Simplicial sets

Last week, I started a series on sieves and Grothendieck topoi. This is a short intermezzo on a well-known lemma from category theory that I will need for next week’s instalment.

Let $\mathscr C$ be a small category. Recall that a presheaf on $\mathscr C$ is a functor $\mathscr C^{\operatorname{op}} \to \mathbf{Set}$ . Examples include the representable presheaves $h_X$ for $X \in \mathscr C$ , given by $\operatorname{Hom}(-,X)$ . The Yoneda lemma says that for any presheaf $F \colon \mathscr C^{\operatorname{op}} \to \mathbf{Set}$ and any $X \in \mathscr C$ , the map

$\begin{align*}\operatorname{Nat}(h_X,F) &\to F(X) \\\alpha &\mapsto \alpha_X(\operatorname{id}_X)\end{align*}$

is an isomorphism. Applying this to $F = h_Y$ shows that the Yoneda embedding

$\begin{align*}h \colon \mathscr C &\to [\mathscr C^{\operatorname{op}},\mathbf{Set}] \\X &\mapsto h_X\end{align*}$

is fully faithful.

Given a functor $F \colon \mathscr C^{\operatorname{op}} \to \mathbf{Set}$ , write $(h \downarrow F)$ for the comma category whose objects are pairs $(X,\alpha)$ where $X \in \mathscr C$ and $\alpha \colon h_X \to F$ is a morphism (natural transformation) in $[\mathscr C^{\operatorname{op}},\mathbf{Set}]$ . A morphism $f \colon (X,\alpha) \to (Y,\beta)$ is a morphism $f \colon X \to Y$ such that the triangle

(1) $\begin{array}{ccccc}\!\!h_X\!\!\!\!\! & & \!\!\!\!\!\stackrel{h_f}\longrightarrow\!\!\!\!\! & & \!\!\!\!\!h_Y\!\! \\ & \!\!\!\!{\underset{\alpha}{}}\!\!\searrow\!\!\!\! & & \!\!\!\!\swarrow\!\!{\underset{\beta}{}}\!\!\!\! & \\[-.3em] & & F\! & & \end{array}$

of natural transformations commutes, where $h_f \colon \operatorname{Hom}(-,X) \to \operatorname{Hom}(-,Y)$ denotes postcomposition by $f$ . Note that $(h \downarrow F)$ is again a small category, and there is a forgetful functor $U \colon (h \downarrow F) \to \mathscr C$ taking $(X,\alpha)$ to $X \in \mathscr C$ .

By the Yoneda lemma, the category $(h \downarrow F)$ is isomorphic (not just equivalent!) to the category $\int F$ of pairs $(X,s)$ with $X \in \mathscr C$ and $s \in F(X)$ with morphisms $f \colon (X,s) \to (Y,t)$ given by morphisms $f \colon X \to Y$ in $\mathscr C$ such that $F(f)(t) = s$ . It’s convenient to keep both points of view.

Lemma. Let $\mathscr C$ be a small category, and let $F \colon \mathscr C^{\operatorname{op}} \to \mathbf{Set}$ be a functor.

The object $F \in [\mathscr C^{\operatorname{op}},\mathbf{Set}]$ is naturally a cocone under $hU \colon (h \downarrow F) \to [\mathscr C^{\operatorname{op}},\mathbf{Set}]$ via the morphisms $\alpha \colon h_X \to F$ .
This cocone makes $F$ the colimit of the diagram $hU$ of representable functors.

In particular, any presheaf on a small category is a colimit of representable presheaves.

Proof. A cocone under $hU$ is a presheaf $G$ with a natural transformation $\phi \colon hU \to G$ to the constant diagram $(h \downarrow F) \to [\mathscr C^{\operatorname{op}},\mathbf{Set}]$ with value $G$ . This means every $(X,\alpha) \in (h \downarrow F)$ is taken to a natural transformation $\phi_{(X,\alpha)} \colon h_X \to G$ , such that for any morphism $f \colon (X,\alpha) \to (Y,\beta)$ , the square

$\begin{array}{ccc} h_X & \stackrel{\phi_{(X,\alpha)}}\longrightarrow & G \\ \!\!\!\!\!\!\!\!\!{h_f}\downarrow & & |\!| \\ h_Y & \underset{\phi_{(Y,\beta)}}\longrightarrow & G\end{array}$

commutes. By the Yoneda lemma, such a datum corresponds to an association of elements $\phi_{(X,s)} \in G(X)$ for all $(X,s) \in \int F$ such that for every $f \colon (X,s) \to (Y,t)$ with $F(f)(t) = s$ , we have $G(f)(\phi_{(Y,t)}) = \phi_{(X,s)}$ .

(1) To make $F$ a cocone under $hU$ , simply take $\phi_{(X,s)} = s \in F(X)$ .

(2) Given any other cocone $\phi \colon hU \to G$ under $hU$ , define the natural transformation $\eta \colon F \to G$ by

$\begin{align*}\eta_X \colon F(X) &\to G(X) \\s &\mapsto \phi_{(X,s)}.\end{align*}$

Naturality follows since $F(f)(t) = s$ implies $G(f)(\phi_{(Y,t)}) = \phi_{(X,s)}$ . It is clear that $\eta$ is the unique natural transformation of cocones $F \to G$ under $hU$ , showing that $F$ is the colimit. $\qedsymbol$

One can also easily rewrite this argument in terms of natural transformations $h_X \to G$ . For instance, the universal cocone $hU \to F$ is the natural transformation $\phi \colon hU \to F$ of functors $(h \downarrow F) \to [\mathscr C^{\operatorname{op}},\mathbf{Set}]$ given on $(X,\alpha) \in (h \downarrow F)$ by $\alpha \colon h_X \to F$ . Naturality of $\phi$ follows at once from (1). But checking that this thing is universal is a bit more tedious in this language.

Example. A standard example where this point of view is useful is simplicial sets. Let $\Delta$ be the category of finite nonempty totally ordered sets, with (weakly) monotone increasing functions as morphisms. A simplicial set is a functor $\Delta^{\operatorname{op}} \to \mathbf{Set}$ , and we often think of them as combinatorial models for topological spaces. The representable ones are the standard $n$ -simplices $\Delta^n = \operatorname{Hom}(-,[n])$ , where $[n]$ is the totally ordered set $\{0,\ldots,n\}$ for $n \in \mathbf Z_{\geq 0}$ .

If $X \colon \Delta^{\operatorname{op}} \to \mathbf{Set}$ is a simplicial set, its value at $[n]$ is called the $n$ -simplices $X_n$ of $X$ . By the Yoneda lemma, this is $\operatorname{Hom}(\Delta^n,X)$ . Then the story above is saying that a simplicial set is the colimit over all its $n$ -simplices for $n \in \mathbf Z_{\geq 0}$ . This is extremely useful, as many arguments proceed by attaching simplices one at a time.

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’

Category Archives: Simplicial sets

Presheaves are colimits of representables (topologies 2/6)