Algebraic weak factorisation systems I: Accessible AWFS

• John Bourke ^[1] ; Richard Garner ^[2]
  1. [1] Masaryk University

     Masaryk University

     Chequia

     
  2. [2] Macquarie University

     Macquarie University

     Australia

     
• Localización: Journal of pure and applied algebra, ISSN 0022-4049, Vol. 220, Nº 1 (January 2016), 2016, págs. 108-147
• Idioma: inglés
• DOI: 10.1016/j.jpaa.2015.06.002
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• Resumen

  • Algebraic weak factorisation systems (awfs) refine weak factorisation systems by requiring that the assignations sending a map to its first and second factors should underlie an interacting comonad–monad pair on the arrow category. We provide a comprehensive treatment of the basic theory of awfs—drawing on work of previous authors—and complete the theory with two main new results. The first provides a characterisation of awfs and their morphisms in terms of their double categories of left or right maps. The second concerns a notion of cofibrant generation of an awfs by a small double category; it states that, over a locally presentable base, any small double category cofibrantly generates an awfs, and that the awfs so arising are precisely those with accessible monad and comonad. Besides the general theory, numerous applications of awfs are developed, emphasising particularly those aspects which go beyond the non-algebraic situation.