Nerve, unknown subscript, Categories, Segal condition

Question

What does it mean the index $(NC)_0$ in $$\sigma_m:(NC)_m\to(NC)_1\times_{(NC)_0}...\times_{(NC)_0}(NC)_1?$$

This equation is from page xiii of Carlos Simpson's book "Homotopy Theory of Higher Categories : From Segal Categories to $n$-Categories and beyond".

Answer 1

It denotes a (generalized) pullback. Basically, if $C$ is a category, then $(NC)_0$ is the set of objects of $C$, $(NC)_1$ is the set of arrows, $d_0:(NC)_1\to (NC)_0$ (resp. $d_1$) is the function that assigns its domain (resp. codomain) to each arrow. Then the right-hand term is the limit of the diagram $$\require{AMScd} \begin{CD} & & & & & & (NC)_1 \\ & & & & & @V{d_1}VV \\& & & & (NC)_1@>{d_0}>> (NC)_0 \\ & & & & @V{d_1}VV \\ & & (NC)_1@>{d_0}>> (NC)_0 \\ & &\vdots \\ (NC)_1@>{d_0}>> (NC)_0\end{CD}$$with $m$ copies of $(NC)_1$. It is the set of $m$-tuple of arrows $(f_1,f_2,\dots,f_m)$ such that the domain of $f_i$ is the codomain of $f_{i+1}$ for all $0<i<m$.

So for example for $m=2$ you have the pullback $$\begin{CD}(NC)_1\times_{(NC)_0}(NC)_1 @>>> (NC)_1\\ @VVV @VV{d_1}V\\ (NC)_1@>>{d_0}> (NC)_0\end{CD}$$ which is the set of pairs of arrows $(f,g)$ such that the composition $f\circ g$ can be defined.

For more information, you can check out the nLab pages of the nerve and of the Segal condition.

Answer 2

To write an object under a product usually indicates a fibre product (aka pullback) or a generalisation thereof, and here it is done multiple times with $(NC)_0$, the $0$-simplices of the simplicial object $NC$.