Algebraic weak factorisation systems II: categories of weak maps

TL;DR

This paper explores categories of weak maps in algebraic weak factorisation systems, revealing their structure as homotopy categories and connecting them to comonads, monads, and various applications in category theory.

Contribution

It provides a new characterization of categories of weak maps as homotopy categories and establishes a fully faithful adjoint between AWFS and cofibrant replacement comonads.

Findings

Categories of weak maps are described as homotopy categories with sections for acyclic fibrations.

The functor from AWFS to cofibrant replacement comonads has a fully faithful right adjoint.

Applications include generalized sketches, 2D monad theory, and dg-categories.

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.