Algebraic weak factorisation systems II: Categories of weak maps

作者：Bourke John; Garner Richard^*

来源：Journal of Pure and Applied Algebra, 2016, 220(1): 148-174.

DOI：10.1016/j.jpaa.2015.06.003

We investigate the categories of weak maps associated to an algebraic weak factorisation system (AwFs) in the sense of Grandis-Tholen [14]. 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 [22], to the two-dimensional monad theory of Blackwell-Kelly-Power [4], and to the theory of dg-categories [19].

• 出版日期2016-1