# Contents

## Idea

The simplicial identities encode the relationships between the face and degeneracy maps in a simplicial object, in particular, in a simplicial set.

## Definition

The simplicial identities are the duals to the simplicial relations of coface and codegeneracy maps described at simplex category:

Let $S\in$ sSet with

• face maps ${\partial }_{i}:{S}_{n}\to {S}_{n-1}$ obtained by omitting the $i$th vertex;

• degeneracy maps ${s}_{i}:{S}_{n}\to {S}_{n+1}$ obtained by repeating the $i$th vertex.

###### Definition

The simplicial identities satisfied by face and degeneracy maps as above are (whenever these maps are composable as indicated):

1. ${\partial }_{i}\circ {\partial }_{j}={\partial }_{j-1}\circ {\partial }_{i}$ if $i,

2. ${s}_{i}\circ {s}_{j}={s}_{j}\circ {s}_{i-1}$ if $i>j$.

3. ${\partial }_{i}\circ {s}_{j}=\left\{\begin{array}{cc}{s}_{j-1}\circ {\partial }_{i}& \mathrm{if}\phantom{\rule{thickmathspace}{0ex}}ij+1\end{array}$

## Properties

### Relation to nilpotency of differentials

The simplicial identities of def. 1 can be understood as a non-abelian or “unstable” generalization of the identity

$\partial \circ \partial =0$\partial \circ \partial = 0

satisfied by differentials in chain complexes (in homological algebra).

Write $ℤ\left[S\right]$ be the simplicial abelian group obtained form $S$ by forming degreewise the free abelian group on the set of $n$-simplices, as discussed at chains on a simplicial set.

Then using these formal linear combinations we can sum up all the $\left(n+1\right)$ face maps ${\partial }_{i}:{S}_{n}\to {S}_{n-1}$ into a single map:

###### Definition

The alternating face map differential in degree $n$ of the simplicial set $S$ is the linear map

$\partial :ℤ\left[{S}_{n}\right]\to ℤ\left[{S}_{n-1}\right]$\partial : \mathbb{Z}[S_n] \to \mathbb{Z}[S_{n-1}]

defined on basis elements $\sigma \in {S}_{n}$ to be the alternating sum of the simplicial face maps:

(1)$\partial \sigma ≔\sum _{k=0}^{n}\left(-1{\right)}^{k}{\partial }_{k}\sigma \phantom{\rule{thinmathspace}{0ex}}.$\partial \sigma \coloneqq \sum_{k = 0}^n (-1)^k \partial_k \sigma \,.

This is the differential of the alternating face map complex of $S$:

###### Proposition

The simplicial identity def. 1 (1) implies that def. 2 indeed defines a differential in that $\partial \circ \partial =0$.

###### Proof

By linearity, it is sufficient to check this on a basis element $\sigma \in {S}_{n}$. There we compute as follows:

$\begin{array}{rl}\partial \partial \sigma & =\partial \left(\sum _{j=0}^{n}\left(-1{\right)}^{j}{\partial }_{j}\sigma \right)\\ & =\sum _{j=0}^{n}\sum _{i=0}^{n-1}\left(-1{\right)}^{i+j}{\partial }_{i}{\partial }_{j}\sigma \\ & =\sum _{0\le i\begin{aligned} \partial \partial \sigma & = \partial \left( \sum_{j = 0}^n (-1)^j \partial_j \sigma \right) \\ & = \sum_{j=0}^n \sum_{i = 0}^{n-1} (-1)^{i+j} \partial_i \partial_j \sigma \\ & = \sum_{0 \leq i \lt j \leq n} (-1)^{i+j} \partial_i \partial_j \sigma + \sum_{0 \leq j \leq i \lt n} (-1)^{i + j} \partial_i \partial_j \sigma \\ & = \sum_{0 \leq i \lt j \leq n} (-1)^{i+j} \partial_{j-1} \partial_i \sigma + \sum_{0 \leq j \leq i \lt n} (-1)^{i + j} \partial_i \partial_j \sigma \\ & = - \sum_{0 \leq i \leq j \lt n} (-1)^{i+j} \partial_{j} \partial_i \sigma + \sum_{0 \leq j \leq i \lt n} (-1)^{i + j} \partial_i \partial_j \sigma \\ & = 0 \end{aligned} \,.

Here

1. the first equality is (1);

2. the second is (1) together with the linearity of $d$;

3. the third is obtained by decomposing the sum into two summands;

4. the fourth finally uses the simplicial identity def. 1 (1) in the first summand;

5. the fifth relabels the summation index $j$ by $j+1$;

6. the last one observes that the resulting two summands are negatives of each other.

## References

For instance definition 1.1 in

Revised on November 7, 2012 14:25:00 by Urs Schreiber (82.169.65.155)