Simplicial group

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QUICK PHRASES: simplicial object in category of groups, group object in category of simplicial sets

Definition

Abstract definition

A simplicial group can be defined in the following equivalent ways:

  1. Explicitly, it is a contravariant functor from the simplex category to the category of groups.
  2. It is a simplicial object in the category of groups.
  3. It is a group object in the category of simplicial sets.

Explicit definition

A simplicial group G is the following data:

  1. For every nonnegative integer, a group G_n
  2. For every positive integer n and all i with 0 \le i \le n, a group homomorphism \partial_i:G_n \to G_{n-1} called a face map
  3. For every nonnegative integer n and all i with 0 \le i \le n, a group homomorphism \sigma_i:G_n \to G_{n+1} called a degeneracy map

(Note that there are actually multiple maps \partial_i and \sigma_i corresponding to different values of n \ge i. However, on any given G_n, there is only one map, so we can typically avoid specifying the n-value).

with the following conditions:

  1. \partial_i \circ \partial_j = \partial_{j-1} \circ \partial_i\ \forall \ i < j for each n where either side makes sense
  2. \sigma_i \circ \sigma_j = \sigma_{j+1}\circ \sigma_i \ \forall \ i \le j for each n where either side makes sense
  3. \partial_i \circ \sigma_j = \sigma_{j-1} \circ \partial_i \ \forall \ i < j for each n where either side makes sense
  4. \partial_i \circ \sigma_j = \operatorname{id} \ \forall \ i = j \mbox{ or } i = j + 1 for each n where either side makes sense
  5. \partial_i \circ \sigma_j = \sigma_j \partial_{i-1} \ \forall \ i > j + 1 for each n where either side makes sense

Note that all these conditions follow from the relations originally present in the simplex category. They can be constructed systematically using the trapezoid rule.

Related notions