Let be an -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 -category of spectra. A *simplicial object* of is a functor

that is, it is a morphism of simplicial sets from the nerve of the opposite of the simplex category to . 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 be a simplicial set, thus defining a homotopy type and thus an object of the -category of spaces. Alternatively, can be regarded as a simplicial object in sets, so a simplicial object in (discrete) spaces. In other words, has two incarnations:

- .
- .

The connection is that is the geometric realization (in the -category of spaces) of the simplicial object . More generally, whenever one has a bisimplicial set , defining an object of , then the geometric realization of in is the diagonal simplicial set . These are model categorical observations: one chooses a presentation for (e.g., the usual Kan model structure on simplicial sets), and then uses the fact that -categorical colimits in 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 -category “really” means, though, so let’s do some more examples. (more…)