Chaitanya Leena Subramaniam: The small object argument for enriched factorisation systems in enriched ∞-categories

Abstract

Given a complete and cocomplete closed symmetric monoidal category 𝒱, a weak factorisation system (A,B) on 𝒱 is monoidal when its left class A is closed under Leibniz tensors (“pushout products”) and contains the morphism ∅→I from the initial object to the monoidal unit. Then maps in any 𝒱-category ℂ have an orthogonality relation relative to the right class B — a map f in ℂ is B-orthogonal to a map g in ℂ just when the Leibniz hom (“pullback hom”) ⟨f,g⟩ is in B. We can use the relation of B-orthogonality to say when a pair of classes of maps (L,R) in ℂ form a (𝒱,(A,B))-enriched factorisation system.

Quillen's Small Object Argument constructs weak factorisation systems in a 1-category, starting with the data of a generating set of maps. Quillen’s S.O.A. can be generalised to construct weak factorisation systems in ∞-categories [Lurie, 1.4.7].

In this talk, I will show a variant of the S.O.A. that constructs enriched factorisation systems in enriched ∞-categories, starting from a generating small diagram of arrows. Quillen's S.O.A. can be verified, by hand, to work in enriched 1-categories [Jurka], but this reasoning does not work well in ∞-category theory. I will present a different argument that does work for enriched ∞-categories. This more general construction defines (and uses) a slightly more general relation on arrows than B-orthogonality. Moreover, this construction produces a functorial factorisation when the weak factorisation system (A,B) is suitably functorial.

There are many examples of monoidal weak factorisation systems and enriched factorisation systems, in 1-category theory and in ∞-category theory.

For instance, in any ∞-topos (and in any 1-topos), the surjections (= effective epimorphisms) and the split surjections are both right classes of monoidal weak factorisation systems. Ordinary (non-functorial) weak factorisation systems are exactly those that are enriched over the surjections of the ∞-topos 𝒮 of spaces, and functorial weak factorisation systems are related to an enrichment over split surjections in 𝒮.

Another interesting example is given by the uniform Kan fibrations in the 1-topos of symmetric semicartesian (a.k.a. "BCH") cubical sets. These can be described using a general enriched orthogonality relation.

The talk will be "1/∞"-agnostic, and can be viewed entirely through the lens of 1-category theory to recover the corresponding results for enriched 1-categories.

This is joint work with Mathieu Anel.

References:

[Jurka] J. Jurka, "An enriched small object argument over a cofibrantly generated base" (arXiv:2401.05974)
[Lurie] J. Lurie, "Derived Algebraic Geometry X: Formal Moduli Problems"