Simple transitive 2-representations via (co)algebra 1-morphisms

Abstract

For any fiat 2-category C, we show how its simple transitive 2-representations can be constructed using coalgebra 1-morphisms in the injective abelianization of C . Dually, we show that these can also be constructed using algebra 1-morphisms in the projective abelianization of C. We also extend Morita-Takeuchi theory to our setup and work out several examples, including that of Soergel bimodules for dihedral groups, explicitly.

Pyramids and 2-representations

“…The weak Jordan-Hölder theory developed in [MM5] motivates the study of so-called simple transitive 2-representations which are suitable 2-analogues of simple modules. It turns out that, in many cases, simple transitive 2-representations can be explicitly classified, see [MM5,MM6,MaMa,KMMZ,MMMT,MMZ,MT,MZ1,MZ2,Zh,Zi1,Zi3] for the results and [Maz2] for a detailed survey on the subject. In many, but not all, cases, simple transitive 2-representations are exhausted by so-called cell 2-representations defined already in [MM1].…”

Coidempotent Subcoalgebras and Short Exact Sequences of Finitary 2-Representations

“…It follows from [MMMT,Theorem 22] that the as an object of C R (i, i), the coalgebra 1-morphism C L := [R, R] is isomorphic to F. For C ½ , the corresponding coalgebra 1-morphism C ½ := [k, k] can be calculated via the defining adjunction isomorphisms Hom C R (C ½ , G) ∼ = Hom C ½ (i) (k, Gk) for all indecomposable 1-morphisms G, which yield that it is is isomorphic to the simple socle L ½ of ½ in C R (i, i). In fact, C ½ is a quotient 2-representation of P, so C ½ is a subcoalgebra of C P , which implies that the counit and comultiplication maps defining C ½ are both the identity map on L ½ .…”

“…We can (injectively) abelianise both the 2-category C and, for a 2-representation M, the category M := i∈C M(i) and use the notation (−) for the injective abelianisation (2)-functor. For the 2-category, this needs to be done in a rather technical way, see [MMMT,Section 3.2] to preserve strictness of horizontal composition. Note that, provided that C is weakly fiat, composition in C is left exact in both variables.…”

“…For M, it is equivalently possible to use the classical diagrammatic abelianisation, see [Fr], or [MMMT,Section 3.1] for a presentation adapted to our notation. This induces an abelian 2-representation M on M.…”

Trihedral Soergel bimodules

“…To conclude, one could say that Elias' quantum Satake correspondence [Eli16], [Eli17] categorifies the relation between the Verlinde algebra and the small dihedral quotient, while the results from [KMMZ19], [MT16], [MMMT19] categorify the relations between their N-representations.…”