∞-category with weak equivalences and fibrations

Definition 2.2.4 (\(\infty \)-category with weak equivalences and fibrations). An \(\infty \)-category with weak equivalences and fibrations is a triple \((C,W,F)\), where \(C\) is an \(\infty \)-category with a terminal object, and \(W\) and \(F\) are collections of morphisms satisfying the following conditions:

(1)

(2-out-of-3) For composable morphisms \(f\) and \(g\), if two of \(f\), \(g\), \(gf\) are in \(W\), then so is the third.

(2)

(Fibrations) The class \(F\) is a class of fibrations.

(3)

(Pullback axiom) For any pullback square

in which \(p\) is a fibration between fibrant objects, and \(Y'\) is fibrant, if \(p\) belongs to \(W\) then so does \(p'\).

(4)

(Factorization axiom) For any morphism \(f\colon X \to Y\) in \(C\) with fibrant codomain, there exists a morphism \(w\colon X \to X'\) in \(W\) and a fibration \(p\colon X' \to Y\) such that \(f = p \circ w\).

The morphisms in \(W\) are called weak equivalences. Morphisms in \(W \cap F\) are trivial fibrations. We similarly define an \(\infty \)-category with weak equivalences and cofibrations by dualizing (replacing \(F\) with a class of cofibrations \(I\) and appropriately modifying axioms (3) and (4) to involve pushouts and cofibrant objects).