Strongly saturated classes | Higher Topos Theory notes

A.3. Strongly saturated classes

When studying localizations of presentable categories, we require a stronger stability property.

Let $$L$$ be a class of morphisms of a presentable category $$C$$ . We say $$L$$ is strongly saturated if it is saturated and closed under 2-out-of-3.

As with saturated classes, we say a strongly saturated class $$L$$ is of small generation if it is the smallest strongly saturated class containing some set of morphisms $\Sigma$ . The following two results identify these classes with kernels of accessible Bousfield localizations.

Definition A.13.

Let $\varphi\colon C \to D$ be a cocontinuous functor between presentable categories. We define its kernel as the collection of morphisms in $$C$$ that is inverted by $\varphi$ :

\ker(\varphi) \quad := \quad \{f\colon X \to Y \in \Ar(C) \mid \varphi(f)\text{ is an isomorphism}\}.

Proposition A.14. ([Lurie 2009, Proposition 5.5.4.16])

Let $\varphi\colon C \to D$ be a cocontinuous functor between presentable categories. Then $\ker(\varphi)$ is strongly saturated and of small generation.

Proposition A.15. ([Lurie 2009, Propositions 5.2.7.12 and 5.5.4.15])

Let $$C$$ be a presentable category and let $\Sigma$ be a strongly saturated class of small generation.

The inclusion $\Loc_{\Sigma}(C) \hookrightarrow C$ of the $\Sigma$ -local objects in $$C$$ admits an accessible left adjoint $L\colon C \to \Loc_{\Sigma}(C)$ .
The category $\Loc_{\Sigma}(C)$ is presentable.
We have $\ker(L) = \Sigma$ , i.e. a morphism in $$C$$ is inverted by $$L$$ if and only if it lies in $\Sigma$ .
For every category $$E$$ , precomposition with $$L$$ induces a fully faithful functor
$\Fun(\Loc_{\Sigma}(C),E)\longrightarrow\Fun(C,E)$
whose essential image consists of the functors that invert every morphism in $\Sigma$ .

References

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