ScalingStacks

We give a brief description of the classifying diagram and classification diagram constructions here, in order to motivate the definition of a complete Segal space. These constructions are discussed in detail in Section 3.

To any category $C$ one may associate its classifying space $BC$; this is a space obtained by taking a vertex for each object of $C$, attaching a $1$-simplex for each morphism of $C$, attaching a $2$-simplex for each commutative triangle in $C$, and so forth. It is well-known that if the category $C$ is in fact a groupoid, then it is characterized (up to equivalence of categories) by its classifying space; for a groupoid $C$ the classifying space $BC$ has the homotopy type of a disjoint union of spaces $K(\pi_{X},1)$, where $X$ ranges over the representatives of isomorphism classes of objects in $C$ and each $\pi_{X}$ is the group of automorphisms of the object $X$ in $C$.

A general category cannot be recovered from its classifying space. Instead, let $\iso C$ denote the subcategory of $C$ consisting of all objects and all isomorphisms between them; thus $\iso C$ is just the maximal subgroupoid of the category $C$. From the homotopy type of the classifying space $B(\iso C)$ of this groupoid one can recover some information about the category $C$, namely the set of isomorphism classes of objects in $C$ and the group of automorphisms of any object. For this reason one may view $B(\iso C)$ as a kind of “moduli space” for the category $C$.

Although a category $C$ is not determined by its classification space, it turns out (3.7) that it is determined, up to equivalence, by a simplicial diagram of spaces $[n]\mapsto B\iso(C^{[n]})$ which we call the classifying diagram of $C$; here $[n]$ denotes the category consisting of a sequence of $(n+1)$ objects and $n$ composable arrows, and $C^{[n]}$ denotes the category of functors from $[n]\rightarrow C$. The classifying diagram of a category is in fact a complete Segal space.

The homotopy theoretic analogue of $B(\iso C)$ is Dwyer and Kan’s notion of the classification space of a model category. Given a closed model category ${\operatorname{\mathbf{M}}}$, let $\we{\operatorname{\mathbf{M}}}\subset{\operatorname{\mathbf{M}}}$ denote the subcategory consisting of all objects and all weak equivalences between them. The classification space of ${\operatorname{\mathbf{M}}}$ is denoted $\class({\operatorname{\mathbf{M}}})$, and is defined to be $B(\we{\operatorname{\mathbf{M}}})$, the classifying space of the category of weak equivalences of ${\operatorname{\mathbf{M}}}$. The classification space of a model category is in many ways analogous to the space $B(\iso C)$ considered above. For example, $\class({\operatorname{\mathbf{M}}})$ has the homotopy type of a disjoint union of spaces $B(\haut{X})$, where $X$ ranges over appropriate representatives of weak equivalence classes of objects in ${\operatorname{\mathbf{M}}}$, and $\haut{X}$ denotes the simplicial monoid of self-homotopy equivalences of $X$ (8.7). Classification spaces arise naturally in the study of realization problems, e.g., the problem of realizing a diagram in the homotopy category of spaces by an actual diagram of spaces; see [DK84b], [DK84a].

Given a closed model category ${\operatorname{\mathbf{M}}}$, form a simplicial space $[n]\mapsto\class({\operatorname{\mathbf{M}}}^{[n]})$, called the classification diagram of ${\operatorname{\mathbf{M}}}$. We show (8.3) that the classification diagram of a closed model category is essentially a complete Segal space. (“Essentially” means up to an easy fibrant replacement.)