Presheaves are colimits of representables (topologies 2/6)

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.

  1. 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.
  2. 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.

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