Algebraic weak factorisation systems II: categories of weak maps

Abstract

We investigate the categories of weak maps associated to an algebraic weak factorisation system (AWFS) in the sense of Grandis-Tholen. For any AWFS on a category with an initial object, cofibrant replacement forms a comonad, and the category of (left) weak maps associated to the AWFS is by definition the Kleisli category of this comonad. We exhibit categories of weak maps as a kind of "homotopy category", that freely adjoins a section for every "acyclic fibration" (=right map) of the AWFS; and using this characterisation, we give an alternate description of categories of weak maps in terms of spans with left leg an acyclic fibration. We moreover show that the 2-functor sending each AWFS on a suitable category to its cofibrant replacement comonad has a fully faithful right adjoint: so exhibiting the theory of comonads, and dually of monads, as incorporated into the theory of AWFS. We also describe various applications of the general theory: to the generalised sketches of Kinoshita-Power-Takeyama, to the two-dimensional monad theory of Blackwell-Kelly-Power, and to the theory of dg-categories.

30 pages, final journal version

References in corpus (3)

• Higher Topos Theory
• Algebraic weak factorisation systems I: accessible AWFS
• The Coalgebraic Structure of Cell Complexes

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• Hom weak $\omega$-categories of a weak $\omega$-category
• A semi-model structure for Grothendieck weak 3-groupoids
• 2-categorical opfibrations, Quillen's Theorem B, and $S^{-1}S$