The homotopy theory of bialgebras over pairs of operads

TL;DR

This paper establishes a model category structure for bialgebras over pairs of operads in chain complexes, enabling homotopical algebra techniques in various algebraic contexts.

Contribution

It introduces a cofibrantly generated model structure on the category of bialgebras over pairs of operads in chain complexes, extending homotopical methods to new algebraic structures.

Findings

Model structure on bialgebras over operads established

Applicable to associative, Lie, and Poisson bialgebras

Enables homotopical algebra in these categories

Abstract

We endow the category of bialgebras over a pair of operads in distribution with a cofibrantly generated model category structure. We work in the category of chain complexes over a field of characteristic zero. We split our construction in two steps. In the first step, we equip coalgebras over an operad with a cofibrantly generated model category structure. In the second one we use the adjunction between bialgebras and coalgebras via the free algebra functor. This result allows us to do classical homotopical algebra in various categories such as associative bialgebras, Lie bialgebras or Poisson bialgebras in chain complexes.