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Stream: learning: questions

Topic: The colimit of a certain simplicial object

fosco (Sep 28 2020 at 10:30):

Let $\cal C$ be a small category; consider a sequence of objects

• $A_0$, an object of $$\cal C$;
• $f:A_1\to A_0$, an arrow of $\cal C$ (i.e., an object of ${\cal C}/A_0$;
• $A_2 \overset{h}\to A_1 \overset{f}\to A_0$, an object of the slice $({\cal C}/A_0)/f$
• ...

It is evident that this yields a sequence of $n$-simplices $x_n$ in the nerve of $\cal C$, one for each $n\ge 0$, in such a way that the last face of $x_n$ is $x_{n-1}$. Altogether, this piece of data allows to define inductively
$\begin{cases}{\cal C}_0 := \cal C \\{\cal C}_1 := {\cal C}/A_0 \\{\cal C}_n := {\cal C}_{n-1}/x_{n-1} \\ \end{cases}$
blurring the distinction between $x_n$ and a tuple of composable morphisms of $\cal C$.

What is the colimit of this sequence of categories (that, I think, glues to a simplicial object $\Delta^{op} \to {\sf Cat}$?

sarahzrf (Oct 01 2020 at 17:59):

$(\mathcal C / A_0) / f \simeq \mathcal C / A_1$