nLab
category of simplices

Definition

The category of simplices of a simplicial set $X_{•}:{\Delta }^{\mathrm{op}}\to \mathrm{Set}$ is the category of elements of the presheaf $X_{•}$.

Remarks

We spell this out in detail:

The category of simplices is the comma category $(Y,X_{•})$, where $Y$ is the Yoneda embedding, hence the functor $Y:[n]\to {\Delta }^n$

The objects of the category of simplices are therefore morphisms $c:{\Delta }^n\to X_{•}$ from the standard simplicial simplex $\Delta$ to $X_{•}$, hence are $n$-cells of $X_{•}$, while morphisms $c\to c\prime$ are morphisms $f:{\Delta }^n\to {\Delta }^{n\prime }$ in the simplex category $\Delta$ such that

$$\begin{array}{ccccc}{\Delta }^n& & \stackrel{f}{\to }& & {\Delta }^{n\prime }\\ & {}_c\searrow & & {\swarrow }_{c\prime }\\ & & X_{•}\end{array}.$$\array{ \Delta^n &&\stackrel{f}{\to}&& \Delta^{n'} \\ & {}_{c}\searrow && \swarrow_{c'} \\ && X_\bullet } \,.