Model structures for coalgebras

TL;DR

This paper unifies two classical model category structures on coalgebras over a field by showing they are extremes of a continuum of structures parameterized by a partially ordered set.

Contribution

It introduces a unified framework that encompasses both existing model structures on coalgebras, extending the understanding of their relationships.

Findings

Identifies a continuum of model structures on coalgebras over a cooperad.

Shows the classical structures are extremal cases within this continuum.

Provides a new perspective on the homotopical properties of coalgebras.

Abstract

Classically, there are two model category structures on coalgebras in the category of chain complexes over a field. In one, the weak equivalences are maps which induce an isomorphism on homology. In the other, the weak equivalences are maps which induce a weak equivalence of algebras under the cobar functor. We unify these two approaches, realizing them as the two extremes of a partially ordered set of model category structures on coalgebras over a cooperad satisfying mild conditions.