Corollary 2.2.9. Let \((C,W,I)\) be an \(\infty \)-category with weak equivalences and cofibrations, let \(C_c \subseteq C\) be the full subcategory of cofibrant objects, and write \(W_c\) for the weak equivalences between cofibrant objects.
- (1)
-
The inclusion \(\iota \colon C_c \hookrightarrow C\) induces an equivalence \[ \overline \iota \colon C_c[W_c^{-1}] \xrightarrow {\simeq } C[W^{-1}]. \]
- (2)
-
The localization \(C[W^{-1}]\) has finite colimits (i.e., an initial object and pushouts), and the localization functor \(\gamma \colon C \to C[W^{-1}]\) is right exact.
- (3)
-
For any \(\infty \)-category \(D\) with finite colimits, composition with \(\gamma \) induces an equivalence \[ \Fun ^{\rex }(C[W^{-1}],D) \iso \Fun ^{\rex }_W(C,D). \]
Proof. Apply Theorem 2.2.8 to the opposite category \(C\catop \). □
Generated from the authoritative LaTeX source.