A necessary and sufficient condition for induced model structures

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.Comment: 49 pages; final journal version to appear in the Journal of Topolog

Lifting accessible model structures

“…Our terminology and approach will largely follow that of ; we begin by recalling the necessary background. Given a class of maps in a category , we write and for the classes of maps with the left, respectively, right, lifting property against each map in , and given a functor , we write for the class of morphisms in which are mapped into by .…”

“…Although the main result of is correct, the proof given there turns out to contains a subtle error: in some cases, it exhibits ‘lifted factorizations’ which are not those of the desired left‐ or right‐lifted weak factorization systems, but of slightly different ones. The purpose of this note is to fix this error.…”

“…These results were generalized in to obtain left and right liftings of factorizations for what the authors term accessible model structures. The simplest formulation of what this means is that given in : a model structure is accessible if its factorizations into the classes can be realized by accessible functors – ones which preserve ‐filtered colimits for some regular cardinal .…”

“…This means that, as well as accessible functors that realize the factorizations in each case, there should also be provided fillers: subject to axioms which, among other things, cause these data to endow the functors and with the structure of a comonad and a monad , respectively; see Definition . While this is apparently stronger than the notion of accessible model category used in , it turns out that, starting from the less elaborate definition, one can always derive the data required for the more elaborate one; this is the content of [, Remark 3.1.8].…”

“…In this way, the above procedure may produce factorizations for an incorrect lifting of one of the original weak factorization systems. The author who is responsible for this error was well aware of this issue when she proved [, Theorem 3.10] – indeed, an important part of that argument explains why it does not arise in the cofibrantly generated and right‐lifting context – but had fallen out of touch with that awareness when writing [, Section 3].…”