Algebraic weak factorisation systems I: Accessible AWFS

Abstract

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.

Lax orthogonal factorisation systems

“…The basic theory of awfss appeared in [9] with the name of natural weak factorisation system, and was later expanded in [8], especially with respect to the construction of cofibrantly generated awfs. Further study appeared recently in [2]. From Section 4 onwards, the present paper expands the theory in another direction, that of awfs in 2-categories whose lifting operations, or diagonal fillers, satisfy a universal property with respect to 2-cells.…”

“…One can easily characterise the awfs obtained in this way. Furthermore, if R is idempotent, then so is L, a proof of which can be found in [2].…”

“…The underlying wfs of an awfs pL, Rq is an orthogonal factorisation system precisely when L and R are idempotent [9]; for this, it is enough if either is idempotent [2].…”

“…From Section 4 onwards, the present paper expands the theory in another direction, that of awfs in 2-categories whose lifting operations, or diagonal fillers, satisfy a universal property with respect to 2-cells. Before all that we need to collect present basics of the theory of awfs, mostly following [8,2].…”

“…The first is the 2-category LaripK q, whose objects are morphisms f in K equipped with a right adjoint coretract r f , ie a left adjoint right inverse or lari, in the terminology used in [2]; we may write an object of this 2-category as pf, rq, omitting the unit and counit of the adjunction, since the unit is an identity 2-cell and the counit is, therefore, the unique 2-cell that satisfies the adjunction triangle axioms for f % r. A morphism pf, rq Ñ pf 1 q is canonically a lari. In this way, LaripK q has a double category structure, and furthermore, the composition is obviously compatible with 2-cells, so the double category structure extends to an internal category in the category of 2-categories.…”