From Partial Model Categories to ∞-categories

In this article we study the problem of extracting an ∞-category from a relative category. We introduce partial model categories, which are relative categories that satisfy mild versions of the axioms of a model category. Since these axioms involve only the weak equivalences, they are general enough to include the vast majority of the relative categories one encounters in practice. We show that the simplicial nerve of a partial model category is “essentially” a complete Segal space, generalizing a result of Charles Rezk. To prove this, we must introduce a significant generalization of a Quillen’s Theorem B. We show also that, conversely, any complete Segal space is dimensionwise equivalent to the simplicial nerve of a partial model category.