Projects per year

### 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 |

## Fingerprint 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 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

## Cite this

*Journal of Pure and Applied Algebra*,

*220*(1), 148-174. https://doi.org/10.1016/j.jpaa.2015.06.003