∞-Category with weak equivalences and fibrations

An ∞-Category with weak equivalences and fibrations is a Quasicategory C 𝐶 C equipped with

Such that the following properties hold:

  1. For any Cartesian square of C 𝐶 C

    x′superscript𝑥′{x^{\prime}}x𝑥{x}y′superscript𝑦′{y^{\prime}}y𝑦{y}p′superscript𝑝′\scriptstyle{p^{\prime}}u𝑢\scriptstyle{u}p𝑝\scriptstyle{p}v𝑣\scriptstyle{v}

    in which p 𝑝 p is a fibration between Fibrant objects, and y ′ superscript 𝑦 ′ y^{\prime} is fibrant, if p 𝑝 p belongs to W 𝑊 W , then so does p ′ superscript 𝑝 ′ p^{\prime} .

  2. For any map with fibrant codomain f : x → y : 𝑓 → 𝑥 𝑦 f:x\to y in C 𝐶 C , there exists a map w : x → x ′ : 𝑤 → 𝑥 superscript 𝑥 ′ w:x\to x^{\prime} and a fibration p : x ′ → y : 𝑝 → superscript 𝑥 ′ 𝑦 p:x^{\prime}\to y such that f 𝑓 f is a composite of p 𝑝 p and w 𝑤 w .

The condition (2) is reminiscent of the normal factorization property we'd find in a Model category, but here we've restricted ourselves to only factorizing maps with fibrant codomain.

References