Simplicial Sets - Core Definitions

Simplicial sets form the combinatorial foundation of higher category theory, providing a discrete model for studying homotopy-theoretic phenomena. They generalize the notion of simplicial complexes from topology while maintaining strong categorical properties that make them amenable to algebraic manipulation.

DefinitionThe Simplex Category

The simplex category $\Delta$ has objects the finite ordered sets $[n] = \{0, 1, \ldots, n\}$ for $n \geq 0$ , with morphisms being order-preserving maps. The basic generating morphisms are:

Face maps $d^i: [n-1] \to [n]$ for $0 \leq i \leq n$ , which skip the element $i$
Degeneracy maps $s^i: [n+1] \to [n]$ for $0 \leq i \leq n$ , which repeat the element $i$

These satisfy the simplicial identities: $d^j d^i = d^i d^{j-1} \text{ for } i < j$ $s^j s^i = s^i s^{j+1} \text{ for } i \leq j$ $d^j s^i = \begin{cases} s^{i-1} d^j & \text{if } i > j+1 \\ \text{id} & \text{if } i = j, j+1 \\ s^i d^{j-1} & \text{if } i < j \end{cases}$

DefinitionSimplicial Sets

A simplicial set is a functor $X: \Delta^{\text{op}} \to \mathbf{Set}$ . Equivalently, it consists of:

A sequence of sets $X_n$ for each $n \geq 0$ (the n-simplices)
Face maps $d_i: X_n \to X_{n-1}$ for $0 \leq i \leq n$
Degeneracy maps $s_i: X_n \to X_{n+1}$ for $0 \leq i \leq n$

satisfying the dual simplicial identities. We denote the category of simplicial sets by $\mathbf{sSet}$ .

ExampleThe Standard Simplex

For each $n \geq 0$ , the standard n-simplex $\Delta^n$ is the representable functor $\Delta^n = \text{Hom}_\Delta(-, [n])$ . By the Yoneda lemma, we have a natural bijection: $\text{Hom}_{\mathbf{sSet}}(\Delta^n, X) \cong X_n$

This means n-simplices in any simplicial set $X$ correspond exactly to maps from the standard simplex into $X$ .

RemarkGeometric Realization

The geometric realization functor $|-|: \mathbf{sSet} \to \mathbf{Top}$ provides a bridge to classical topology. For a simplicial set $X$ , we define: $|X| = \coprod_{n \geq 0} X_n \times |\Delta^n_{\text{top}}| / \sim$

where $\Delta^n_{\text{top}} = \{(t_0, \ldots, t_n) \in \mathbb{R}^{n+1}: \sum t_i = 1, t_i \geq 0\}$ is the topological standard simplex. This construction is left adjoint to the singular complex functor, establishing a Quillen equivalence between simplicial sets and topological spaces.

The power of simplicial sets lies in their ability to encode higher-dimensional coherence data combinatorially. Each n-simplex represents a potential n-dimensional cell, while face and degeneracy maps encode the boundary and collapse operations. This structure makes simplicial sets the natural setting for defining $\infty$ -categories as "simplicial sets with composition."