Let’s do some more examples of cofinality. In the previous post, I erroneously claimed that the map

$\displaystyle \Delta^{op}_{inj \leq n} \rightarrow \Delta^{op}_{\leq n}$

was cofinal: that is, taking a colimit of an ${n}$-truncated simplicial object in an ${\infty}$-category was the same as taking the colimit of the associated ${n}$-truncated semisimplicial object. (The claim has since been deleted.) This is false, even when ${n = 1}$. In fact, the map of categories

$\displaystyle \Delta^{op}_{inj \leq 1} \rightarrow \Delta^{op}_{\leq 1}$

is not even a weak homotopy equivalence. (While this is not obvious, one of the statements of “Theorem A” is that a cofinal map is automatically a homotopy equivalence.)

In fact, ${\Delta^{op}_{inj \leq 1}}$ looks like ${\bullet \rightrightarrows \bullet}$. This is not contractible: if we take ${\pi_1}$ of the nerve, that’s the same as taking the category itself and inverting all the morphisms. So that gives us a free groupoid on two morphisms. However, ${\Delta^{op}_{\leq 1}}$ is contractible. We’ll see that this is true in general for any ${\Delta^{op}_{\leq n}}$, but the $\Delta^{op}_{inj, \leq n}$ only become “asymptotically” contractible.

The purpose of this post is to work through a few examples of Theorem A, discussed in the previous post. This will show that the colimit of an $n$-truncated simplicial object can in fact be recovered from the semisimplicial restriction, but in a somewhat more subtle way than one might expect. We will need this lemma in the discussion of the Dold-Kan correspondence. (more…)

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…)

The following result is useful in algebraic K-theory.

Theorem 1 Let ${F: \mathcal{C} \rightarrow \mathcal{D}}$ be a functor between categories. Suppose ${\mathcal{C}/d}$ is contractible for each ${d \in \mathcal{D}}$. Then ${F: N\mathcal{C} \rightarrow N \mathcal{D}}$ is a weak homotopy equivalence.

I don’t really know enough to give a good justification for the usefulness, but in essence, what Quillen did in the 1970s was to show that the Grothendieck group of an “exact category” could be interpreted homotopically as the fundamental group of the nerve of the “Q-category” built from the exact category. As a result, Quillen was able to define higher K-groups as the higher homotopy groups of this space. He then proved a lot of results that were proved by ad hoc, homological means for the Grothendieck group of a category for the higher K-groups as well, by interpreting them in terms of homotopy theory. This result (together with the extension, “Theorem B”) is a key homotopical tool he used to analyze these nerves.

Here ${N \mathcal{C}}$ denotes the nerve of the category ${\mathcal{C}}$: it is the simplicial set whose ${n}$-simplices consist of composable strings of ${n+1}$ morphisms of ${\mathcal{C}}$. The overcategory ${\mathcal{C}/d}$ has objects consisting of pairs ${(c, f)}$ for ${c \in \mathcal{C}}$, ${f: Fc \rightarrow d}$ a morphism in ${\mathcal{D}}$; morphisms in ${\mathcal{C}/d}$ are morphisms in ${\mathcal{C}}$ making the natural diagram commute. We say that a category is contractible if its nerve is weakly contractible as a simplicial set.

There are other reasons to care. For instance, in higher category theory, the above condition on contractibility of over-categories is the analog of cofinality in ordinary category theory. Anyway, this result is pretty important.

But what I want to explain in this post is that “Theorem A” (and Theorem B, but I’ll defer that) is really purely formal. That is, it can be deduced from some standard and not-too-difficult manipulations with model categories (which weren’t all around when Quillen wrote “Higher algebraic K-theory I”).

To prove this, we shall obtain the following expression for a category:

$\displaystyle N \mathcal{C} = \mathrm{colim}_d N (\mathcal{C}/d),$

where ${d}$ ranges over the objects of ${\mathcal{D}}$. This expresses the nerve of ${\mathcal{C}}$ as a colimit of simplicial sets arising as the nerves of ${\mathcal{C}/d}$. We will compare this with a similar expression for the nerve of $\mathcal{D}$, that is ${N \mathcal{D} = \mathrm{colim}_d N(\mathcal{D}/d)}$. Then, the point will be that ${N(\mathcal{C}/d) \rightarrow N(\mathcal{D}/d)}$ is a weak equivalence for each ${d}$; this by itself does not imply that the induced map on colimits is a weak equivalence, but it will in this case because both the colimits will in fact turn out to be homotopy colimits. I’ll start by explaining what those are.