The cohomology of coalgebras in species

TL;DR

This paper develops a new approach to the cohomology of twisted coalgebras in species, enabling effective computations and structural insights, including a Künneth formula and multiplicative structure analysis.

Contribution

It introduces an alternative description of the cohomology theory for twisted coalgebras using Koszul duality, facilitating explicit calculations and structural understanding.

Findings

Complete cohomology computations for key examples

Identification of the multiplicative structure in cohomology

Establishment of a K"unneth formula for the theory

Abstract

Aguiar and Mahajan introduced a cohomology theory for the twisted coalgebras of Joyal, with particular interest in the computation of their second cohomology group, which gives rise to their deformations. We use the Koszul duality theory between twisted algebras and coalgebras on the twisted coalgebra that gives rise to their cohomology theory to give a new alternative description of it which, in particular, allows for its effective computation. We compute it completely in various examples, including those proposed by Aguiar and Mahajan, and obtain structural results: in particular, we study its multiplicative structure and provide a K\"unneth formula.