Holmstrom Homotopy cartesian

In a model category, a commutative square “X,Y,W,Z” is homotopy cartesian if for every factorization of the left map $f: Y \to Z$ into a trivial cofibration followed by a fibration, the induced map $X \to W \times_Z \tilde{Y}$ is a weak equivalence. Here $\tilde{Y}$ is the thing in the middle of the factorization.

In fact, to show that a diagram is homotopy cartesian, it suffices to find one factorization such that the relevant map is a WE.

A homotopy fibre sequence is homotopy cartesian diagram of simplicial sets, such that the lower map is the inclusion of the base point. Example: Every fibration sequence.

See Jardine-Goerss for more details, section II.9 (p 128). This section also discusses categories of cofibrant/fibrant objects.

Jardine-Goerss chapter IV discusses “detection principles” for homotopy cartesian diagrams.

nLab page on Homotopy cartesian