useful.

Let $p>0$, and consider the increasing maps $e_i : [1] \to [p]$ given by $e_i(0) = i$ and $e_i(1) = i+1$. For any simplicial set $A$ considered as a functor $A : \mathbf{\Delta}^{op} \to Sets$, we can evaluate $A$ on these $e_i$ and, noting that $e_i(1) = e_{i+1}(0)$, we get a family of functions $A_p \to A_1$, which yield a cone diagram, for instance, for $p =3$: $$\xymatrix{A_p \ar[drrr]^{A(e_1)}\ar[ddrr]^{A(e_2)}\ar[dddr]_{A(e_3)}&&&\\
&&&A_1\ar[d]^{d_0}\\
&&A_1\ar[r]^{d_1}\ar[d]^{d_0}&A_0\\
&A_1\ar[r]_{d_1}&A_0&&}$$ and in general, thus yield a map $$\delta[p]: A_p \to A_1\times_{A_0}A_1  \times_{A_0}\ldots \times_{A_0}A_1.$$ The maps, $\delta[p]$, have been called the *Segal maps* and will recur throughout the rest of these notes.

 \
If $A = Ner(\mathbb{C})$ for some small category $\mathbb{C}$, then for $A$, the Segal maps are bijections.

**Proof**

A simplex $\sigma \in Ner(\mathbb{C})_p$ corresponds uniquely to a composable $p$-chain of arrows in $\mathbb{C}$, and hence exactly to its image under the relevant Segal map. $\blacksquare$

Better than this is true:

\[GrotSegal\]  \
If $A$ is a simplicial set such that the Segal maps are bijections then there is a category structure on the directed graph $$\xymatrix{A_1 \ar[r]<1ex> \ar[r]&A_0\ar[l]<1ex>}.$$making it a category whose nerve is isomorphic to the given $A$.

**Proof**

To get composition you use 