Coidempotent Subcoalgebras and Short Exact Sequences of Finitary 2-Representations

Abstract

In this article, we study short exact sequences of finitary 2representations of a weakly fiat 2-category. We provide a correspondence between such short exact sequences with fixed middle term and coidempotent subcoalgebras of a coalgebra 1-morphism defining this middle term. We additionally relate these to recollements of the underlying abelian 2-representations.

Finitary birepresentations of finitary bicategories

“…as follows e.g. from [ChMi,Lemma 3]. However, this does not contradict Theorem 4.19, since a coalgebra C which is strictly in C will correspond to a birepresentation M that is either not transitive or has smaller apex.…”

“…If M ∈ C -stmod J , then the coalgebra C satisfying (4.22) is cosimple by the generalization of [MMMZ,Corollary 12] to bicategories. For the converse statement, first observe that for cosimple C, the birepresentation inj C (C) is transitive by the generalization of [ChMi,Theorem 20 (ii)] to bicategories. The generalization of [MMMZ,Corollary 12] to bicategories then implies that it is simple transitive.…”

“…Finitary 2-representation theory of finitary 2-categories, which is the categorical analog of finite dimensional representation theory of finite dimensional algebras, has evolved considerably in the last decade, see e.g. [MM1,MM2,MM3,MM5,MM6,ChMa,ChMi,KMMZ,MMMT1,MMMZ] and references therein.…”