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.

1. Example: the diagonal of $\Delta^{op}$

Let’s start with an important example in practice: the (opposite to) the simplex category. This has the handy property that the diagonal is cofinal.

Proposition 6 The diagonal map

$\displaystyle \Delta^{op} \rightarrow \Delta^{op} \times \Delta^{op}$

is cofinal.

Simplicial sets with the property that the diagonal is cofinal are called sifted; these simplicial sets are a generalization of filtered categories. They have the following property. In spaces, filtered colimits commute with finite limits. In spaces, sifted colimits commute with finite products. I’ve heard that one can make this analogy a lot more precise, pairing types of colimits and the types of limits they commute with, but I don’t know the details.

Proof: The strategy, again, is to use Theorem A. We need to show that if ${m, n}$ are fixed, the category

$\displaystyle \Delta^{op} \times_{\Delta^{op} \times \Delta^{op}} (\Delta^{op} \times \Delta^{op})_{[m] \times [n] /}$

is contractible. This is the opposite to the category ${\mathcal{C}}$ of objects ${[p], p \in \mathbb{Z}_{ \geq 0}}$ equipped with maps ${[p] \rightarrow [m], [p] \rightarrow [n]}$. We’ll show that ${\mathcal{C}}$ is contractible, which will complete the proof.

There are various tricks for showing that a category like this is contractible.

One such trick is to note that an adjunction between categories is automatically a weak homotopy equivalence. In fact, if ${F, G: \mathcal{C} \rightleftarrows \mathcal{D}}$ is an adjunction, then we have unit and counit natural transformations

$\displaystyle 1 \rightarrow GF, \quad FG \rightarrow 1,$

which act as categorical homotopies: we find that ${F}$ and ${G}$ are inverse to each other.

So the strategy here is to use an adjunction to reduce the above category to a category whose nerve is more familiar. Namely, let’s consider the category ${\mathcal{D}}$ whose objects are monomorphisms of posets ${[p] \rightarrow [m] \times [n]}$. Then there is an adjunction ${\mathcal{C} \rightleftarrows \mathcal{D}}$ whose left adjoint sends a map ${f: [p] \rightarrow [m ] \times [n]}$ to the injective map ${\mathrm{im} f \hookrightarrow [m] \times [n]}$. The right adjoint is the forgetful functor.

It follows from this that we need only show that ${\mathcal{D}}$ is contractible. Here’s where the next trick comes in: one recognizes these categories as subdivisions of appropriate simplicial sets. In this case, (see below), the nerve of ${\mathcal{D}}$ is the subdivision of the poset ${[m] \times [n]}$, which means that its homotopy type is that of ${\Delta^m \times \Delta^n}$—i.e., it is contractible.

$\Box$

Let’s state a few facts about subdivision: if we can recognize the nerve of some category as a subdivision, then we can probably figure out its homotopy type. Given a poset ${\mathcal{P}}$, we define a category ${C_{\mathcal{P}}}$ whose objects are injections ${[n] \hookrightarrow \mathcal{P}}$ and whose morphisms are commutative triangles; then we set

$\displaystyle \mathrm{sd} \mathcal{P} = N (C_{\mathcal{P}}),$

that is, ${\mathrm{sd} \mathcal{P}}$ is the nerve of the category of injections ${[n] \hookrightarrow \mathcal{P}}$.

This is functorial in ${\mathcal{P}}$ in an appropriate sense: one takes the image. Observe that ${\mathrm{sd} \mathcal{P}}$ is the colimit of ${\mathrm{sd} [n]}$ over all maps ${[n] \rightarrow \mathcal{P}}$. This procedure enables one to define a simplicial set

$\displaystyle \mathrm{sd} X_\bullet, \quad X_\bullet \in \mathbf{sSet}.$

As Goerss-Jardine prove, ${\mathrm{sd}}$ preserves the homotopy type. So for instance, ${\mathrm{sd} (\Delta^n \times \Delta^m)}$, which we encountered previously, is a contractible homotopy type.

2. A lemma for Dold-Kan

In the course of proving the ${\infty}$-categorical version of the Dold-Kan correspondence (by this I mean the equivalence between filtered and simplicial objects in a stable ${\infty}$-category), we’ll need to be able to take certain colimits over ${\Delta_{\leq n}^{op}}$. As we saw above, the cofinality of the inclusion ${\Delta^{op}_{inj} \hookrightarrow \Delta^{op}}$ does not generalize to the inclusions ${\Delta^{op}_{inj , \leq n} \rightarrow \Delta^{op}_{\leq n}}$: we can’t compute the colimit of an ${n}$-truncated simplicial object by restricting to the semisimplicial object. But we almost can: the truncated semisimplicial object does determine the colimit, just in a slightly unusual way.

Proposition 7 (Lurie) The forgetful functor ${\Delta^{op}_{inj, [n]/} \rightarrow \Delta^{op}_{\leq n}}$ is cofinal.

Let’s unwind what this means. Here ${\Delta^{op}_{inj [n]/}}$ looks scary, but it’s actually something familiar. By definition, it is opposite to the nerve of the category of all injections ${[k] \hookrightarrow [n]}$. In other words, ${\Delta^{op}_{inj, [n]/}}$ is the nerve of the opposite of the poset of nonempty subsets of ${[n]}$. This has a forgetful functor to ${\Delta^{op}_{\leq n}}$.

Unfortunately, the argument that used to be here is wrong. A correct argument is in Lurie’s book; I was trying unsuccessfully to find one which made no reference to topological spaces.

In the next post, we’re going to apply this to derive the higher categorical version of the Dold-Kan correspondence.