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Abstract
We investigate the categories of weak maps associated to an algebraic weak factorisation system (awfs) in the sense of GrandisTholen [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 2functor 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 KinoshitaPowerTakeyama [22], to the twodimensional monad theory of BlackwellKellyPower [4], and to the theory of dgcategories [19].
Original language  English 

Pages (fromto)  148174 
Number of pages  27 
Journal  Journal of Pure and Applied Algebra 
Volume  220 
Issue number  1 
DOIs  
Publication status  Published  1 Jan 2016 
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Dive into the research topics of 'Algebraic weak factorisation systems II: categories of weak maps'. Together they form a unique fingerprint.Projects
 2 Finished

Structural homotopy theory: a categorytheoretic study
Street, R., Lack, S., Verity, D., Garner, R., MQRES, M., MQRES 3 (International), M. 3., MQRES 4 (International), M. & MQRES (International), M. (.
1/01/13 → 31/12/16
Project: Research
