3❈

④ 1×∆×1 ④④④ ④ ④ }④④

X4

✤

µ

❈❈ ❈❈1×!×1=π13 ❈❈ ❈! / X2

X❉ ❉❉ ③③ ❉❉∆ ③ ③ ❉❉ ③ ③ ❉! ③ ③ }③ ✤ / X2 !

1

(18)

η

We call this the groupoid monoid structure on X 2 . The reverse in Span, namely δ = (π13 , 1 × ∆ × 1) : X 3 → X 2 × X 4 and υ = (∆, !) : X → X 2 × 1 is called the groupoid comonoid structure on X 2 . Similarly, an arbitrary groupoid G = ( G1 st // G0 ) with set of objects G0 , set of morphisms G1 , and s, t the source and target maps, gives rise to a monoid (and a comonoid) in Span with multiplication and counit as follows qq qqq q q q x qq q (s,t)

G1 × G1

G2 ▼▼ ▼▼▼ ▼▼m ▼▼▼ ▼▼▼ & ✤ /

G0 ❇

G1

1

⑤ ! ⑤⑤ ⑤ ⑤⑤ ~⑤ ⑤ ✤

❇❇ ❇❇e ❇❇ ❇ /

(19) G1

where G2 = G1s ×t G1 are the composable morphisms of G and e : G0 → G1 gives the identity. 4.3. Frobenius monoid structure. Recall that a Frobenius pseudomonoid in any monoidal 2-category (or bicategory) is an object with a pseudomonoid and pseudocomonoid

14


structure (M, m, j, d, ǫ) together with isomorphisms d⊗1

M ⊗M

∼ =

m 1⊗δ

∼ = M ⊗M ⊗M

M ⊗M ⊗M

M

(20)

1⊗m d

M ⊗M

m⊗1

satisfying certain coherence conditions. For a detailed study of such objects and equivalent formulations see [LSW11, p. 2.6] in a 2-categorical context. Regarding morphisms, if f : A → B is a 1-cell between Frobenius pseudomonoids we may choose any of the four combinations of lax or oplax structures (Definition 2.5.1) between the pseudo(co)monoids, or their strict versions. For example, we can denote by Frobopl,opl the category of (strict) Frobenius monoids with oplax monoid, oplax comonoid morphisms (f, φ, φ0, ψ, ψ0 ) between them. The following proposition exhibits how Frobenius monoids in the monoidal category Span arise naturally from arbitrary sets and groupoids. Proposition 4.3.1. (i) For any set X, the trivial monoid and trivial comonoid structure make X into a Frobenius monoid in Span. (ii) For any set X, the groupoid monoid and comonoid structures on X 2 make it a Frobenius monoid in Span. (iii) For any groupoid G, the groupoid monoid and comonoid structures on G1 make it a Frobenius monoid in Span. Proof. The first statement is easily verified. We prove the second statement, whereas the proof of the third is along the same lines. In order to verify the (strict) Frobenius condition (20), the middle composite is X4 π134

p

π124

X3

X3

1×∆×1

1×∆×1

π13 π13 µ

X2 × X2

δ

X2

X2 × X2

which explicitly acts as (a, b, b, d) →7 (a, b, c, d) 7→ (a, c, c, d). The downside composite on the other hand is X4

12 ×∆×1

p

1×∆×12

12 ×π13

X2 × X3

X3 × X2 13 ×∆×1

X2 × X2

1×δ

π13 ×12

1×∆×13

X2 × X2 × X2

µ×1

X2 × X2

which is the span. The upper composite can be verified in a similar way.

OPLAX HOPF ALGEBRAS

15

4.4. Oplax bimonoid and oplax Hopf monoid structures. The following establishes the oplax bimonoid structure of X 2 in Span. Proposition 4.4.1. (i) For any set X, the groupoid monoid and trivial comonoid structures on X 2 make it an oplax bimonoid in Span. (ii) For any set X, the trivial monoid and groupoid comonoid structures on X 2 make it an oplax bimonoid in Span. (iii) For any groupoid G, the trivial monoid and groupoid comonoid structures on G1 make it an oplax bimonoid in Span. Proof. Since Span is isomorphic to Spanop , the first two statements are equivalent, whereas the third statement can be proved in essentially the same way. The proof of the second statement is verified as follows. Span is a symmetric monoidal (2-)category and X 2 is a monoid, hence X 2 × X 2 is also a monoid with multiplication and unit deduced from (11). The data for the oplax bimonoid structure (Definition 3.1.1) are span morphisms θ, θ0 , χ, χ0 defined below. X 3❊

1∆1

❊❊ ❊❊ ❊❊ ∆ ❊❊ ❊❊ ❊❊ "

②② ② |②

X 4❊

②② ② |②

❊❊∆2 ❊"

❊❊∆2 ❊"

id ②② |②②

id

X 4❊

id ②② |②②

X 4 b❊

X 8❊

X8

❊❊ ❊❊ ❊❊ ❊❊ 1∆1 ❊❊❊ ❊

X8

θ=id