Homotopy complete category with fibrations

Definition 2.2.10. Let \(C\) be an \(\infty \)-category with weak equivalences and fibrations. We say that \(C\) is homotopy complete if the following two conditions are satisfied:

(1)

The product \(\prod _{i \in I} X_i\) of any small collection of fibrant objects \((X_i)_{i \in I}\) exists in \(C\) and is again fibrant;

(2)

Given a small collection \((f_i\colon X_i \to Y_i)\) of morphisms between fibrant objects, if each \(f_i\) is a fibration (resp. trivial fibration), then also \(\prod _{i} f_i\) is a fibration (resp. trivial fibration).

A functor between homotopy complete \(\infty \)-categories with weak equivalences and fibrations is called homotopy continuous if it is left exact and preserves small products of fibrant objects.

Dually, we say an \(\infty \)-category with weak equivalences and cofibrations \(C\) is homotopy cocomplete if \(C\catop \) is homotopy complete; we obtain the analogous notion of a homotopy cocontinuous functor.