Algebraic weak factorisation systems II: categories of weak maps

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Authors

BOURKE John Denis GARNER Richard Henry George

Year of publication 2016
Type Article in Periodical
Magazine / Source Journal of Pure and Applied Algebra
MU Faculty or unit

Faculty of Science

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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