Simplicial Set - Face and Degeneracy Maps

Face and Degeneracy Maps

In Δop, there are two particularly important classes of maps called face maps and degeneracy maps which capture the underlying combinatorial structure of simplicial sets.

The face maps di : n → n − 1 are given by

di (0 → … → n) = (0 → … → i − 1 → i + 1 → … → n).

The degeneracy maps si : n → n + 1 are given by

si (0 → … → n) = (0 → … → i − 1 → i → i → i + 1 → … → n).

By definition, these maps satisfy the following simplicial identities:

  1. di dj = dj−1 di if i < j
  2. di sj = sj−1 di if i < j
  3. dj sj = id = dj+1 sj
  4. di sj = sj di−1 if i > j + 1
  5. si sj = sj+1 si if i ≤ j.

The simplicial category Δ has as its morphisms the monotonic non-decreasing functions. Since the morphisms are generated by those that 'skip' or 'add' a single element, the detailed relations written out above underlie the topological applications. It can be shown that these relations suffice.

Read more about this topic:  Simplicial Set

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