Left exact localization via fibrant objects

Theorem 2.2.8 (Cisinski (2019), Proposition 7.5.6, Theorem 7.5.18). Let \((C,W,F)\) be an \(\infty \)-category with weak equivalences and fibrations, let \(C_f \subseteq C\) be the full subcategory of fibrant objects, and write \(W_f\) for the weak equivalences between fibrant objects.

(1)

The inclusion \(\iota \colon C_f \hookrightarrow C\) induces an equivalence \[ \overline \iota \colon C_f[W_f^{-1}] \xrightarrow {\simeq } C[W^{-1}]. \]

(2)

The localization \(C[W^{-1}]\) has finite limits (i.e., a terminal object and pullbacks), and the localization functor \(\gamma \colon C \to C[W^{-1}]\) is left exact (when \(C[W^{-1}]\) is given the trivial structure as in Example 2.2.5).

(3)

For any \(\infty \)-category \(D\) with finite limits, composition with \(\gamma \) induces an equivalence \[ \Fun ^{\lex }(C[W^{-1}],D) \iso \Fun ^{\lex }_W(C,D) \] between left exact functors \(C[W^{-1}] \to D\) and left exact functors \(C \to D\) that invert the morphisms in \(W\). In particular, every left exact functor \(C \to D\) which inverts the morphisms \(W\) induces a left exact functor \(C[W^{-1}] \to D\).