Let ${\mathcal{C}}$ be a stable ${\infty}$-category. For us, this means that we have three important properties:

1. ${\mathcal{C} }$ admits finite limits and colimits.
2. ${\mathcal{C}}$ has a zero object: that is, the initial object is also final.
3. A square in ${\mathcal{C}}$ is a pull-back if and only if it is a push-out.

This is equivalent to the stability of ${\mathcal{C}}$. Actually, stability is usually defined using slightly weaker conditions, and then it takes a little work to show that these stronger ones are implied. We’ll just work with these. Stability can be thought of as a higher-categorical version of being triangulated; a general stable $\infty$-category has many of the properties (in a higher categorical sense) of the homotopy category of spectra, or the (classical) derived category.

Our goal is to show that in this case, we have an equivalence of ${\infty}$-categories

$\displaystyle \mathrm{Fun}(\Delta^{op}, \mathcal{C}) \simeq \mathrm{Fun}(\mathbb{Z}_{\geq 0}, \mathcal{C})$

between simplicial objects in ${\mathcal{C}}$ and filtered (nonnegatively) objects in ${\mathcal{C}}$. The idea here is that the geometric realization of a simplicial object comes with a canonical filtration, given by geometric realizing the ${n}$-truncations for each ${n}$. This is going to give the associated filtered object. (We don’t know that the geometric realization exists, but the realizations of the truncations will.)

We will actually prove something stronger: for each ${n}$, there is an equivalence

$\displaystyle \mathrm{Fun}(\Delta^{op}_{\leq n}, \mathcal{C}) \simeq \mathrm{Fun}( [0, n], \mathcal{C}), \ \ \ \ \ (1)$

where ${[0, n] \subset \mathbb{Z}_{\geq 0}}$ is the subcategory of elements ${\leq n}$. In other words, ${n}$-truncated simplicial objects are the same as ${n}$-filtered objects of ${\mathcal{C}}$. (Note that, as a simplicial set, the nerve of ${[0, n]}$ is ${\Delta^n}$.) These equivalences will be compatible, and taking inverse limits will give the Dold-Kan correspondence. (more…)

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…)