Lax orthogonal factorisation systems

Abstract

Abstract. This paper introduces lax orthogonal algebraic weak factorisation systems on 2-categories and describes a method of constructing them. This method rests in the notion of simple 2-monad, that is a generalisation of the simple reflections studied by Cassidy, Hébert and Kelly. Each simple 2-monad on a finitely complete 2-category gives rise to a lax orthogonal algebraic weak factorisation system, and an example of a simple 2-monad is given by completion under a class of colimits. The notions of kz lifting operation, lax natural lifting operation and lax orthogonality between morphisms are studied.

On factorisation systems for Ord-enriched categories and categories of partial maps

“…This diagram resembles the distributivity transformation described in the next section, even if it carries less structure. In fact, it is not in general a natural transformation and it does not satisfy any distributivity law as described in [Bourke andGarner, 2016, Clementino andLópez Franco, 2016] or in [Clementino and López Franco, 2020, Section 4].…”

Lax orthogonal factorisations in monad-quantale-enriched categories

“…An awfs (L, R) is a lax orthogonal factorisation system, abbreviated lofs, if L and R are lax idempotent. These factorisations were introduced by the authors in [14] and further studied in the Ord-enriched categories setting, as used here, in [15].…”

“…Simple monads and their lofss. The notion of simple monad we present here, studied in [14,15], is the Ord-enriched version of simple reflection of [8]. In an Ord-enriched category C with comma-objects, given an Ord-monad S = (S, η, µ), we construct a monad R on C 2 by considering the comma-object Kf = Sf ↓ η Y and defining Rf : Kf → Y as the second projection.…”

“…In 1985 Cassidy-Hébert-Kelly [8] studied orthogonal factorisations systems induced by reflective subcategories, with particular emphasis in the case when the reflection is simple. Among the lax orthogonal factorisation systems (lofss), that generalise the orthogonal ones in 2-categories, those arising from simple monads -as defined by the authors of this paper in [14,15] -have particular relevance. This paper intends to give a systematic way of producing simple monads in (some) topological categories over Set using the presheaf monads of (T, V )-Cat studied in [23,11].…”

“…A version for algebraic factorisation systems was introduced in [19]. The construction we employ, introduced in [14], gives rise to weak factorisations systems that need not be cofibrantly generated (see [29]).…”