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.)

A strange contractible space

Here’s a strange phenomenon that I ran into when writing a MathOverflow answer a few years ago.

Lemma. Let X be a set endowed with the cofinite topology, and assume X is path connected. Then X is contractible.

The assumption is for example satisfied when |X| \geq |\mathbf R|, for then any injection f \colon [0,1] \hookrightarrow X is a path from x_0 = f(0) to x_1 = f(1). Path connectedness of cofinite spaces is related to partitioning the interval into disjoint closed subsets; see the remark below for some bounds on the cardinalities.

Proof. The result is trivial if X is finite, for then both are equivalent to |X| = 1. Thus we may assume that X is infinite. Choose a path f \colon [0,1] \to X from some x_0 = f(0) to some x_1 = f(1) \neq x_0. This induces a continuous map [0,1] \times X \to X \times X. Choose a bijection

    \[g \colon (X \setminus \{x_0,x_1\}) \times X \stackrel \sim\to X,\]

and extend to a map \bar g \colon X \times X \to X by g(x_0,x) = x and g(x_1,x) = x_1 for all x \in X. Then \bar g is continuous: the preimage of x \in X is g^{-1}(x) \cup (x_0,x) if x \neq x_1, and g^{-1}(x) \cup (x_0,x) \cup x \times X if x = x_1, both of which are closed. Thus \bar g \circ (f \times \mathbf 1_X) is a homotopy from \mathbf 1_X to the constant map x_1, hence a contraction. \qedsymbol

I would love to see an animation of this contraction as t goes from 0 to 1… I find especially the slightly more direct argument for |X| \geq |\mathbf R| given here elusive yet somehow strangely visual.

Remark. If X is countable (still with the cofinite topology), then X is path connected if and only if |X| = 1. In the finite case this is clear (because then X is discrete), and in the infinite case this is a result of Sierpiński. See for example this MO answer of Timothy Gowers for an easy argument.

There’s also some study of path connectedness of cofinite topological spaces of cardinality strictly between \aleph_0 = |\mathbf N| and \mathfrak c = |\mathbf R|, if such cardinalities exist. See this MO question for some results. In particular, it is consistent with ZFC that the smallest cardinality for which X is path connected is strictly smaller than \mathfrak c.