# When is an $\infty$-categorical localization left exact?

Let $$L: \mathcal C^\to_\leftarrow L\mathcal C : i$$ be an adjunction with $$i$$ fully faithful. In ordinary category theory, $$L$$ is left exact iff the class of $$L$$-local morphisms is stable under base change [1].

This appears to be true $$\infty$$-categorically, as well. Is there a proof in the literature?

But $$\infty$$-categorically, this is no longer true:

Example: Let $$\mathcal C$$ be the $$\infty$$-category of spaces and $$L\mathcal C$$ the full subcategory of $$n$$-truncated spaces for some fixed $$n$$, so that $$L$$ is the $$n$$-truncation functor and the $$L$$-local morphisms are those with $$(n+1)$$-connected fibers. Then $$L$$ is not left exact (failing to preserve, for example, the pullback square $$K(\mathbb Z, n) \rightrightarrows \ast, \ast \rightrightarrows K(\mathbb Z, n+1)$$), but the $$L$$-local morphisms are stable under base change.

Question: Let $$L: \mathcal C^\to_\leftarrow L\mathcal C: i$$ be an adjunction of finitely-complete $$\infty$$-categories with $$i$$ fully faithful. Let $$\mathcal W = L^{-1}(\{\textrm{isos}\}) \subseteq \textrm{Mor} \mathcal C$$ be the class of $$L$$-local morphisms. What are necessary and sufficient closure conditions on $$\mathcal W$$ ensuring that $$L$$ is left exact?

I'm happy to assume that $$\mathcal C$$ is presentable, or even an $$\infty$$-topos, and that $$\mathcal W$$ is of small generation.

[1] Here we assume that $$\mathcal C$$ is finitely complete. A morphism $$f$$ is said to be $$L$$-local if $$L(f)$$ is an isomorphism, and a class of morphisms $$\mathcal W$$ is stable under base change if $$f \in \mathcal W$$ implies $$f' \in \mathcal W$$ where $$f'$$ is any pullback of $$f$$ along an arbitrary morphism.

• Is the class of morphisms of spaces inverted by $n$-truncation stable under pullback? Isn't your pullback square a counterexample?: $\ast \to K(\mathbb{Z},n+1)$ is inverted by $n$-truncation, but $K(\mathbb{Z},n) \to \ast$ isn't, right? – Alexander Campbell Jan 8 at 21:21
• @AlexanderCampbell -- yes, I was getting confused. This criterion does hold $\infty$-categorically. I've just written up a proof over here. – Tim Campion Jan 8 at 21:30