Categorical centers and Yetter Drinfel`d-modules as 2-categorical (bi)lax structures

Abstract: Joint work with Sebastian Halbig

Center categories of monoidal categories ${\mathcal C}$ and of bimodule categories ${\mathcal M}$ are very well known and studied in the literature. We consider the (weak) center category ${\mathcal Z}(F,{\mathcal M},G)$ of a ${\mathcal C}\text{-} {\mathcal D}$-bimodule category ${\mathcal M}$ twisted by two lax monoidal functors $F:{\mathcal E}\to {\mathcal D}$ and $G:{\mathcal E}\to {\mathcal C}$, for another monoidal category ${\mathcal E}$. (The weakness corresponds to dealing with half-braidings, while with strongness we allude to (invertible) braidings.)

We show how the 2-categorical viewpoint provides a deeper insight on such center categories. Namely, for fixed bicategories ${\mathcal B}$ and ${\mathcal B}'$ there are bicategories $\operatorname{Lax}_{lx}({\mathcal B},{\mathcal B}')$ and $\operatorname{Lax}_{clx}({\mathcal B},{\mathcal B}')$ of lax functors ${\mathcal B} \to {\mathcal B}'$, lax (resp. colax) transformations and their modifications. We reveal how in a specific case of ${\mathcal B}$ and ${\mathcal B}'$ we can identify the hom-categories of these two bicategories with the weak left (resp. right) twisted centers, so that the horizontal composition in the bicategories corresponds to the composition of weak twisted center categories between themselves. In this way we obtain a bicategory of weak left (resp. right) centers categories. We show how a full sub-bicategory of both of them recovers the bicategory $TF({\mathcal C},{\mathcal D})$ from [Shim, Section 3]. Moreover, we prove a more general result in bicategories by which the rigidity of $TF({\mathcal C},{\mathcal D})$ is recovered. 

On the other hand, we introduce a 2-category ${\rm Bilax}({\mathcal K},{\mathcal K}')$ of bilax functors (among 2-categories ${\mathcal K}$ and ${\mathcal K}'$), bilax natural transformations and bilax modifications. Its 0-cells are a 2-categorification of bilax functors of [Agui] and of bimonoidal functors of [CS]. We show how bilax functors generalize the notions of bialgebras in braided monoidal categories, $bimonads$ in 2-categories (with respect to Yang-Baxter operators, YBO's), and preserve bimonads (w.r.t. YBO's), $module$ $comonads$ and $comodule$ $monads$, and $relative$ $bimonad$ $modules$. Moreover, the component functors of a bilax functor on hom-categories factor through the category of $Hopf$ $bimodules$ (w.r.t. YBO's). (The 2-categorical notions in italic letters are introduced in our work.) 

We finally show that there is a 2-category equivalence ${\rm Bilax} (1, \Sigma{\mathcal C})\simeq{\mathcal YD}(\Sigma{\mathcal C})$ and a faithful 2-functor ${\rm Bilax}(1,{\mathcal K})\hookrightarrow\operatorname{Dist}({\mathcal K})$. Here ${\mathcal YD}(\Sigma{\mathcal C})$ is a 2-category of Yetter-Drinfel`d modules in a braided monoidal category ${\mathcal C}$ and $\operatorname{Dist}({\mathcal K})$ is the 2-category of mixed distributive laws of [PW].

[Agui] M. Aguiar, S. Mahajan, Monoidal functors, species and Hopf algebras, CRM Monograph Series 29 Amer. Math. Soc. (2010).

[CS] M. B. McCurdy, R. Street, What Separable Frobenius Monoidal Functors Preserve, Cahiers de Topologie et Geometrie Differentielle Categoriques 51/1 (2010).

[Shim] K. Shimizu: Ribbon structures of the Drinfel`d center, arXiv:1707.09691 (2017a)

[PW] J. Power, H. Watanabe, Combining a monad and a comonad, Theoretical Computer Science 280 (2002), 137--262.

Mathematics

Audience: researchers in the topic

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European Quantum Algebra Lectures (EQuAL)

Series comments: EQuAL is an online seminar series on quantum algebra and related topics such as topological and conformal field theory, operator algebra, representation theory, quantum topology, etc. sites.google.com/view/equalseminar/home

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