Invariance Properties of Davydov-Yetter Cohomology
Peter Mader
TL;DR
This work proves invariance properties for Davydov-Yetter cohomology, showing that cohomology is unchanged when freely adjoining a unit to a semigroupal category and when passing to common cocompletions such as additive envelopes, pseudo-abelian envelopes, Ind-completions, or presheaf completions. The key strategy adapts Hochschild cohomology arguments to the monoidal, non-unital setting and leverages Day convolution and the Yoneda embedding to relate the cohomology of a category to that of its cocompletion. The results imply that constructions like Ind-completion and monoidal abelian envelopes do not alter the DY cohomology, enabling stable deformation-theoretic conclusions across categorical enhancements. Methodologically, the paper develops a robust functorial picture for DY cohomology via correctly constrained squares of semigroupal functors, and provides concrete isomorphism results for the cochain complexes under unit-adjunction and colimit adjunctions, with broad implications for deformation theory in monoidal and pseudo-tensor categories.
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.