Saturated classes | Higher Topos Theory notes

A.2. Saturated classes

In practice, factorization systems are often constructed by specifying a set of generators for the left class.

Let $$L$$ be a class of morphisms of a presentable category $$C$$ . We say that $$L$$ is saturated if it contains all isomorphisms, is closed under composition, and the subcategory of $\Ar(C)$ spanned by the morphisms in $$L$$ is closed under colimits.

Remark A.9.

Given a factorization system $$(L,R)$$ on $$C$$ , it immediately follows from Proposition A.3 that $$L$$ is saturated.

Note that the intersection of any collection of saturated classes is again saturated. This allows us to make the following definition:

Definition A.10.

Let $\Sigma$ be a class of morphisms in $$C$$ . The saturation of $\Sigma$ , denoted $\Sigma^s$ , is the smallest saturated class containing $\Sigma$ . We say that a saturated class $$L$$ is of small generation if $L = \Sigma^s$ for some set of morphisms $\Sigma$ .

The following proposition provides the bridge between saturated classes and factorization systems.

Proposition A.11. ([Lurie 2009, Proposition 5.5.5.7])

Let $$C$$ be a presentable category and let $$L$$ be a saturated class of small generation. Then the pair $(L,L^{\perp})$ forms a factorization system on $$C$$ .

References

Jacob Lurie. Higher topos theory. Ann. Math. Stud. 170, Princeton, NJ: Princeton University Press. 2009.