The Stacks project

Example 14.11.4. Consider the category $\Delta /[n]$ of objects over $[n]$ in $\Delta $, see Categories, Example 4.2.13. There is a functor $p : \Delta /[n] \to \Delta $. The fibre category of $p$ over $[k]$, see Categories, Section 4.34, has as objects the set $\Delta [n]_ k$ of $k$-simplices in $\Delta [n]$, and as morphisms only identities. For every morphism $\varphi : [k] \to [l]$ of $\Delta $, and every object $\psi : [l] \to [n]$ in the fibre category over $[l]$ there is a unique object over $[k]$ with a morphism covering $\varphi $, namely $\psi \circ \varphi : [k] \to [n]$. Thus $\Delta /[n]$ is fibred in sets over $\Delta $. In other words, we may think of $\Delta /[n]$ as a presheaf of sets over $\Delta $. See also, Categories, Example 4.37.7. And this presheaf of sets agrees with the simplicial set $\Delta [n]$. In particular, from Equation (14.4.0.1) and Lemma 14.11.3 above we get the formula

\[ \mathop{Mor}\nolimits _{\textit{PSh}(\Delta )}(\Delta /[n], U) = U_ n \]

for any simplicial set $U$.


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