ScalingStacks

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.