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.

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.)

We believe that the most interesting feature of the theory of complete Segal spaces described above is that constructions of new homotopy theories from old ones can be made entirely inside the setting of the theory. We have already described one example: diagrams categories in a model category can be modeled as the internal function complex in the category of simplicial spaces. (We only give the proof here for the case where the model category is simplicial sets, however.)

A related construction is that of homotopy inverse limits of homotopy theories. We give one example here, without proof, to illustrate the ideas. Let $W=\class({\operatorname{\mathcal{T}}}_{*})$, the classification diagram of the category of pointed topological spaces; $W$ is a complete Segal space. Let $\omega\colon W\rightarrow W$ be the self-map associated to the loop-space functor $\Omega\colon{\operatorname{\mathcal{T}}}_{*}\rightarrow{\operatorname{\mathcal{T}}}_{*}$. Then we can form the homotopy inverse limit $W_{\infty}$, in the category of simplicial spaces, of the tower: $\dots\rightarrow W\xrightarrow{\omega}W\xrightarrow{\omega}W\xrightarrow{\omega}W$. One discovers that $W_{\infty}$ is again a complete Segal space, and that it is weakly equivalent to the classification space of the category of spectra! One should understand this example as a reinterpretation of the definition of the notion of $\Omega$-spectra.

Another example is that of sheaves of homotopy theories. There is a model category for sheaves of spaces (= sheaves of simplicial sets) over a base space (or more generally a Grothendieck topology) [Jar87], [Jar96]. Thus there is a model category structure for sheaves of simplicial spaces. Say a sheaf of simplicial spaces $W$ is a complete Segal sheaf if each stalk is a complete Segal space in the sense of this paper. This would appear to provide an adequate notion of “sheaves of homotopy theories”, and is worth investigation.

We note that several other abstract models of homotopy theory have been proposed. One has been proposed by W. Dwyer and D. Kan, as was noted above. Since the complete Segal spaces described in our work are themselves objects in a certain closed model category, our construction gives another model for a homotopy theory of homotopy theory. We believe that our model is “equivalent” to that of Dwyer and Kan, via a suitable notion of equivalence; in particular, there should be constructions which take complete Segal spaces to simplicially enriched categories and vice versa, and these constructions should be inverses to each other (modulo appropriate notions of equivalence.) We hope to give a proof of this in the future.

Another model has been proposed by A. Heller [Hel88]. He suggests that a homotopy theory be modeled by a certain type of contravariant $2$-functor from the category of small categories to the category of large categories. For example, from a closed model category ${\operatorname{\mathbf{M}}}$ there is a construction which assigns to each small category $C$ the homotopy category $\ho({\operatorname{\mathbf{M}}}^{C})$ of the category of $C$-diagrams in ${\operatorname{\mathbf{M}}}$, and which associates to each functor $C\rightarrow D$ restriction functors $\ho({\operatorname{\mathbf{M}}}^{D})\rightarrow\ho({\operatorname{\mathbf{M}}}^{C})$ which themselves admit both left and right adjoints, arising from “homotopy Kan extensions”. Because Heller’s models require the existence of such homotopy Kan extensions, they seem to be less general than the models considered in this paper, and we do not know the proper relationship between his theory and the others.

In Section 2 we set up notation for simplicial spaces and discuss the Reedy model category structure for simplicial spaces. In Section 3 we define the classification diagram construction, which produces a simplicial space from category theoretic data. In Section 4 we define the notion of a Segal space, and in Section 5 we discuss in elementary terms how one can view a Segal space as a model for a homotopy theory. In Section 6 we define the notion of a complete Segal space. In Section 7 we present our main theorems. In Section 8 we show how the classification diagram of a simplicial closed model category gives rise to a complete Segal space.

In Sections 9 through 14 we give proofs for the more technical results from earlier sections.

I would like to thank Dan Kan for his encouragement and hospitality, and Bill Dwyer for his beautiful talk at the 1993 Čech conference, where I first learned about the homotopy theory of homotopy theory. I would also like to thank Phil Hirschhorn, Mark Johnson, and Brooke Shipley for their helpful comments on the manuscript.