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…)