ScalingStacks

Complete Segal spaces arise naturally in situations where one can do homotopy theory. Any category gives rise to a complete Segal space by means of a classifying diagram construction, to be described below. A Quillen closed model category can give rise to a complete Segal space by means of a classification diagram construction, which is a generalization of the classifying diagram. More generally, a pair $(C,W)$ consisting of a category $C$ and a subcategory $W$ gives rise to a complete Segal space by means of a localization of the classification diagram.

Given a closed model category ${\operatorname{\mathbf{M}}}$ and a small category $C$, it is often the case that the category ${\operatorname{\mathbf{M}}}^{C}$ of functors from $C$ to ${\operatorname{\mathbf{M}}}$ is again a closed model category. In this case, one can ask whether the classification diagram of ${\operatorname{\mathbf{M}}}^{C}$ is equivalent to the complete Segal space obtained as the internal hom-object of maps from the classifying diagram of $C$ to the classification diagram of ${\operatorname{\mathbf{M}}}$. A consequence (8.12) of a result of Dwyer and Kan tells us that this equivalence holds at least when ${\operatorname{\mathbf{M}}}$ is the category of simplicial sets, or more generally a category of diagrams of simplicial sets; it presumably holds for a general closed model category, but we do not prove that here.