Finitary birepresentations of finitary bicategories

Abstract

In this paper, we discuss the generalization of finitary 2-representation theory of finitary 2-categories to finitary birepresentation theory of finitary bicategories. In previous papers on the subject, the classification of simple transitive 2-representations of a given 2-category was reduced to that for certain subquotients. These reduction results were all formulated as bijections between equivalence classes of 2-representations. In this paper, we generalize them to biequivalences between certain 2-categories of birepresentations. Furthermore, we prove an analog of the double centralizer theorem in finitary birepresentation theory.

Basic Hopf algebras and symmetric bimodules

“…Here we recall some basic definitions from finitary birepresentation theory using the conventions from [MMMTZ2].…”

“…A bicategory C is quasi (multi)fiab if it is (multi)finitary and each 1-morphism has a left and a right adjoint. If these are moreover isomorphic, C is (multifiab), see [MMMTZ2,Definition 2.5] for more detail.…”

“…By a mild generalisation of [ChMa,Subsection 3.2], every finitary birepresentation M of a (multi)finitary bicategory has an apex, which is the unique maximal two-sided cell not annihilated by M. To each left cell L, we can associate the cell birepresentation C L , which is the quotient of the left 2-ideal generated by the 1-morphisms in L by its unique maximal C-stable ideal. By construction, this is simple transitive and its apex is the two-sided cell containing L. For more details on cells and cell birepresentations, see [MMMTZ2,Section 2.5].…”

“…The most powerful tool for classifying simple transitive 2-representations of a general fiat 2category C is given by the so-called H-cell reduction results in [MMMZ] and [MMMTZ2]. These reduce the problem to the classification of simple transitive 2-representations of certain subquotients C H of C .…”

Adjunction in the absence of identity

“…finitary and having a generator) correspond to coalgebra 1-morphisms in C, and the equivalence of bilax 2-representations gives Morita-Tacheuchi equivalence of the corresponding coalgebras. If C is a fiat 2-category, then all coalgebra 1-morphisms in C arise in this way since inj C -C is finitary for any coalgebra C in C (we do not know a reference for this exact statement, so we prove it in Corollary 3.39, another proof is expected to appear in [MMMTZ2]), so that we have a correspondence between finitary 2-represntations of C and all coalgebras in C. For a fiax category C , however, it seems that we need to restrict to certain coalgebras in C to have such a correspondence. To characterize such coalgebras, let C be a coalgebra in C pi, iq and consider the bilax 2-representation C C :" add C tFC | F P C pi, jq, j P Ob C u of C .…”