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@article{1317571,
author = {Bourke, John Denis and Garner, Richard Henry George},
article_number = {1},
doi = {http://dx.doi.org/10.1016/j.jpaa.2015.06.003},
keywords = {Algebraic weak factorisation system. Weak maps.},
language = {eng},
issn = {0022-4049},
journal = {Journal of Pure and Applied Algebra},
title = {Algebraic weak factorisation systems II: categories of weak maps},
volume = {220},
year = {2016}
}
TY - JOUR
ID - 1317571
AU - Bourke, John Denis - Garner, Richard Henry George
PY - 2016
TI - Algebraic weak factorisation systems II: categories of weak maps
JF - Journal of Pure and Applied Algebra
VL - 220
IS - 1
SP - 148-174
EP - 148-174
PB - North Holland
SN - 00224049
KW - Algebraic weak factorisation system. Weak maps.
N2 - 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].
ER -
BOURKE, John Denis a Richard Henry George GARNER. Algebraic weak factorisation systems II: categories of weak maps. \textit{Journal of Pure and Applied Algebra}. North Holland, 2016, roč.~220, č.~1, s.~148-174. ISSN~0022-4049. doi:10.1016/j.jpaa.2015.06.003.
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