Invariance Properties of Davydov-Yetter Cohomology

TL;DR

This paper investigates the invariance properties of Davydov-Yetter cohomology, demonstrating its stability under certain categorical constructions like adjoining units and colimits, which has implications for related categorical completions.

Contribution

The paper proves that Davydov-Yetter cohomology remains invariant when a unit object is added or colimits are introduced, extending understanding of its robustness.

Findings

DY cohomology is invariant under adjoining a unit object

DY cohomology is invariant under adjoining colimits

Implications for invariance in categorical constructions

Abstract

Davydov-Yetter (DY) cohomology is a cohomology theory for linear semigroupal (i.e.~monoidal but not necessarily categories and functors, measuring deformations of their coherence isomorphisms. We show that DY cohomology is invariant under freely adjoining a unit object, and under adjoining colimits. This implies that constructions such as Ind-completion and monoidal abelian envelope do not change the cohomology.