Algebraic weak factorisation systems II: categories of weak maps

John Bourke, Richard Garner*

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    12 Citations (Scopus)


    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].

    Original languageEnglish
    Pages (from-to)148-174
    Number of pages27
    JournalJournal of Pure and Applied Algebra
    Issue number1
    Publication statusPublished - 1 Jan 2016


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