A simplicial approach to multiplier bimonoids
aa r X i v : . [ m a t h . C T ] A p r A SIMPLICIAL APPROACH TO MULTIPLIER BIMONOIDS
GABRIELLA B ¨OHM AND STEPHEN LACK
Abstract.
Although multiplier bimonoids in general are not known to correspondto comonoids in any monoidal category, we classify them in terms of maps from theCatalan simplicial set to another suitable simplicial set; thus they can be regardedas (co)monoids in something more general than a monoidal category (namely, thesimplicial set itself). We analyze the particular simplicial maps corresponding tothat class of multiplier bimonoids which can be regarded as comonoids. Introduction
The recent paper [6] showed that monoids, as well as many generalizations, includ-ing monads, monoidal categories, skew monoidal categories [9], and internal versionsof these, can be classified as simplicial maps from the
Catalan simplicial set C toappropriately chosen simplicial sets. For the constructions in the current paper themost relevant observation in [6] is a bijective correspondence between monoids in amonoidal category M and simplicial maps from C to the nerve N ( M ) of M . Bialgebras — over a field or, more generally, in a braided monoidal category C —can be defined as comonoids in the monoidal category M of monoids in C . Thusapplying the results of [6], they are classified by simplicial maps from C to the nerve N ( M op ) of the category M regarded with the opposite composition.Classically, Hopf algebras over a field are defined as bialgebras with a further prop-erty.
Multiplier Hopf algebras [10] generalize Hopf algebras beyond the case when thealgebra has a unit. The typical motivating example of a multiplier Hopf algebra con-sists of finitely supported functions on an infinite group with values in the base field.The analogous notion of multiplier bialgebra was introduced later, in [2], together withits ‘weak’ generalization.For many applications it is important to work with Hopf algebras and bialgebras notonly over fields but, more generally, in braided monoidal categories which are differentfrom the symmetric monoidal category of vector spaces. The formulation of multiplierbialgebras in braided monoidal categories is our longstanding project initiated in [3].Generalizing some constructions in [8] to any braided monoidal category C , wedescribed in [4] how certain multiplier bimonoids in C can be seen as certain comonoidsin an appropriately constructed monoidal category M . In light of the findings of [6],this gives rise to a correspondence between these multiplier bialgebras and certainsimplicial maps from the Catalan simplicial set C to N ( M op ).The aim of this paper is to go beyond that characterization and prove a bijectionbetween arbitrary multiplier bimonoids in C and arbitrary simplicial maps from C toa suitable simplicial set M . The simplicial maps C → N ( M op ) that correspond, via Date : August 24, 2018. the results of [4] and [6], to nice enough multiplier bimonoids, turn out to factorizethrough a canonical embedding of a simplicial subset of M into N ( M op ). Regular multiplier bimonoids constitute a distinguished class of multiplier bimonoids.In order to classify them as well, we also present a simplicial set M with the propertythat simplicial maps C → M correspond bijectively to regular multiplier bimonoids in C . Notation.
Throughout the paper, C denotes a braided monoidal category. We donot assume that it is strict but — relying on coherence — we omit explicitly denotingthe associativity and unit isomorphisms. The monoidal unit is denoted by I and themonoidal product is denoted by juxtaposition (we also use the power notation for theiterated monoidal product of the same object). The braiding is denoted by c . Thecomposite of morphisms f : A → B and g : B → C in C is denoted by g.f : : A → C and we write 1 for the identity morphisms in C . Acknowledgement.
We gratefully acknowledge the financial support of the Hun-garian Scientific Research Fund OTKA (grant K108384), as well as the AustralianResearch Council Discovery Grant (DP130101969) and an ARC Future Fellowship(FT110100385). The second named author is grateful for the warm hospitality of hishosts during visits to the Wigner Research Centre in Sept-Oct 2014 and Aug-Sept2015. 2.
Preliminaries on the Catalan simplicial set
In this section we briefly recall from [6] an explicit description of the Catalan sim-plicial set and its role in the classification of monads in bicategories; thus in particularof monoids in monoidal categories.2.1.
Simplicial sets.
Consider the simplex category ∆ whose objects are non-emptyfinite ordinals and whose morphisms are the order preserving functions. By definition,a simplicial set is a presheaf on ∆. Explicitly, a simplicial set W is given by a collection { W n } of sets labelled by the natural numbers n — the sets of n -simplices — togetherwith the face maps d i : W n → W n − and the degeneracy maps s i : W n → W n +1 , for0 ≤ i ≤ n , obeying the simplicial relations : d i d j = d j − d i if i < j s i s j = s j +1 s i if i ≤ jd i s j = s j − d i if i < j i ∈ { j, j + 1 } s j d i − if i > j + 1 . An n -simplex is said to be degenerate if it belongs to the image of one of the degeneracymaps, otherwise it is non-degenerate .We often draw an n -simplex w as an n -dimensional oriented geometric simplexwhose n − d i ( w ). For example, for n = 2 we draw an oriented SIMPLICIAL APPROACH TO MULTIPLIER BIMONOIDS 3 triangle w := d d ( w ) = d d ( w ) w := d ( w ) * * ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ w w := d d ( w ) = d d ( w ) w := d ( w ) ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ w := d ( w ) / / w := d d ( w ) = d d ( w ) , for n = 3 we draw an oriented tetrahedron, and so on.A simplicial map is a natural transformation between such presheaves. Explicitly,a simplicial map W → V is a collection of functions { f n : W n → V n } labelled by thenatural numbers n which commute with the face and degeneracy maps in the sensethat s i f n = f n +1 s i and d i f n = f n − d i for all possible values of i .2.2. The Catalan simplicial set.
The
Catalan simplicial set C has a single 0-simplex ∗ . It has two 1-simplices: s ( ∗ ) and a non-degenerate one to be called α .There are three degenerate 2-simplices s s ( ∗ ) = s s ( ∗ ), s ( α ) and s ( α ) and twonon-degenerate ones to be denoted by ∗ α (cid:31) (cid:31) ❄❄❄❄❄❄❄ τ ∗ α ? ? ⑧⑧⑧⑧⑧⑧⑧ α / / ∗ ∗ s ( ∗ ) ❆❆❆❆❆❆❆❆ ε ∗ s ( ∗ ) ? ? ⑧⑧⑧⑧⑧⑧⑧ α / / ∗ . All higher simplices are generated coskeletally , meaning that for any natural number n >
2, and for any n -boundary (that is, n + 1-tuple { w , . . . , w n } of n − d j ( w i ) = d i ( w j +1 ) for all 0 ≤ i ≤ j < n ) there is a unique filler (that is,an n -simplex w obeying d i ( w ) = w i for all 0 ≤ i ≤ n ). In this situation we write w = ( w , . . . , w n ).From this property of the Catalan simplicial set one can deduce that there are fournon-degenerate 3-simplices φ = ( τ, τ, τ, τ ) , λ = ( ε, s ( α ) , τ, s ( α )) , ̺ = ( s ( α ) , τ, s ( α ) , ε ) , κ = ( ε, s ( α ) , s ( α ) , ε )corresponding to the four tetrahedra drawn below. ∗ α / / α (cid:15) (cid:15) α ❄❄❄ ττ (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄ ∗ α (cid:15) (cid:15) ττ ⑧⑧⑧⑧⑧⑧⑧ α ⑧⑧⑧ (cid:127) (cid:127) ⑧⑧⑧ ∗ ∗ α o o ∗ α / / α (cid:15) (cid:15) α ❄❄❄ s ( α ) s ( α ) (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄ ∗ s ( ∗ ) (cid:15) (cid:15) τ ε ⑧⑧⑧⑧⑧⑧⑧ α ⑧⑧⑧ (cid:127) (cid:127) ⑧⑧⑧ ∗ ∗ s ( ∗ ) o o ∗ s ( ∗ ) / / α (cid:15) (cid:15) α ❄❄❄ ετ (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄ ∗ s ( ∗ ) (cid:15) (cid:15) s ( α ) s ( α ) ⑧⑧⑧⑧⑧⑧⑧ α ⑧⑧⑧ (cid:127) (cid:127) ⑧⑧⑧ ∗ ∗ α o o ∗ s ( ∗ ) / / α (cid:15) (cid:15) α ❄❄❄ εs ( α ) (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄ ∗ s ( ∗ ) (cid:15) (cid:15) εs ( α ) ⑧⑧⑧⑧⑧⑧⑧ α ⑧⑧⑧ (cid:127) (cid:127) ⑧⑧⑧ ∗ ∗ s ( ∗ ) o o For some equivalent, more conceptual, descriptions of C consult [6].2.3. The nerve of a bicategory.
Any bicategory B determines a simplicial set N ( B )known as the nerve of B . The 0-simplices of N ( B ) are the objects of B . The 1-simplicesare the 1-cells of B , with faces provided by the source and the target maps. For a GABRIELLA B ¨OHM AND STEPHEN LACK given 2-boundary { w , w , w } , the 2-simplices w w ! ! ❈❈❈❈❈❈❈❈ w w w = = ④④④④④④④④ w / / w are 2-cells w : w w → w in B . For a given 3-boundary { w , w , w , w } ,there is precisely one filler if the diagram( w w ) w ∼ = / / w (cid:15) (cid:15) w ( w w ) w / / w w w (cid:15) (cid:15) w w w / / w commutes and there is no filler otherwise. If the filler exists then it is denoted by( w , w , w , w ). All higher simplices are determined coskeletally.The degenerate 1-simplex on a 0-simplex A is the identity 1-cell 1 : A → A . Thedegenerate 2-simplices s ( a ) and s ( a ) on a 1-simplex a are the coherence isomorphism2-cells a → a and 1 a → a , respectively. On higher simplices the degeneracy mapsare determined by the uniqueness of the filler for a given boundary.As was observed in [6], a simplicial map C → N ( B ) is the same thing as a monad in B . The 1-cell underlying the monad is the image of α , with multiplication and unitprovided by the images of τ and ε , respectively.Monoidal categories can be seen as bicategories with a single object. Thus theabove considerations apply in particular to them. In particular, the above used symbol N ( M ) stands for the monoidal nerve of a monoidal category M (rather than the nerveof the underlying ordinary category). Since a monad in a one-object bicategory is thesame as a monoid in the corresponding monoidal category, simplicial maps C → N ( M )classify the monoids in M .3. A simplicial description of multiplier bimonoids
Multiplier bimonoids in braided monoidal categories are defined as compatible pairsof counital fusion morphisms [3]. Thus it is not too surprising that the first step inour simplicial characterization of multiplier bimonoids is a simplicial treatment ofcounital fusion morphisms. In Section 3.2 we shall associate to the braided monoidalcategory C a simplicial set M such that a simplicial map C → M is the same thingas a multiplier bimonoid in C . As a preparation for that, first we construct in Section3.1 a simplicial set M and analyze the relation of simplicial maps C → M to counitalfusion morphisms in C . The simplicial set M and its symmetric counterpart M willbe used as building blocks of M .3.1. Counital fusion morphisms.
Recall that a fusion morphism on an object A of C is a morphism t : A → A making commutative the first diagram of A t / / t (cid:15) (cid:15) A t / / A A c / / A t / / A c − / / A t O O A t / / e ! ! ❇❇❇❇❇❇❇❇ A e (cid:15) (cid:15) A. (3.1) SIMPLICIAL APPROACH TO MULTIPLIER BIMONOIDS 5
The morphism e : A → I is a counit of t if it makes commutative the second diagramof (3.1).The simplicial set M has a single 0-simplex ∗ . Its 1-simplices are the semigroups in C ; that is, objects A equipped with an associative multiplication m : A → A . The 2-simplices with given 2-boundary { A , A , A } are morphisms ϕ : A A → A A in C rendering commutative the diagrams A A A ϕ / / m (cid:15) (cid:15) A A A m (cid:15) (cid:15) A A ϕ / / A A A A A ϕ (cid:15) (cid:15) m / / A A ϕ / / A A A A A c / / A A A
12 1 ϕ / / A A A c − / / A A A . m O O For a given 3-boundary { ϕ , ϕ , ϕ , ϕ } there is precisely one filler if the fusionequation A A A ϕ (cid:15) (cid:15) ϕ / / A A A
23 1 ϕ / / A A A A A A c / / A A A
23 1 ϕ / / A A A c − / / A A A ϕ O O (3.2)commutes — in which case we write ( ϕ , ϕ , ϕ , ϕ ) for the filler — and thereis no filler otherwise. All higher simplices are generated coskeletally.The action of the face maps should be clear from the above presentation. The uniquedegenerate 1-simplex is the monoidal unit I of C (regarded as a trivial semigroup),while for a 1-simplex ( A, m ) the degenerate 2-simplices s ( A, m ) and s ( A, m ) aregiven by ∗ A (cid:31) (cid:31) ❄❄❄❄❄❄❄ m ∗ I ? ? ⑧⑧⑧⑧⑧⑧⑧ A / / ∗ and ∗ I (cid:31) (cid:31) ❄❄❄❄❄❄❄ ∗ A ? ? ⑧⑧⑧⑧⑧⑧⑧ A / / ∗ respectively. On higher simplices the degeneracy maps are determined by the unique-ness of the filler for a given boundary. Remark . If the simplicial set M looks contrived, we can motivate it as follows,based on the fusion equation (3.2). Suppose we were to try to define a simplicialset with a unique 0-simplex, with objects of C as 1-simplices, with morphisms of theform ϕ : A A → A A as 2-simplices, and with 4-tuples ( ϕ , ϕ , ϕ , ϕ ) of2-simplices satisfying the fusion equation as 3-simplices, and with higher simplicesdefined coskeletally. Then we can define the last degeneracy map in each degree as inthe definition of M , but the other degeneracy maps will not exist. For a 1-simplex A , the degenerate 2-simplex s ( A ) should be a morphism AA → IA in C , so it willbe available if we require that, instead of being mere objects A of C , each 1-simplexbe equipped with a morphism m : A → A . For a 2-simplex ϕ as above, the existenceof 3-simplices with the boundary appropriate for s ( ϕ ) and s ( ϕ ) amounts to thecommutativity of the two diagrams in the definition of 2-simplices in M . Furthermorethe morphism AA → IA induced by m will satisfy these equations if and only if m isassociative. Thus in this sense, the form of the simplicial set M is forced upon us bythe fusion equation. GABRIELLA B ¨OHM AND STEPHEN LACK
More formally, we could proceed as follows. Any morphism in the simplex category∆ has an epi-mono factorization. Considering only those morphisms in ∆ in whosefactorization only the last fibre of the epimorphism has more than one element, weobtain a subcategory to be denoted by ∇ . The construction of the previous paragraphdetermines a presheaf P C on ∇ .The inclusion functor J : ∇ → ∆ induces a functor J ∗ : [∆ op , Set ] → [ ∇ op , Set ]between the presheaf categories, possessing a right adjoint given by the right Kanextension
Ran J . Explicitly, J ∗ takes a simplicial set to the presheaf on ∇ obtained byforgetting all but the last degeneracy map; and Ran J takes a presheaf X on ∇ to thesimplicial set whose n -simplices are families { x f ∈ X j } labelled by morphisms f : j → n in ∆ such that for all morphisms w of codomain j in ∇ , the identity w ∗ ( x f ) = w f.w holds (where w ∗ denotes the image of w under the functor X : ∇ op → Set ).The value of
Ran J at the presheaf P C on ∇ is precisely M . Proposition 3.2.
For any braided monoidal category C , consider the associated sim-plicial set M above. To give a simplicial map C → M is the same as specifying anobject A in C equipped with both a semigroup structure with multiplication m : A → A and a fusion morphism t : A → A with counit e : A → I , subject to the followingcompatibility relations. A t / / m (cid:15) (cid:15) (a) A m (cid:15) (cid:15) A t / / A A m / / t (cid:15) (cid:15) (b) A t / / A A c / / A t / / A c − / / A m O O A e / / m (cid:15) (cid:15) (c) A e (cid:15) (cid:15) A e / / I A m / / m (cid:15) (cid:15) (d) A m (cid:15) (cid:15) A t / / A e / / A Proof.
A simplicial map C → M is given by the images of the non-degenerate 1-simplex α and of the non-degenerate 2-simplices τ and ε . This means, respectively, asemigroup ( A, m ), a morphism t : A → A in C making the diagrams (a) and (b) inthe claim commute, and a morphism e : A → I making diagram (c) commute. Thesimplicial map can be defined on the non-degenerate 3-simplex φ of C if and only if t obeys the fusion equation in the first diagram of (3.1). It can be defined on the3-simplex λ if and only if the counitality condition in the second diagram of (3.1)holds. It can be defined on ̺ if and only if diagram (d) of the claim commutes, whilefor κ we get the same condition encoded in diagram (c). (cid:3) Remark . If t : A → A is a fusion morphism with counit e : A → I in C , thenwe have a semigroup ( A, m := e .t ) in C for which all diagrams of Proposition 3.2commute: (a), (b) and (c) can be found in (3.6), (3.5) and (3.4) in [5], respectively,and (d) follows by the associativity of m = e .t . Hence there is a correspondingsimplicial map C → M .However, there may be more general simplicial maps C → M for which the mul-tiplication of the corresponding semigroup ( A, m ) is different from the multiplication e .t coming from the counital fusion morphism ( A, t, e ).Let us consider the particular kind of simplicial maps C → M for which the mul-tiplication m happens to be non-degenerate in the sense that both maps C ( X, AY ) → C ( AX, AY ) f m . f C ( X, Y A ) → C ( XA, Y A ) g m.g SIMPLICIAL APPROACH TO MULTIPLIER BIMONOIDS 7 are injective, for any objects X and Y . Since by identities (a) and (d) in Proposition3.2 m.e .t e . m.t e .t. m = m.m , we conclude that in this case m = e .t .By the associativity of m and commutativity of (d), m = e .t also follows if1 m : A → A is an epimorphism.By the same construction as above, we can associate a simplicial set to the monoidalcategory C rev , obtained from C by reversing the monoidal product and using the samebraiding c . The opposite of this simplicial set is called M . Explicitly, M also has asingle 0-simplex ∗ and the semigroups in C as 1-simplices. The 2-simplices of a given2-boundary { A , A , A } are now morphisms ψ : A A → A A in C making thediagrams A A A
02 1 ψ / / m (cid:15) (cid:15) A A A m (cid:15) (cid:15) A A ψ / / A A A A A ψ (cid:15) (cid:15) m / / A A ψ / / A A A A A
02 1 c / / A A A ψ / / A A A c − / / A A A m O O commute. For a given 3-boundary { ψ , ψ , ψ , ψ } there is precisely one filler if A A A ψ (cid:15) (cid:15) ψ / / A A A ψ / / A A A A A A
03 1 c / / A A A ψ / / A A A
12 1 c − / / A A A ψ O O commutes — in which case we write ( ψ , ψ , ψ , ψ ) for the filler — and there isno filler otherwise. All higher simplices are generated coskeletally. The unique degen-erate 1-simplex is again the monoidal unit I of C (regarded as a trivial semigroup),while for a 1-simplex ( A, m ) the degenerate 2-simplices s ( A, m ) and s ( A, m ) aregiven by ∗ A (cid:31) (cid:31) ❄❄❄❄❄❄❄ ∗ I ? ? ⑧⑧⑧⑧⑧⑧⑧ A / / ∗ and ∗ I (cid:31) (cid:31) ❄❄❄❄❄❄❄ m ∗ A ? ? ⑧⑧⑧⑧⑧⑧⑧ A / / ∗ respectively; note that the roles of s and s have been interchanged. As before, onthe higher simplices the degeneracy maps are determined by the uniqueness of thefiller for a given boundary.Since the simplicial set C is isomorphic to its opposite, Proposition 3.2 then char-acterizes the simplicial maps C → M as objects A carrying the compatible structuresof a semigroup, and a counital fusion morphism in C rev ; once again, the roles of λ and ρ have been interchanged relative to the case of M .As a further possibility, we can use the above construction to associate a simplicialset M to the monoidal category C , obtained from C by keeping the same monoidalproduct but replacing the braiding c with c − , and using the twisted multiplication m.c − for a 1-simplex (that is, semigroup) ( A, m ), so that in particular the degenerate2-simplex s ( A, m ) is given by m.c − . Proposition 3.2 can also be used to describethe simplicial maps C → M . GABRIELLA B ¨OHM AND STEPHEN LACK
Finally, applying the above construction to the braided monoidal category ( C ) rev = C rev we obtain a simplicial set M .3.2. Multiplier bimonoids. A multiplier bimonoid [3] in a braided monoidal cate-gory C consists of a fusion morphism t in C and a fusion morphism t in C rev with acommon counit e : A → I such that the diagrams A t / / t (cid:15) (cid:15) A t (cid:15) (cid:15) A t / / A A t / / t (cid:15) (cid:15) A e (cid:15) (cid:15) A e / / A (3.3)commute. Thus by Remark 3.3, it can be thought of as a pair of simplicial maps C → M and C → M subject to compatibility conditions expressing the fact that theunderlying semigroups and the counits are equal, and the diagrams in (3.3) commute,with the common diagonal of the second of these given by the multiplication. Guidedby this fact, we construct below a simplicial set M whose simplices are suitablycompatible pairs consisting of a simplex in M and a simplex in M . We prove that asimplicial map C → M is the same thing as a multiplier bimonoid in C .The simplicial set M has a single 0-simplex ∗ and the semigroups of C as 1-simplices. The 2-simplices are pairs ( ϕ | ψ ) consisting of a 2-simplex ϕ of M and a2-simplex ψ of M with common boundary { A , A , A } , such that the diagram A A A
12 1 ϕ / / ψ (cid:15) (cid:15) A A A m (cid:15) (cid:15) A A A
12 1 m / / A A (3.4)commutes. The 3-simplices are pairs ( ϕ , ϕ , ϕ , ϕ | ψ , ψ , ψ , ψ ) con-sisting of a 3-simplex ( ϕ , ϕ , ϕ , ϕ ) in M and a 3-simplex ( ψ , ψ , ψ , ψ )in M such that ( φ ijk | ψ ijk ) is a 2-simplex in M for each ijk , and the diagram A A A
23 1 ϕ / / ψ (cid:15) (cid:15) A A A ψ (cid:15) (cid:15) A A A
23 1 ϕ / / A A A (3.5)commutes. The face and degeneracy maps act on the pairs componentwise, and highersimplices are defined coskeletally. Remark . We explained in Remark 3.1 a sense in which the simplicial set M isdictated by the fusion equation; in particular, the associativity of the multiplicationsin the 1-simplices and the commutativity of the diagrams in the definition of the 2-simplices are required in order for various degenerate 3-simplices to satisfy the fusionequation.The case of M is similar: it has the same 1-simplices as M , and once we imposecommutativity of (3.5) on the 3-simplices, any 2-simplex ( ϕ | ψ ) must obey (3.4) inorder to have a 3-simplex with the boundary of s ( ϕ | ψ ). SIMPLICIAL APPROACH TO MULTIPLIER BIMONOIDS 9
Theorem 3.5.
There is a bijection between simplicial maps C → M and multiplierbimonoids in C .Proof. Again, a simplicial map C → M is given by the images ( A, m ) of the non-degenerate 1-simplex α , ( t | t ) of the non-degenerate 2-simplex τ and ( e | e ) of thenon-degenerate 2-simplex ε . By Proposition 3.2 ( t , e ) is a counital fusion morphismin C obeying conditions (a)-(d); and ( t , e ) is a counital fusion morphism in C rev obeying symmetric counterparts of conditions (a)-(d). Furthermore, there are com-patibility conditions between them: ( t | t ) is a 2-simplex of M if and only if diagram(e) in A t / / t (cid:15) (cid:15) (e) A m (cid:15) (cid:15) A m / / A A e / / e (cid:15) (cid:15) (f) II I A t / / t (cid:15) (cid:15) (g) A t (cid:15) (cid:15) A t / / A A t (cid:15) (cid:15) (h) A m (cid:15) (cid:15) A e / / A A t / / (i) A e (cid:15) (cid:15) A m / / A commutes and ( e | e ) is a 2-simplex of M if and only if (f) does so. The simplicialmap is well-defined on the non-degenerate 3-simplices φ , λ , ̺ and κ if and only if therespective diagrams (g), (h), (i) and (f) again commute.From (f) we infer that the counits e and e are equal so we will denote them simplyby e . Then (h) and (i) take the equivalent form in the second diagram of (3.3), withcommon diagonal m . As observed in Remark 3.3, from this it follows that all identities(a)-(d), as well as their symmetric counterparts, hold true. Diagram (g) is identicalto the first diagram of (3.3); this implies (e) upon postcomposing by 1 e C → M is the same thing as a pair of counital fusionmorphism ( t , e ) in C and a counital fusion morphism ( t , e ) in C rev (with commoncounit e ) rendering commutative the diagrams of (3.3). (cid:3) Applying the construction of this section to the braided monoidal category C , andusing the reversed multiplications m.c − of the semigroups ( A, m ), we obtain a sim-plicial set M .3.3. Regular multiplier bimonoids. A regular multiplier bimonoid [3] in a braidedmonoidal category C is a tuple ( A, t , t , t , t , e ) such that ( A, t , t , e ) is a multi-plier bimonoid in C and ( A, t , t , e ) is a multiplier bimonoid in C , and such that thefollowing diagrams commute, where m stands for the common diagonal of the firstdiagram. A t / / c (cid:15) (cid:15) A e (cid:15) (cid:15) A t / / c (cid:15) (cid:15) A c (cid:15) (cid:15) A t / / t (cid:15) (cid:15) A t (cid:15) (cid:15) A t / / c (cid:15) (cid:15) A c (cid:15) (cid:15) A t / / t (cid:15) (cid:15) A t (cid:15) (cid:15) A t (cid:15) (cid:15) A t (cid:15) (cid:15) A m (cid:15) (cid:15) A t (cid:15) (cid:15) A m (cid:15) (cid:15) A e / / A A m / / A A t / / A A m / / A A t / / A (3.6)We shall classify regular multiplier bimonoids via simplicial maps from C to a simplicialset M which we now describe. The simplicial set M has a single 0-simplex ∗ , and its 1-simplices are the semigroupsin the braided monoidal category C . The 2-simplices are pairs ( ϕ | ψ || ϕ ′ | ψ ′ ) consistingof a 2-simplex ( ϕ | ψ ) of M and a 2-simplex ( ϕ ′ | ψ ′ ) of M with common boundary( A, B, C ), obeying the compatibility conditions
CBA ϕ / / ψ ′ (cid:15) (cid:15) C A c − (cid:15) (cid:15) C A m (cid:15) (cid:15) CA m / / CA CBA ψ / / ϕ ′ (cid:15) (cid:15) CA c − (cid:15) (cid:15) CA m (cid:15) (cid:15) C A m / / CA ABA ϕ / / c (cid:15) (cid:15) ACA c (cid:15) (cid:15) BA ϕ ′ (cid:15) (cid:15) CA m (cid:15) (cid:15) CA m / / CA CBC ψ / / c (cid:15) (cid:15) CAC c (cid:15) (cid:15) C B ψ ′ (cid:15) (cid:15) C A m (cid:15) (cid:15) C A m / / CA.
The 3-simplices are pairs( ϕ , ϕ , ϕ , ϕ | ψ , ψ , ψ , ψ || ϕ ′ , ϕ ′ , ϕ ′ , ϕ ′ | ψ ′ , ψ ′ , ψ ′ , ψ ′ )consisting of a 3-simplex ( ϕ , ϕ , ϕ , ϕ | ψ , ψ , ψ , ψ ) of M and a 3-simplex ( ϕ ′ , ϕ ′ , ϕ ′ , ϕ ′ | ψ ′ , ψ ′ , ψ ′ , ψ ′ ) of M for which ( ϕ ijk | ψ ijk || ϕ ′ ijk | ψ ′ ijk )is a 2-simplex in M for every 0 ≤ i < j < k ≤ A A A
23 1 ϕ / / ψ ′ (cid:15) (cid:15) A A A ψ ′ (cid:15) (cid:15) A A A
23 1 ϕ / / A A A A A A ψ / / ϕ ′ (cid:15) (cid:15) A A A ϕ ′ (cid:15) (cid:15) A A A ψ / / A A A (3.7)commute. The higher simplices are generated coskeletally and the face and the de-generacy maps act on the pairs memberwise. Theorem 3.6.
There is a bijection between simplicial maps C → M and regularmultiplier bimonoids in C .Proof. For a simplicial map C → M , denote the image of the 2-simplex α by ( A, m ),and denote the images of the 3-simplices τ and ε by ( t | t || t | t ) and ( e | e || e ′ | e ′ ), respec-tively. By Theorem 3.5, ( t , t , e ) is a multiplier bimonoid in C for which m = e .t =1 e.t , and ( t , t , e ′ ) is a multiplier bimonoid in C rev for which m.c − = e ′ .t = 1 e ′ .t .Furthermore, from the requirements that ( t | t || t | t ) and ( e | e || e ′ | e ′ ) be 2-simplices of M we obtain the following identities. A t / / t (cid:15) (cid:15) (j) A c − (cid:15) (cid:15) A m (cid:15) (cid:15) A m / / A A t / / t (cid:15) (cid:15) (k) A c − (cid:15) (cid:15) A m (cid:15) (cid:15) A m / / A A t / / c (cid:15) (cid:15) (l) A c (cid:15) (cid:15) A t (cid:15) (cid:15) A m (cid:15) (cid:15) A m / / A A t / / c (cid:15) (cid:15) (m) A c (cid:15) (cid:15) A t (cid:15) (cid:15) A m (cid:15) (cid:15) A m / / A A e / / e ′ (cid:15) (cid:15) (n) II I.
SIMPLICIAL APPROACH TO MULTIPLIER BIMONOIDS 11
The boundaries of the images of the 3-simplices φ , λ , ̺ and κ under a simplicial mapare determined. The fillers to be their images exist if and only if the diagrams A t / / t (cid:15) (cid:15) (o) A t (cid:15) (cid:15) A t / / A A t / / t (cid:15) (cid:15) (p) A t (cid:15) (cid:15) A t / / A A t / / t (cid:15) (cid:15) m ❇❇❇ (q)(r) ❇❇❇❇❇❇❇ A e ′ (cid:15) (cid:15) A e ′ / / A A t / / t (cid:15) (cid:15) m.c − ❇❇ (s)(t) ❇❇❇❇❇❇ A e (cid:15) (cid:15) A e / / A commute (in the case of κ the same condition (n) occurs again).Conditions (l), (o), (m) and (p) are identical to the last four diagrams of (3.6). Inlight of (n), conditions (q), (r), (s) and (t) are redundant. Conditions (j) and (k) arealso redundant: they follow from (o) and (p), respectively, postcomposing them with1 e e ′
1. Finally (n) implies the commutativity of the first diagram of (3.6), withcommon diagonal m . (cid:3) Remark . We discussed in Remark 3.1 and Remark 3.4 a sense in which the defi-nitions of 2-simplices in M and M are dictated by the definitions of the respective3-simplices.When it comes to M , however, this property breaks down. While commutativityof the first two diagrams in the definition of 2-simplices in M is needed in order fordegenerate 3-simplices to exist in M , commutativity of the other two diagrams is not.Commutativity of these last two diagrams would be needed, though, if we were tomodify the definition of 3-simplices so as to require that, in addition to the diagramsof (3.7), also A A A
23 1 ϕ / / c (cid:15) (cid:15) A A A c / / A A A
23 1 ϕ / / A A A ϕ ′ (cid:15) (cid:15) A A A ϕ ′ / / A A A
23 1 ϕ / / A A A A A A ψ / / c (cid:15) (cid:15) A A A
02 1 c / / A A A ψ / / A A A ψ ′ (cid:15) (cid:15) A A A
03 1 ψ ′ / / A A A ψ / / A A A commute.Simplicial maps from C to the resulting simplicial set would now correspond to astronger notion of regular multiplier bimonoid, in which the second diagram of (3.6)is replaced by the fusion equation in the second diagram of [3, Remark 3.10], andwith an analogous change to the fourth diagram of (3.6). Although in general thiswould result in a stronger notion of regular multiplier bimonoid, the difference woulddisappear in the case where the multiplication is non-degenerate, and it was alreadyanticipated in [3, Remark 3.10] that in the not necessarily non-degenerate case suchstrengthenings might be needed.3.4. Multiplier bimonoids which are comonoids.
In our paper [4], followingsome ideas in [8], we associated to any braided monoidal category C a monoidalcategory M , and we described a correspondence between certain multiplier bimonoids in C and certain comonoids in M [4, Theorem 5.1]. In this section, we explain thiscorrespondence in terms of simplicial maps and the Catalan simplicial set.A comonoid in the monoidal category M is the same as a monoid in the monoidalcategory M op . We now form the nerve N ( M op ) of the monoidal category M op , asin Section 2.3; this is not to be confused with the nerve of the underlying categoryof M op . As explained in [6], simplicial maps c : C → N ( M op ) can be identified withmonoids in M op , and so in turn with comonoids in M .On the other hand, we have shown that simplicial maps a : C → M can be identifiedwith multiplier bimonoids in C . In order to compare these, we construct a simplicialset Q which is contained in both M and N ( M op ). Now a multiplier bimonoid in C corresponds to a comonoid in M just when there is a common factorization of thecorresponding simplicial maps as in the following diagram. C a c " " Q / / (cid:15) (cid:15) M N ( M op )We now turn to the details. To define the monoidal category M , in [4] we fixed aclass Q of regular epimorphisms in C which is closed under composition and monoidalproduct, contains the isomorphisms, and is right-cancellative in the sense that if s : A → B and t.s : A → C are in Q , then so is t : B → C . Since each q ∈ Q isa regular epimorphism, it is the coequalizer of some pair of morphisms. Finally wesuppose that this pair may be chosen in such a way that the coequalizer is preservedby taking the monoidal product with any fixed object.The objects of the associated category M are those semigroups in C whose multi-plication is non-degenerate and belongs to Q . The morphisms f : A B are pairs( f : AB → B ← BA : f ) of morphisms in Q such that the first two, equivalently, thelast two diagrams in A B f / / m (cid:15) (cid:15) AB f (cid:15) (cid:15) AB f / / B BAB f / / f (cid:15) (cid:15) B m (cid:15) (cid:15) B m / / B BA f / / m (cid:15) (cid:15) BA f (cid:15) (cid:15) BA f / / B commute. The composite g • f of morphisms f : A B and g : B C is defined byuniversality of the coequalizer in the top row of either diagram in ABC g / / f (cid:15) (cid:15) AC ( g • f ) (cid:15) (cid:15) BC g / / C CBA g / / f (cid:15) (cid:15) CA ( g • f ) (cid:15) (cid:15) CB g / / C. The identity morphism A A is the pair ( m : A → A ← A : m ). SIMPLICIAL APPROACH TO MULTIPLIER BIMONOIDS 13
This category M is monoidal. The monoidal product of semigroups A and C is( AC ) c / / A C mm / / AC and the monoidal product of morphisms f : A B and g : C D is the pair ACBD c / / ABCD f g / / BD BADC f g o o BDAC. c o o If C is a closed braided monoidal category with pullbacks, then for any semigroup B with non-degenerate multiplication one can define its multiplier monoid M ( B ), see[4]. It is a monoid in C and a universal object characterized by the property thatmorphisms ( f , f ) : A B in M correspond bijectively to multiplicative morphisms f : A → M ( B ) in C such that AB f / / M ( B ) B i / / B and BA f / / B M ( B ) i / / B are in Q , where i : M ( B ) → M ( B ) is the identity morphism in C ; regarded as amorphism ( i , i ) : M ( B ) B in M . In the category of vector spaces M ( B ) reducesto the multiplier algebra of B as defined in [7].Take a multiplier bimonoid ( A, t , t , e ) in C for which • the underlying semigroup has a non-degenerate multiplication m := e .t =1 e.t • m, e , and the morphisms d and d defined by A c / / A t / / A c − / / A m / / A A c / / A t / / A c − / / A m / / A all belong to Q .The correspondence in [4, Theorem 5.1] associated a multiplier bimonoid of this typeto the comonoid in M with underlying object ( A, m ), with comultiplication A A having components d and d , and with counit A I whose components are both e .The simplicial set M has a simplicial subset Q as follows. The only 0-simplex ∗ of M is a 0-simplex also in Q . The 1-simplices of Q are those semigroups ( A, m ) in C whose multiplication is non-degenerate and belongs to Q . The 2-simplices of Q arethose 2-simplices ( ϕ | ψ ) of M whose faces belong to Q and for which the morphisms b ϕ : A A A
12 1 c − / / A A A ϕ / / A A A
01 1 c / / A A A m / / A A b ψ : A A A c − / / A A A
02 1 ψ / / A A A c / / A A A
12 1 m / / A A are in Q . The 3-simplices of Q are all those 3-simplices of M whose faces belong to Q . Clearly a simplicial map from C to Q is the same thing as a multiplier bimonoidin C having the properties listed above.The desired simplicial map Q → N ( M op ) sends the 0-simplex ∗ to the single 0-simplex of the nerve N ( M op ). It sends the 1-simplex ( A, m ) in Q to its underlyingobject A . It sends a 2-simplex ( ϕ | ψ ) to the morphism A A A in M whosecomponents are b ϕ and b ψ . On the higher simplices it is unambiguously defined by theuniqueness of the filler of any boundary in N ( M op ). This assignment is injective bynon-degeneracy of the relevant multiplications. References [1] G. B¨ohm,
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