Holmstrom Path object

In a model category, a path object for an object $X$ is a factorisation of the diagonal map $X \to X \times X$ into a WE followed by a fibration.

A path object is the dual of a cylinder object.

The functorial factorisations provide a functorial path object, denoted by $X^I$.

In situations where the category is not (“yet”) a model category, one knows that path objects exist whenever the category is a simplicial category.

nLab page on Path object