Comonad Cohomology of Track Categories

TL;DR

This paper introduces a new comonad cohomology for track categories, linking it to existing Dwyer-Kan-Smith cohomology and providing an algebraic perspective, especially for 2-groupoids.

Contribution

It defines a novel comonad cohomology for track categories and establishes its relationship with Dwyer-Kan-Smith cohomology, including conditions for their equivalence.

Findings

Comonad cohomology is linked to Dwyer-Kan-Smith cohomology via a long exact sequence.

Under certain conditions, comonad cohomology coincides with Dwyer-Kan-Smith cohomology up to a dimension shift.

Results are specialized to the case of 2-groupoids, providing algebraic insights.

Abstract

We define a comonad cohomology of track categories and we show it is linked by a long exact sequence to its Dwyer-Kan-Smith cohomology . Under mild hypothesis on the track category, we show that its comonad cohomology coincides, up to dimension shift, with its Dwyer-Kan-Smith cohomology, therefore obtaining an algebraic formulation of the latter. We also specialize our results to the case where the track category is a $2$-groupoid.