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 theĀ 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.)

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