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.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (5)

Comment #253 by Keenan Kidwell on

Comment #254 by Keenan Kidwell on

Comment #258 by Johan on

Comment #3284 by Kevin Buzzard on

Comment #3371 by Johan on