Simplicial sets

A few weeks ago, I finally struck up the courage to take some baby steps reading Lurie’s Higher topos theory. In a series of posts mostly written for my own benefit, I will untangle some of the basic definitions and provide some easy examples. The first one is one I was already somewhat familiar with: simplicial sets.

Definition. For each $n \in \mathbf N$ , write $[n]$ for the poset $0 \leq \ldots \leq n$ . The full subcategory of $\mathbf{Poset}$ on these $[n]$ is denoted $\Delta$ , the simplex category. Concretely, it has objects $[n]$ for all $n \in \mathbf N$ , and morphisms

$\operatorname{Hom}\big([m],[n]\big) = \left\{f \colon [m] \to [n]\ \Big|\ i \leq j \Rightarrow f(i) \leq f(j)\right\}.$

A simplicial set is a functor $X \colon \Delta^{\operatorname{op}} \to \mathbf{Set}$ . This can be described rather concretely using the objects $X_n = X([n])$ and the face and degeneracy maps between them; see e.g. Tag 0169. The category of simplicial sets is usually denoted $[\Delta^{\operatorname{op}}, \mathbf{Set}]$ , $\mathbf{sSet}$ , or $\mathbf{Set}_{\Delta}$ (in analogy with cosimplicial sets $\mathbf{Set}^\Delta = [\Delta,\mathbf{Set}]$ ).

The representable simplicial set $\operatorname{Hom}(-,[n])$ is usually denoted $\Delta^n$ or $\Delta[n]$ . Then the Yoneda lemma shows that the functor $\mathbf{sSet} \to \mathbf{Set}$ given by $X \mapsto X_n$ is represented by $\Delta^n$ , i.e.

$X_n = \operatorname{Hom}_{\mathbf{sSet}}\big(\Delta^n,X).$

Definition. The geometric realisation functor $| \cdot | \colon \mathbf{sSet} \to \mathbf{Top}$ is defined as follows: for $\Delta^n$ , the geometric realisation $|\Delta^n|$ is the standard $n$ -simplex

$|\Delta^n| := \left\{(x_0,\ldots,x_n) \in \mathbf R^{n+1}\ \Bigg|\ x_i \geq 0, \sum_{i=0}^n x_i = 1\right\} \subseteq \mathbf R^{n+1}.$

(If no confusion arises, it may also be denoted $\Delta^n$ .) This is functorial in $[n]$ : for a map $a \colon [m] \to [n]$ (equivalently, by the Yoneda lemma, a map $a \colon \Delta^m \to \Delta^n$ ) we get a continuous map $a \colon |\Delta^m| \to |\Delta^n|$ by

$a(x_0,\ldots,x_m)_j = \sum_{i \in a^{-1}(j)} x_i.$

For an arbitrary simplicial set $X$ , write

$|X| := \underset{\Delta^n \to X}{\operatorname{colim}}\ |\Delta^n|,$

where the transition map $|\Delta^m| \to |\Delta^n|$ corresponding to a map $\Delta^m \to \Delta^n$ over $X$ is defined via

$\operatorname{Hom}_{\mathbf{sSet}}\big(\Delta^m,\Delta^n\big) \stackrel\sim\leftarrow \operatorname{Hom}_\Delta\big([m],[n]\big) \to \operatorname{Hom}_{\mathbf{Top}}\big(|\Delta^m|,|\Delta^n|\big).$

This is functorial in $X$ , and when $X = \Delta^n$ it coindices with the previous definition because the identity $\Delta^n \to \Delta^n$ is terminal in the index category.

Remark. In a fancier language, $| \cdot |$ is the left Kan extension of the functor $[n] \mapsto |\Delta^n|$ along the Yoneda embedding $\Delta \to \mathbf{sSet}$ . (Those of you familiar with presheaves on spaces will recognise the similarity with the definition of $f^{-1}\mathscr F$ for $f \colon X \to Y$ a continuous map of topological spaces, which is another example of a left Kan extension.)

Remark. It is a formal consequence of the definitions that geometric realisation preserves arbitrary colimits (“colimits commute with colimits”). This also follows because it is a left adjoint to the singular set functor, but we won’t explore this here.

Wisdom. The most geometric way to think about a simplicial set is through its geometric realisation.

For example, we can define the $i^{\text{th}}$ horn $\Lambda_i^n$ in $\Delta^n$ as the union of the images of the maps $\Delta^{n-1} \to \Delta^n$ coming from the face maps $\delta^j_n \colon [n-1] \to [n]$ for $j \neq i$ . Since geometric realisation preserves colimits (alternatively, stare at the definitions), we see that the geometric realisation of $\Lambda^i_n$ is obtained in the same way from the maps $|\Delta^{n-1}| \to |\Delta^n|$ , so it is the $n$ -simplex with its interior and face opposite the $i^{\text{th}}$ vertex removed.

The geometric realisation is a good first approximation for thinking about a simplicial set. However, when thinking about $\infty$ -categories (e.g. in the next few posts), this is actually not the way you want to think about a simplicial set. Indeed, homotopy of simplicial sets (equivalently their geometric realisations) is stronger than equivalence of $\infty$ -categories. (More details later, hopefully.)

Lovely little lemmas

Or maybe ‘amusing observations’

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