Definition 14.2.1. For any integer $n\geq 1$, and any $0\leq j \leq n$ we let $\delta ^ n_ j : [n-1] \to [n]$ denote the injective order preserving map skipping $j$. For any integer $n\geq 0$, and any $0\leq j \leq n$ we denote $\sigma ^ n_ j : [n + 1] \to [n]$ the surjective order preserving map with $(\sigma ^ n_ j)^{-1}(\{ j\} ) = \{ j, j + 1\} $.
14.2 The category of finite ordered sets
The category $\Delta $ is the category with
objects $[0], [1], [2], \ldots $ with $[n] = \{ 0, 1, 2, \ldots , n\} $ and
a morphism $[n] \to [m]$ is a nondecreasing map $\{ 0, 1, 2, \ldots , n\} \to \{ 0, 1, 2, \ldots , m\} $ between the corresponding sets.
Here nondecreasing for a map $\varphi : [n] \to [m]$ means by definition that $\varphi (i) \geq \varphi (j)$ if $i \geq j$. In other words, $\Delta $ is a category equivalent to the “big” category of nonempty finite totally ordered sets and nondecreasing maps. There are exactly $n + 1$ morphisms $[0] \to [n]$ and there is exactly $1$ morphism $[n] \to [0]$. There are exactly $(n + 1)(n + 2)/2$ morphisms $[1] \to [n]$ and there are exactly $n + 2$ morphisms $[n] \to [1]$. And so on and so forth.
Lemma 14.2.2. Any morphism in $\Delta $ can be written as a composition of the morphisms $\delta ^ n_ j$ and $\sigma ^ n_ j$.
Proof. Let $\varphi : [n] \to [m]$ be a morphism of $\Delta $. If $j \not\in \mathop{\mathrm{Im}}(\varphi )$, then we can write $\varphi $ as $\delta ^ m_ j \circ \psi $ for some morphism $\psi : [n] \to [m - 1]$. If $\varphi (j) = \varphi (j + 1)$ then we can write $\varphi $ as $\psi \circ \sigma ^{n - 1}_ j$ for some morphism $\psi : [n - 1] \to [m]$. The result follows because each replacement as above lowers $n + m$ and hence at some point $\varphi $ is both injective and surjective, hence an identity morphism. $\square$
Lemma 14.2.3. The morphisms $\delta ^ n_ j$ and $\sigma ^ n_ j$ satisfy the following relations.
If $0 \leq i < j \leq n + 1$, then $\delta ^{n + 1}_ j \circ \delta ^ n_ i = \delta ^{n + 1}_ i \circ \delta ^ n_{j - 1}$. In other words the diagram
\[ \xymatrix{ & [n] \ar[rd]^{\delta ^{n + 1}_ j} & \\ [n - 1] \ar[ru]^{\delta ^ n_ i} \ar[rd]_{\delta ^ n_{j - 1}} & & [n + 1] \\ & [n] \ar[ru]_{\delta ^{n + 1}_ i} & } \]commutes.
If $0 \leq i < j \leq n - 1$, then $\sigma ^{n - 1}_ j \circ \delta ^ n_ i = \delta ^{n - 1}_ i \circ \sigma ^{n - 2}_{j - 1}$. In other words the diagram
\[ \xymatrix{ & [n] \ar[rd]^{\sigma ^{n - 1}_ j} & \\ [n - 1] \ar[ru]^{\delta ^ n_ i} \ar[rd]_{\sigma ^{n - 2}_{j - 1}} & & [n - 1] \\ & [n - 2] \ar[ru]_{\delta ^{n - 1}_ i} & } \]commutes.
If $0 \leq j \leq n - 1$, then $\sigma ^{n - 1}_ j \circ \delta ^ n_ j = \text{id}_{[n - 1]}$ and $\sigma ^{n - 1}_ j \circ \delta ^ n_{j + 1} = \text{id}_{[n - 1]}$. In other words the diagram
\[ \xymatrix{ & [n] \ar[rd]^{\sigma ^{n - 1}_ j} & \\ [n - 1] \ar[ru]^{\delta ^ n_ j} \ar[rd]_{\delta ^ n_{j + 1}} \ar[rr]^{\text{id}_{[n - 1]}} & & [n - 1] \\ & [n] \ar[ru]_{\sigma ^{n - 1}_ j} & } \]commutes.
If $0 < j + 1 < i \leq n$, then $\sigma ^{n - 1}_ j \circ \delta ^ n_ i = \delta ^{n - 1}_{i - 1} \circ \sigma ^{n - 2}_ j$. In other words the diagram
\[ \xymatrix{ & [n] \ar[rd]^{\sigma ^{n - 1}_ j} & \\ [n - 1] \ar[ru]^{\delta ^ n_ i} \ar[rd]_{\sigma ^{n - 2}_ j} & & [n - 1] \\ & [n - 2] \ar[ru]_{\delta ^{n - 1}_{i - 1}} & } \]commutes.
If $0 \leq i \leq j \leq n - 1$, then $\sigma ^{n - 1}_ j \circ \sigma ^ n_ i = \sigma ^{n - 1}_ i \circ \sigma ^ n_{j + 1}$. In other words the diagram
\[ \xymatrix{ & [n] \ar[rd]^{\sigma ^{n - 1}_ j} & \\ [n + 1] \ar[ru]^{\sigma ^ n_ i} \ar[rd]_{\sigma ^ n_{j + 1}} & & [n - 1] \\ & [n] \ar[ru]_{\sigma ^{n - 1}_ i} & } \]commutes.
Proof. Omitted. $\square$
Lemma 14.2.4. The category $\Delta $ is the universal category with objects $[n]$, $n \geq 0$ and morphisms $\delta ^ n_ j$ and $\sigma ^ n_ j$ such that (a) every morphism is a composition of these morphisms, (b) the relations listed in Lemma 14.2.3 are satisfied, and (c) any relation among the morphisms is a consequence of those relations.
Proof. Omitted. $\square$
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