Let {\mathcal{C}} be an {\infty}-category, in the sense of Joyal and Lurie: in other words, a quasicategory or weak Kan complex. For instance, for the purposes of Hopkins-Miller, we’re going to be interested in the {\infty}-category of spectra. A simplicial object of {\mathcal{C}} is a functor

\displaystyle F: N(\Delta^{op}) \rightarrow \mathcal{C} ,

that is, it is a morphism of simplicial sets from the nerve of the opposite {\Delta^{op}} of the simplex category to {\mathcal{C}}. A geometric realization of such a simplicial object is a colimit. A simplicial object is like a reflexive coequalizer (in fact, the 1-skeleton is precisely a reflexive coequalizer diagram) but with extra “higher” data in bigger degrees. Since reflexive coequalizers are a useful tool in ordinary category theory (for instance, in flat descent), we should expect geometric realizations to be useful in higher category theory. That’s what this post is about.

A simple example of a geometric realization is as follows: let {X_\bullet} be a simplicial set, thus defining a homotopy type and thus an object of the {\infty}-category {\mathcal{S}} of spaces. Alternatively, {X_\bullet} can be regarded as a simplicial object in sets, so a simplicial object in (discrete) spaces. In other words, {X_\bullet} has two incarnations:

  1. {X_\bullet \in \mathcal{S}}.
  2. {X_\bullet \in \mathrm{Fun}(\Delta^{op}, \mathcal{S})}.

The connection is that {X_\bullet} is the geometric realization (in the {\infty}-category of spaces) of the simplicial object {X_\bullet}. More generally, whenever one has a bisimplicial set {Y_{\bullet, \bullet}}, defining an object of {\mathrm{Fun}(\Delta^{op}, \mathcal{S})}, then the geometric realization of {Y_{\bullet, \bullet}} in {\mathcal{S}} is the diagonal simplicial set {n \mapsto Y_{n, n}}. These are model categorical observations: one chooses a presentation for {\mathcal{S}} (e.g., the usual Kan model structure on simplicial sets), and then uses the fact that {\infty}-categorical colimits in {\mathcal{S}} are the same as model categorical colimits in simplicial sets. Now, it is a general fact from model category theory that the homotopy colimit of a bisimplicial set is the diagonal.

So we can think of all homotopy types as being built up as geometric realizations of discrete ones. I’ve been trying to understand what a simplicial object in an {\infty}-category “really” means, though, so let’s do some more examples. (more…)