Lax braidings and the lax centre

Day, Brian J.; Elango, P.; Street, Ross

doi:10.1090/conm/441/08497

Cited by 6 publications

(8 citation statements)

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“…In another direction, it seems that we use the fact that the braiding in V has an inverse rather infrequently. Perhaps it is possible to generalise the setting throughout to monoidal categories enriched in a category V equipped with a lax braiding u → u as in [DS07;DPS07]. For now, however, we have no application of such a generalisation, so we have not pursued it.…”

mentioning

confidence: 99%

Monoidal Categories Enriched in Braided Monoidal Categories

Morrison

Penneys

2017

International Mathematics Research Notices

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A. We introduce the notion of a monoidal category enriched in a braided monoidal category V. We set up the basic theory, and prove a classi cation result in terms of braided oplax monoidal functors to the Drinfeld center of some monoidal category T .Even the basic theory is interesting; it shares many characteristics with the theory of monoidal categories enriched in a symmetric monoidal category, but lacks some features. Of particular note, there is no cartesian product of braided-enriched categories, and the natural transformations do not form a 2-category, but rather satisfy a braided interchange relation.Strikingly, our classi cation is slightly more general than what one might have anticipated in terms of strong monoidal functors V → Z (T ). We would like to understand this further; in a future paper we show that the functor is strong if and only if the enriched category is 'complete' in a certain sense. Nevertheless it remains to understand what non-complete enriched categories may look like.One should think of our construction as a generalization of de-equivariantization, which takes a strong monoidal functor Rep(G) → Z (T ) for some nite group G and a monoidal category T , and produces a new monoidal category T //G . In our setting, given any braided oplax monoidal functor V → Z (T ), for any braided V, we produce T //V : this is not usually an 'honest' monoidal category, but is instead V-enriched. If V has a braided lax monoidal functor to Vec, we can use this to reduce the enrichment to Vec, and this recovers de-equivariantization as a special case.

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mentioning

confidence: 99%

Monoidal Categories Enriched in Braided Monoidal Categories

Morrison

Penneys

2017

International Mathematics Research Notices

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“…On the other hand, by Lack's observation in Section 2.1, the fact that 1 X has a left dual (a left dualization) means that 1 X has a right bidual in M X, and hence a right dual in M X. It follows from a result dual to Proposition 3.1 of [7] that the component c 1X ,A is an isomorphism in M X, and hence an equivalence in M X. By the description of equivalences in M X it follows that the 2-cell γ Eq.…”

Section: Proposition 54 Let the Pseudomonoid X Be Left Autonomous Anmentioning

confidence: 89%

“…The centre of a monoidal category was defined in [10], while the lax variant appeared in [21] for ordinary monoidal categories (under the name 'weak center') and in [7] for enriched (pro)monoidal categories. The centre of a monoidal object was introduced in [23], and a lax variant of it was considered in [15,17].…”

Section: Lax Centres Become Centresmentioning

confidence: 99%

Duals Invert

Franco

Street

Wood

2009

Appl Categor Struct

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Monoidal objects (or pseudomonoids) in monoidal bicategories share many of the properties of the paradigmatic example: monoidal categories. The existence of (say, left) duals in a monoidal category leads to a dualization operation which was abstracted to the context of monoidal objects by Day et al. (Appl Categ Struct 11:229-260, 2003). We define a relative version of this called exact pairing for two arrows in a monoidal bicategory; when one of the arrows is an identity, the other is a dualization. In this context we supplement results of Day et al. (Appl Categ Struct 11:229-260, 2003) (and even correct one of them) and only assume the existence of biduals in the bicategory where necessary. We also abstract recent work of Day and Pastro (New York J Math 14:733-742, 2008) on Frobenius monoidal functors to the monoidal bicategory context. Our work began by focusing on the invertibility of The first author was supported by an Internal Graduate Studentship at Trinity College, Cambridge, and by the Fondation Sciences Mathématiques de Paris.The second and third authors gratefully acknowledge financial support from the Australian ARC and the Canadian NSERC. Diagrams typeset using M. Barr's diagram package, diagxy.tex. I. López Franco et al. components at dual objects of monoidal natural transformations between Frobenius monoidal functors. As an application of the abstraction, we recover a theorem of Walters and Wood (Theory Appl Categ 3:25-47, 2008) asserting that, for objects A and X in a cartesian bicategory C , if A is Frobenius then the category MapC (X, A) of left adjoint arrows is a groupoid. Also, the characterization in Walters and Wood (Theory Appl Categ 3:25-47, 2008) of left adjoint arrows between Frobenius objects of a cartesian bicategory is put into our current setting. In the same spirit, we show that when a monoidal object admits a dualization, its lax centre coincides with the centre defined in Street (Theory Appl Categ 13:184-190, 2004). Finally we look at the relationship between lax duals for objects and adjoints for arrows in a monoidal bicategory. 322

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“…The centre of a pseudomonoid in Cat, that is, of a monoidal category, is the usual centre defined in [11]. In fact, lax centres and centres of pseudomonoids in V -Cat exist and are given by the constructions in [5]. Lax centres or (ordinary) monoidal categories were also considered in [18] under the name of 'weak centers'.…”

Section: Centres and Lax Centresmentioning

confidence: 99%

“…In this situation, if we assume J is a map, so thatS exists, we claim that the monadsM andS are isomorphic, or more precisely, that they are isomorphic as monoids in the monoidal V -category by Theorem 4.3, which together with [5,Theorem 4.5] shows that the centre of G in Set-Mod is equivalent to the category called (lax) centre of G in [5].…”

Section: Lax Centres In V -Modmentioning

confidence: 99%

Formal Hopf algebra theory, II: Lax centres

Franco

2009

Journal of Pure and Applied Algebra

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a b s t r a c tThis paper is the second in a series started by [Ignacio L. López Franco, Formal Hopf algebra theory I: Hopf modules for pseudomonoids, J. Pure Appl. Algebra 213 (2009) 1046-1063], aiming to extend the basic theory of Hopf algebras to the context of pseudomonoids in monoidal bicategories. This article concentrates on the notion of lax centre of a pseudomonoid and its relationship with the Drinfel'd or quantum double of a finite Hopf algebra and the centre of a monoidal category. We can distinguish two parts in the present paper. In the first, for a pseudomonoid A with lax centre Z A in a Gray monoid M with certain extra properties, we exhibit an equivalence M (I, Z A) Z (M (I, A)) of categories enriched in M (I, I). In the second, we construct the lax centre of a left autonomous map pseudomonoid A as an Eilenberg-Moore object for a certain opmonoidal monad on A. Moreover, if A is also right autonomous, the lax centre coincides with the centre. As an application, we show that a (left) autonomous monoidal V -category has a (lax) centre in V -Mod, of which we give an explicit description. In another application, we prove that a finite-dimensional coquasi-Hopf algebra H has a centre in the monoidal bicategory Comod(Vect) and it is equivalent to the Drinfel'd double D(H).

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Lax braidings and the lax centre

Cited by 6 publications

References 0 publications

Monoidal Categories Enriched in Braided Monoidal Categories

Monoidal Categories Enriched in Braided Monoidal Categories

Duals Invert

Formal Hopf algebra theory, II: Lax centres

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