On the construction of functorial factorizations for model categories

Abstract

Abstract. We present general techniques for constructing functorial factorizations appropriate for model structures that are not known to be cofibrantly generated. Our methods use "algebraic" characterizations of fibrations to produce factorizations that have the desired lifting properties in a completely categorical fashion. We illustrate these methods in the case of categories enriched, tensored, and cotensored in spaces, proving the existence of Hurewicz-type model structures, thereby correcting an error in earlier attempts by others. Examples include the categories of (based) spaces, (based) G-spaces, and diagram spectra among others.

Algebraic weak factorisation systems I: Accessible AWFS

“…There are two main aspects to the Beck theorem. The first is the following reconstruction result; in the special case where R is idempotent, this is [29, Proposition 5.9], while the general case is given as [6,Theorem 4.15]; for the sake of a self-contained presentation, we include a proof here, which improves on that of [6] only in trivial ways. Proposition 4.…”

“…As shown in [22], any cofibrantly generated weak factorisation system on a well-behaved category may be realised as an awfs, so that the algebraic notions are entirely appropriate for doing homotopy theory; this point of view has been pushed by Riehl, who in [43,44] gives definitions of algebraic model category and algebraic monoidal model category, and in subsequent collaboration has used these notions to obtain non-trivial homotopical results [6,12,15].…”

“…The role of these examples, and others like them, will be to illuminate and justify the main contribution of this paper-that of giving a comprehensive account of the theory of unenriched awfs. Parts of this theory can be found developed across the papers [6,22,25,43,44]; our objective is to draw the most important of these results together, and to complete them with two new theorems that clarify and simplify both the theory and the practice of awfs.…”

Homotopy homomorphisms and the classifying space functor

“…With our choice of weak equivalences the Strøm model structure on Top lifts to Top * so that Ho Top * exists, but in contrast to the Quillen model structure, it is not known that the Strøm model lifts to Mon (there is a model structure on Mon whose weak equivalences are homotopy equivalences in Mon rather than homotopy equivalences of underlying spaces; this follows from work of Cole [8] and Barthel and Riel [2]).…”

Bousfield Localisation and Colocalisation of One-Dimensional Model Structures

“…Another positive consequence of our approach, and our original motivation for developing it, is that allows for an account of (co)localisation for the algebraic model structures of Riehl [25]. These are combinatorially rich presentations of model categories in which, among other things, (acyclic) fibrant replacement constitutes a monad on the category of arrows, and (acyclic) cofibrant replacement a comonad; they have been used to derive non-trivial homotopical results [12,2,7], and are of some importance in the homotopy type theory project [31]. However, there is no account of localisation for algebraic model structures as there seems to be no "algebraic" version of Smith's theorem.…”