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 d_i : n → n − 1 are given by

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

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

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

By definition, these maps satisfy the following simplicial identities:

1. d_i d_j = d_j−1 d_i if i < j
2. d_i s_j = s_j−1 d_i if i < j
3. d_j s_j = id = d_j+1 s_j
4. d_i s_j = s_j d_i−1 if i > j + 1
5. s_i s_j = s_j+1 s_i 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.