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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 .
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- 2 Finished
1/01/13 → 31/12/16