A necessary and sufficient condition for induced model structures

TL;DR

This paper establishes that for accessible model structures, the well-known acyclicity condition is both necessary and sufficient for inducing valid model category structures via adjoint functors, and demonstrates applications in various algebraic contexts.

Contribution

It proves the sufficiency of the acyclicity condition for accessible model structures and applies this to construct new model categories in algebraic settings.

Findings

Acyclicity condition is sufficient for accessible model structures.

Constructed new model structures on differential graded bialgebras and comodules.

Reedy model structures can be viewed as bialgebras.

Abstract

A common technique for producing a new model category structure is to lift the fibrations and weak equivalences of an existing model structure along a right adjoint. Formally dual but technically much harder is to lift the cofibrations and weak equivalences along a left adjoint. For either technique to define a valid model category, there is a well-known necessary "acyclicity" condition. We show that for a broad class of "accessible model structures" - a generalization introduced here of the well-known combinatorial model structures - this necessary condition is also sufficient in both the right-induced and left-induced contexts, and the resulting model category is again accessible. We develop new and old techniques for proving the acyclity condition and apply these observations to construct several new model structures, in particular on categories of differential graded bialgebras, of differential graded comodule algebras, and of comodules over corings in both the differential graded and the spectral setting. We observe moreover that (generalized) Reedy model category structures can also be understood as model categories of "bialgebras" in the sense considered here.