Understanding the Small Object Argument

Abstract

The small object argument is a transfinite construction which, starting from a set of maps in a category, generates a weak factorisation system on that category. As useful as it is, the small object argument has some problematic aspects: it possesses no universal property; it does not converge; and it does not seem to be related to other transfinite constructions occurring in categorical algebra. In this paper, we give an "algebraic" refinement of the small object argument, cast in terms of Grandis and Tholen's natural weak factorisation systems, which rectifies each of these three deficiencies.

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. Before all that we need to collect present basics of the theory of awfs, mostly following [8,2].…”

“…An algebraic weak factorisation system [9,8] is a functorial factorisation where the copointed endofunctor Φ : L ñ 1 is equipped with a comultiplication Σ : L ñ L 2 , making it into a comonad L, and the pointed endofunctor Λ : 1 ñ R is equipped with a multiplication Π : R 2 ñ R, making it into a monad R, plus a distributivity condition. The components of this comultiplication and multiplication will be denoted as follows.…”

“…Furthermore, the monad and comonad must be related by the distributivity condition introduced in [8] that asserts that the natural transformation ∆ : LR ñ RL with components…”

Cyclic multicategories, multivariable adjunctions and mates

“…This example is prototypical, so we explain it further. The sets of maps I = {∂∆ n ∆ n | n ≥ 0} and J = {Λ n k ∆ n | n ≥ 1, 0 ≤ k ≤ n} generate two awfs (C, F t ) and (C t , F ) on sSet by Garner's algebraic small object argument [3]. A simplicial set X is a Kan complex if the unique map X ∆ 0 satisfies the right lifting property with respect to J.…”