Hochschild cohomology for finitary 2-representations

TL;DR

The paper develops Hochschild cohomology for finitary 2-representations of quasi-fiat 2-categories by associating to a cyclic 2-representation an algebra 1-morphism $A$ in the abelianisation and defining $\mathrm{HH}_{\mathscr{C}}^k(A)=\mathrm{Ext}^k_{A\text{-mod}_{\overline{\mathscr{C}}-A}}(A,A)$. It constructs a reduced complex for computation, provides a replacement for the bar resolution when $A$ is not a literal 1-morphism in the 2-category, and proves Morita invariance and quotient-invariance results that allow reduction to cell-theoretic contexts. The authors compute and interpret $\mathrm{HH}^0$, $\mathrm{HH}^1$, and $\mathrm{HH}^2$, relating $\mathrm{HH}^2$ to deformations and describing inflations via $A^{\boxtimes R}$, with exact behavior under inflation $\mathbf{M}^{\boxtimes R}$. They illustrate the theory with examples including strongly regular cells, inflations, and examples in $\mathscr{C}_D$ and $\mathscr{V}ec_G$, showing both vanishing results and instances of nontrivial Hochschild cohomology in low degrees, thereby connecting categorical representations to homological invariants.

Abstract

In this article, we define and investigate Hochschild cohomology for finitary 2-representations of quasi-fiat 2-categories.

Theorems & Definitions (17)

• Lemma 1
• Definition 1
• Proposition 1
• Definition 2
• Proposition 2
• Lemma 2
• Proposition 3
• Proposition 4
• Proposition 5
• Proposition 6