Algebraic weak factorisation systems II: Categories of weak maps

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• 作者：John Bourke^a ; ^{bourkej@math.muni.cz" class="auth_mail" title="E-mail the corresponding author} ; Richard Garner^b ; ^{richard.garner@mq.edu.au" class="auth_mail" title="E-mail the corresponding author}
• 关键词：18A32 ; 55U35
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：220
• 期：1
• 页码：148-174
• 全文大小：636 K

文摘

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].