Limits in localizations of homotopy complete categories

Theorem 2.2.11 (Cisinski (2019), Proposition 7.7.4, Theorem 7.7.6). Let \(C\) be a homotopy complete \(\infty \)-category with weak equivalences and fibrations, and write \(L(C):=C[W^{-1}]\) for its localization.

(1)

The localization \(L(C)\) has small limits, and the localization functor \(\gamma \colon C \to L(C)\) is homotopy continuous;

(2)

If all objects of \(C\) are fibrant, then \(L(C)\) is in fact universal with this property: for any \(\infty \)-category \(E\) with small limits, the functor \(\gamma \) induces an equivalence of \(\infty \)-categories \[ \gamma ^*\colon \Fun ^{\lim }(L(C),E) \iso \Fun ^{\mathrm {hcont}}(C,E), \] where the right-hand side denotes the full subcategory spanned by the homotopy continuous functors.

(3)

Given a homotopy continuous functor \(F\colon C \to D\), the induced functor \(L(F)\colon L(C) \to L(D)\) preserves small limits;

The dual result holds for homotopy cocomplete \(\infty \)-categories with weak equivalences and cofibrations.