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SUMMARY:Bojana Femic (Serbian Academy of Sciences and Arts)
DTSTART:20230327T140000Z
DTEND:20230327T150000Z
DTSTAMP:20260423T035740Z
UID:EQuAL/10
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/EQuAL/10/">C
 ategorical centers and Yetter Drinfel`d-modules as 2-categorical (bi)lax 
 structures</a>\nby Bojana Femic (Serbian Academy of Sciences and Arts) as 
 part of European Quantum Algebra Lectures (EQuAL)\n\n\nAbstract\nJoint wor
 k with Sebastian Halbig\n\nCenter categories of monoidal categories ${\\ma
 thcal C}$ and of bimodule categories ${\\mathcal M}$ are very well known a
 nd studied in the literature. \nWe 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 functor
 s \n$F:{\\mathcal E}\\to {\\mathcal D}$ and $G:{\\mathcal E}\\to {\\mathca
 l C}$\, for another monoidal category ${\\mathcal E}$. (The weakness cor
 responds to dealing with half-braidings\, while with strongness we allude 
 to (invertible) braidings.)\n\nWe show how the 2-categorical viewpoint pro
 vides a deeper insight on such center categories. Namely\, for fixed bica
 tegories ${\\mathcal B}$ and ${\\mathcal B}'$ there are bicategories $\\op
 eratorname{Lax}_{lx}({\\mathcal B}\,{\\mathcal B}')$ and $\\operatorname{L
 ax}_{clx}({\\mathcal B}\,{\\mathcal B}')$ of lax functors ${\\mathcal B} \
 \to {\\mathcal B}'$\, lax (resp. colax) transformations and their modifica
 tions. We reveal how in a specific case of ${\\mathcal B}$ and ${\\mathca
 l B}'$ we can identify the hom-categories of these two bicategories with t
 he weak left (resp. right) twisted centers\, so that the horizontal compos
 ition in the bicategories corresponds to the composition of weak twisted 
 center categories between themselves. In this way we obtain a bicategory o
 f weak left (resp. right) centers categories. We show how a full sub-bicat
 egory of both of them recovers the bicategory $TF({\\mathcal C}\,{\\mathca
 l D})$ 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. \n\nOn the other hand\, we introduce a 2-category ${\\r
 m Bilax}({\\mathcal K}\,{\\mathcal K}')$ of bilax functors (among 2-categ
 ories ${\\mathcal K}$ and ${\\mathcal K}'$)\, bilax natural transformation
 s and bilax modifications. Its 0-cells are a 2-categorification of bilax f
 unctors of [Agui] and of bimonoidal functors of [CS]. We show how bilax f
 unctors generalize the notions of bialgebras in braided monoidal categorie
 s\, $bimonads$ in 2-categories (with respect to Yang-Baxter operators\, 
 YBO's)\, and preserve bimonads (w.r.t. YBO's)\, $module$ $comonads$ and $c
 omodule$ $monads$\, and $relative$ $bimonad$ $modules$. Moreover\, the com
 ponent functors of a bilax functor on hom-categories factor through the ca
 tegory of $Hopf$ $bimodules$ (w.r.t. YBO's). (The 2-categorical notions in
  italic letters are introduced in our work.) \n\nWe finally show that th
 ere is a 2-category equivalence ${\\rm Bilax} (1\, \\Sigma{\\mathcal C})\\
 simeq{\\mathcal YD}(\\Sigma{\\mathcal C})$ and a faithful 2-functor ${\\r
 m Bilax}(1\,{\\mathcal K})\\hookrightarrow\\operatorname{Dist}({\\mathcal 
 K})$. Here ${\\mathcal YD}(\\Sigma{\\mathcal C})$ is a 2-category of Yett
 er-Drinfel`d modules in a braided monoidal category ${\\mathcal C}$ and $
 \\operatorname{Dist}({\\mathcal K})$ is the 2-category of mixed distributi
 ve laws of [PW].\n\n\n[Agui] M. Aguiar\, S. Mahajan\, Monoidal functors\, 
 species and Hopf algebras\, CRM Monograph Series 29 Amer. Math. Soc. (2010
 ).\n\n[CS] M. B. McCurdy\, R. Street\, What Separable Frobenius Monoidal F
 unctors Preserve\,\nCahiers de Topologie et Geometrie Differentielle Categ
 oriques 51/1 (2010).\n\n[Shim] K. Shimizu: Ribbon structures of the Drinfe
 l`d center\, arXiv:1707.09691 (2017a)\n\n[PW] J. Power\, H. Watanabe\, Com
 bining a monad and a comonad\, Theoretical Computer Science 280 (2002)\, 1
 37--262.\n
LOCATION:https://researchseminars.org/talk/EQuAL/10/
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