Algebraic weak factorisation systems II: categories of weak maps
| Authors | |
|---|---|
| Year of publication | 2016 |
| Type | Article in Periodical |
| Magazine / Source | Journal of Pure and Applied Algebra |
| MU Faculty or unit | |
| Citation | |
| Doi | http://dx.doi.org/10.1016/j.jpaa.2015.06.003 |
| Field | General mathematics |
| Keywords | Algebraic weak factorisation system. Weak maps. |
| Attached files | |
| Description | 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]. |
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