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

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

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, looks like . This is not contractible: if we take 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, is contractible. We’ll see that this is true in general for any , but the 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 -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…)