The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

14.11 Simplicial sets

Let $U$ be a simplicial set. It is a good idea to think of $U_0$ as the $0$-simplices, the set $U_1$ as the $1$-simplices, the set $U_2$ as the $2$-simplices, and so on.

We think of the maps $s^ n_ j : U_ n \to U_{n + 1}$ as the map that associates to an $n$-simplex $A$ the degenerate $(n + 1)$-simplex $B$ whose $(j, j + 1)$-edge is collapsed to the vertex $j$ of $A$. We think of the map $d^ n_ j : U_ n \to U_{n - 1}$ as the map that associates to an $n$-simplex $A$ one of the faces, namely the face that omits the vertex $j$. In this way it become possible to visualize the relations among the maps $s^ n_ j$ and $d^ n_ j$ geometrically.

Definition 14.11.1. Let $U$ be a simplicial set. We say $x$ is an $n$-simplex of $U$ to signify that $x$ is an element of $U_ n$. We say that $y$ is the $j$the face of $x$ to signify that $d^ n_ jx = y$. We say that $z$ is the $j$th degeneracy of $x$ if $z = s^ n_ jx$. A simplex is called degenerate if it is the degeneracy of another simplex.

Here are a few fundamental examples.

Example 14.11.2. For every $n \geq 0$ we denote $\Delta [n]$ the simplicial set

\[ \Delta ^{opp} \longrightarrow \textit{Sets},\quad [k] \longmapsto \mathop{Mor}\nolimits _{\Delta }([k], [n]) \]

We leave it to the reader to verify the following statements. Every $m$-simplex of $\Delta [n]$ with $m > n$ is degenerate. There is a unique nondegenerate $n$-simplex of $\Delta [n]$, namely $\text{id}_{[n]}$.

Lemma 14.11.3. Let $U$ be a simplicial set. Let $n \geq 0$ be an integer. There is a canonical bijection

\[ \mathop{Mor}\nolimits (\Delta [n], U) \longrightarrow U_ n \]

which maps a morphism $\varphi $ to the value of $\varphi $ on the unique nondegenerate $n$-simplex of $\Delta [n]$.

Proof. Omitted. $\square$

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 $U$.

Lemma 14.11.5. Let $U$, $V$ be simplicial sets. Let $a, b \geq 0$ be integers. Assume every $n$-simplex of $U$ is degenerate if $n > a$. Assume every $n$-simplex of $V$ is degenerate if $n > b$. Then every $n$-simplex of $U \times V$ is degenerate if $n > a + b$.

Proof. Suppose $n > a + b$. Let $(u, v) \in (U \times V)_ n = U_ n \times V_ n$. By assumption, there exists a $\alpha : [n] \to [a]$ and a $u' \in U_ a$ and a $\beta : [n] \to [b]$ and a $v' \in V_ b$ such that $u = U(\alpha )(u')$ and $v = V(\beta )(v')$. Because $n > a + b$, there exists an $0 \leq i \leq a + b$ such that $\alpha (i) = \alpha (i + 1)$ and $\beta (i) = \beta (i + 1)$. It follows immediately that $(u, v)$ is in the image of $s^{n - 1}_ i$. $\square$


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