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…)