Simplicial Set

Definition

A simplicial set is a contravariant functor $X:{\Delta }^{op}\to \mathbf{Set}$ from the opposite of the simplex category $\Delta$ to the category of sets. The simplex category $\Delta$ has as objects the finite ordered sets $[n]=\{0,1,\dots ,n\}$ for $n\ge 0$, and as morphisms the order-preserving functions.

For each $n\ge 0$, we denote $X_n=X([n])$ as the set of $n$-simplices of $X$. The functor structure gives us:

• Face maps: $d_i:X_n\to X_{n-1}$ for $0\le i\le n$, induced by the injections ${\delta }^i:[n-1]\to [n]$ that skip $i$
• Degeneracy maps: $s_i:X_n\to X_{n+1}$ for $0\le i\le n$, induced by the surjections ${\sigma }^i:[n+1]\to [n]$ that repeat $i$

Simplicial Identities

The face and degeneracy maps satisfy the simplicial identities:

Face-face relations: $d_i{d}_j=d_{j-1}{d}_i\text{if }i<j$

Degeneracy-degeneracy relations: $s_i{s}_j=s_{j+1}{s}_i\text{if }i\le j$

Face-degeneracy relations:

s_{j-1} d_i & \text{if } i < j \\ \text{id} & \text{if } i = j \text{ or } i = j+1 \\ s_j d_{i-1} & \text{if } i > j+1 \end{cases}$$ ## Geometric Realization Every simplicial set $X$ has a *geometric realization* $|X|$, which is a topological space obtained by gluing together standard geometric simplices according to the combinatorial structure of $X$: $$|X| = \bigcup_{n \geq 0} X_n \times \Delta^n / \sim$$ where $\Delta^n$ is the standard $n$-simplex and $\sim$ identifies faces and degeneracies appropriately. ## Examples **Standard $n$-simplex:** $\Delta[n]$ is the simplicial set represented by $[n]$, with $\Delta[n]_k = \text{Hom}_\Delta([k], [n])$. **Singular simplicial set:** For any topological space $Y$, the singular simplicial set $S(Y)$ has $S(Y)_n = \text{Hom}(\Delta^n, Y)$ (continuous maps from the standard $n$-simplex to $Y$). **Nerve of a category:** For a small category $\mathcal{C}$, the nerve $N(\mathcal{C})$ has $N(\mathcal{C})_n$ equal to the set of chains of $n$ composable morphisms in $\mathcal{C}$. ## Kan Complexes A simplicial set $X$ is a *Kan complex* if every horn $\Lambda^n_k \to X$ (for $0 \leq k \leq n$) extends to a simplex $\Delta[n] \to X$. Kan complexes model homotopy types and are the fibrant objects in the standard model structure on simplicial sets. ## Historical Context [[Eilenberg]] and [[Zilber]] introduced semi-simplicial complexes (now called [[Simplicial Set|simplicial sets]]) to compute [[singular homology]]. They defined these objects as functors $X: \Delta^{op} \to \mathbf{Set}$, where $\Delta$ is the [[FinOrd (Category)|category of finite ordinals]]. This established that spaces could be modeled as [[Presheaf|presheaves]] on a [[combinatorial category]]. The [[Eilenberg-Zilber Lemma]] (relating the chain complex of a product to the tensor product of chains) was pivotal in establishing the product structure of these models. This work foreshadowed the interpretation of types as "spaces" or "homotopy types" in modern [[Cubical Type Theory]] and [[Simplicial Type Theory]]. ## Related Concepts - [[Simplicial Complex]] - [[Kan Complex]] - [[Homotopy Type Theory]] - [[Model Category]]