Algebraic weak factorisation systems I: accessible AWFS

TL;DR

This paper develops the theory of algebraic weak factorisation systems (AWFS), providing new characterisations and a cofibrant generation method, with numerous applications beyond traditional non-algebraic contexts.

Contribution

It offers a comprehensive theory of AWFS, introduces characterisation via double categories, and establishes a cofibrant generation framework over locally presentable bases.

Findings

Characterisation of AWFS through double categories of maps

Cofibrant generation of AWFS by small double categories

Accessible monad and comonad characterize generated AWFS

Abstract

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.