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Abstract
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].
Original language | English |
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Pages (from-to) | 148-174 |
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
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Structural homotopy theory: a category-theoretic 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
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