nLab partial model category

Idea

Partial model categories are one of the many intermediate notions between relative categories and model categories.

They axiomatize those properties of model categories that only involve weak equivalences.

Definition

A partial model category is a relative category $(C, W)$ that satisfies the 2-out-of-6 property (if $s r$ and $t s$ are weak equivalences, then so are $r$, $s$, $t$, $t s r$) and admits a 3-arrow calculus?, i.e., there are subcategories $U, V \subseteq W$ of the weak equivalences (which can be thought of as analogues of acyclic cofibrations and acyclic fibrations) such that $U$ is closed under cobase changes along arbitrary morphisms of $C$ (which are required to exist), $V$ is closed under base changes along arbitrary morphisms of $C$, and any weak equivalence can be functorially factored as the composition $v u$ for some $u\in U$ and $v\in V$.

Properties

If $(C,W)$ is a partial model category, then any Reedy fibrant replacement of the Rezk nerve? $N(C,W)$ is a complete Segal space.

Related concepts

• model category

• relative category

• semimodel category

• weak model category

• premodel category

• homotopical category

References

• Clark Barwick, Daniel M. Kan, Partial model categories and their simplicial nerves, arXiv.