simplicial identities



Homotopy theory

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Basic facts




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


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

Let SS \in sSet with

  • face maps i:S nS n1\partial_i : S_n \to S_{n-1} obtained by omitting the iith vertex;

  • degeneracy maps s i:S nS n+1s_i : S_n \to S_{n+1} obtained by repeating the iith vertex.


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

  1. i j= j1 i \partial_i \circ \partial_j = \partial_{j-1} \circ \partial_i if i<ji \lt j,

  2. s is j=s js i1s_i \circ s_j = s_j \circ s_{i-1} if i>ji \gt j.

  3. is j={s j1 i ifi<j id ifi=jori=j+1 s j i1 ifi>j+1\partial_i \circ s_j = \left\{ \array{ s_{j-1} \circ \partial_i & if \; i \lt j \\ id & if \; i = j \; or \; i = j+1 \\ s_j \circ \partial_{i-1} & if\; i \gt j+1 } \right.


Relation to nilpotency of differentials

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

=0 \partial \circ \partial = 0

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

Write [S]\mathbb{Z}[S] be the simplicial abelian group obtained form SS by forming degreewise the free abelian group on the set of nn-simplices, as discussed at chains on a simplicial set.

Then using these formal linear combinations we can sum up all the (n+1)(n+1) face maps i:S nS n1\partial_i : S_n \to S_{n-1} into a single map:


The alternating face map differential in degree nn of the simplicial set SS is the linear map

:[S n][S n1] \partial : \mathbb{Z}[S_n] \to \mathbb{Z}[S_{n-1}]

defined on basis elements σS n\sigma \in S_n to be the alternating sum of the simplicial face maps:

(1)σ k=0 n(1) k kσ. \partial \sigma \coloneqq \sum_{k = 0}^n (-1)^k \partial_k \sigma \,.

This is the differential of the alternating face map complex of SS:


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


By linearity, it is sufficient to check this on a basis element σS n\sigma \in S_n. There we compute as follows:

σ =( j=0 n(1) j jσ) = j=0 n i=0 n1(1) i+j i jσ = 0i<jn(1) i+j i jσ+ 0ji<n(1) i+j i jσ = 0i<jn(1) i+j j1 iσ+ 0ji<n(1) i+j i jσ = 0ij<n(1) i+j j iσ+ 0ji<n(1) i+j i jσ =0. \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} \,.


  1. the first equality is (1);

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

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

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

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

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


For instance definition 1.1 in

Last revised on July 12, 2019 at 19:24:28. See the history of this page for a list of all contributions to it.