Simplicial Sets

sage.categories.simplicial_sets.SimplicialSets

The category of simplicial sets.

A simplicial set \(X\) is a collection of sets \(X_i\), indexed by the non-negative integers, together with maps

\[\begin{split}d_i: X_n \to X_{n-1}, \quad 0 \leq i \leq n \quad \text{(face maps)} \\ s_j: X_n \to X_{n+1}, \quad 0 \leq j \leq n \quad \text{(degeneracy maps)}\end{split}\]

satisfying the simplicial identities:

\[\begin{split}d_i d_j &= d_{j-1} d_i \quad \text{if } i<j \\ d_i s_j &= s_{j-1} d_i \quad \text{if } i<j \\ d_j s_j &= 1 = d_{j+1} s_j \\ d_i s_j &= s_{j} d_{i-1} \quad \text{if } i>j+1 \\ s_i s_j &= s_{j+1} s_{i} \quad \text{if } i \leq j\end{split}\]

Morphisms are sequences of maps \(f_i : X_i \to Y_i\) which commute with the face and degeneracy maps.

EXAMPLES:

sage: from sage.categories.simplicial_sets import SimplicialSets
sage: C = SimplicialSets(); C
Category of simplicial sets