Colimits of presheaves and the density theorem

The goal of this short note is to show how to compute the limit or colimit of a diagram of presheaves, and prove that every presheaf can be written as the colimit of representable presheaves, a result known as the density theorem.

Limits and colimits of presheaves

Let $I$ and $\mathscr{D}$ be categories. Recall that a limit cone of a diagram $F \in [I, \mathscr{D}]$ is the same thing as a universal morphism from the diagonal functor

$$\Delta \colon \mathscr{D} \to [I, \mathscr{D}]; \quad \quad Y \overset{\Delta}{\longmapsto} (i \mapsto Y)$$

to $F$. If a limit exists for each diagram, then they may be assembled into a right adjoint $\lim_I^{\mathscr{D}} \colon [I, \mathscr{D}] \to \mathscr{D}$ to $\Delta$. This functor takes a diagram to the apex of its limiting cone in a functorial way. The rest of the cone can be recovered from the counit $\epsilon$ of the adjunction $\Delta \dashv \lim_I$: for each diagram $F$, its limiting cone is $(\lim_I F, \epsilon_F)$. We call such a functor a limit functor, which is unique up to natural isomorphism if it exists.

To compute the limit or colimit of a diagram of presheaves, we make use the fact that a diagram of presheaves is equivalent to a presheaf of diagrams, and then compute limits objectwise. This equivalence is a consequence of general fact related to currying, which we state without proof.

Using this isomorphism, limits and colimits of diagrams of presheaves may be computed as follows.

Proof. It suffices to show that ${\lim}_I^{[\mathscr{C}^\mathrm{op}, \mathbf{Set}]}$ is the right adjoint to the diagonal functor $\Delta' \colon [\mathscr{C}^\mathrm{op}, \mathbf{Set}] \to [I, [\mathscr{C}^\mathrm{op}, \mathbf{Set}]]$. Since $\Delta' = \Delta_* \circ \mathrm{swap}^{-1}$, an adjunction exists if and only if $\Delta_* \dashv (\lim_I^{\mathbf{Set}})_*$. Now, there exists a natural isomorphism

$$\Theta \colon \mathrm{Hom}_{[I, \mathbf{Set}]} (\Delta [-], -) \overset{\cong}{\longrightarrow} \mathrm{Hom}_{\mathbf{Set}} (-, {\lim}_I^{\mathbf{Set}}[-])$$

via the adjunction $\Delta \dashv \lim_I^{\mathbf{Set}}$, which may be used to construct isomorphisms

$$\widehat{\Theta}_{\mathcal{G}, \widetilde{\mathcal{F}}} \colon \mathrm{Hom}_{[\mathscr{C}^{\mathrm{op}}, [I, \mathbf{Set}]]} (\Delta_*\mathcal{G}, \widetilde{\mathcal{F}}) \longrightarrow \mathrm{Hom}_{[\mathscr{C}^{\mathrm{op}}, \mathbf{Set}]} (\mathcal{G}, ({\lim}_I^{\mathbf{Set}})_*\widetilde{\mathcal{F}}); \quad \quad \widehat{\Theta}_{\mathcal{G}, \widetilde{\mathcal{F}}}(\phi)_X = \Theta_{\mathcal{G}(X), \widetilde{\mathcal{F}}(X)}(\phi_X).$$

This map is well-defined: if $\phi \in \mathrm{Hom}_{[\mathscr{C}^\mathrm{op}, [I, \mathbf{Set}]]}(\Delta_* \mathcal{G}, \widetilde{\mathcal{F}})$ and $\psi \colon X \to Y$ is a morphism in $\mathscr{C}$, then the diagram

commutes, so by naturality of $\Theta$, we have

$$\widehat{\Theta}_{\mathcal{G}, \widetilde{\mathcal{F}}}(\phi)_X \circ \mathcal{G} (\psi) = \Theta_{\mathcal{G}(Y), \widetilde{\mathcal{F}}(X)}(\phi_X \circ [\Delta_* \mathcal{G}](\psi)) = \Theta_{\mathcal{G}(Y), \widetilde{\mathcal{F}}(X)}(\widetilde{\mathcal{F}}(\psi) \circ \phi_Y) = \widetilde{\mathcal{F}}(\psi) \circ \widehat{\Theta}_{\mathcal{G}, \widetilde{\mathcal{F}}}(\phi)_Y.$$

Furthermore, these isomorphisms assemble into a natural isomorphism. Indeed, given $f \colon \widetilde{\mathcal{F}} \to \widetilde{\mathcal{F}}'$, we have

$$\widehat{\Theta}_{\mathcal{G}, \widetilde{\mathcal{F}}'}(f_* \phi)_X = \widehat{\Theta}_{\mathcal{G}, \widetilde{\mathcal{F}}'}(f \circ \phi)_X = \Theta_{\mathcal{G}(X), \widetilde{\mathcal{F}}'(X)}(f_X \circ \phi_X) = (f_X)_* \big[\Theta_{\mathcal{G}(X), \widetilde{\mathcal{F}}(X)}(\phi_X)\big] = (f_X)_*\big[\widehat{\Theta}_{\mathcal{G}, \widetilde{\mathcal{F}}}(\phi)_X\big],$$

and given $g \colon \mathcal{G}' \to \mathcal{G}$, we have

$$\widehat{\Theta}_{\mathcal{G}', \widetilde{\mathcal{F}}}(g^* \phi)_X = \widehat{\Theta}_{\mathcal{G}', \widetilde{\mathcal{F}}}(\phi \circ \Delta_* g)_X = \Theta_{\mathcal{G}'(X), \widetilde{\mathcal{F}}(X)}(\phi_X \circ \Delta g_X) = (g_X)^*\big[\Theta_{\mathcal{G}(X), \widetilde{\mathcal{F}}(X)}(\phi_X)\big] = (g_X)^* \big[\widehat{\Theta}_{\mathcal{G}, \widetilde{\mathcal{F}}}(\phi)_X\big].$$

It follows that there is an adjunction $\Delta' \dashv \lim_I^{[\mathscr{C}^\mathrm{op}, \mathbf{Set}]}$.

The exact same argument may be carried out for colimits. As such, we state the analogous dual result without proof.

Proof. Use the adjunction ${\mathrm{colim}}_{I}^{\mathbf{Set}} \dashv \Delta$ to show that $({\mathrm{colim}}_{I}^{\mathbf{Set}})_* \dashv \Delta_*$.

The density theorem

Let $\mathscr{C}$ be a category and let $\mathcal{F} \colon \mathscr{C}^\mathrm{op} \to \mathbf{Set}$ be a presheaf of sets on $\mathscr{C}$. The Yoneda lemma implies that there exists an isomorphism of presheaves

$$\mathrm{Hom}_{[\mathscr{C}^{\mathrm{op}}, \mathbf{Set}]}(\!\text{よ}[-], \mathcal{F}) \cong \mathcal{F},$$

where $\hspace{-1pt}\text{よ} \colon \mathscr{C} \to [\mathscr{C}^{\mathrm{op}}, \mathbf{Set}]$ is the Yoneda embedding. In particular, morphisms from $\hspace{-1pt}\text{よ}$ to $\mathcal{F}$ correspond to sections of $\mathcal{F}$. As such, we call the comma category $(\!\text{よ} \downarrow \mathcal{F})$ the category of elements of $\mathcal{F}$, which we also denote by $\int_{\mathscr{C}} \mathcal{F}$. The category of elements comes equipped with a canonical projection

$$\pi \colon \int_{\mathscr{C}} \mathcal{F} \to \mathscr{C}; \quad \quad (X,\, \phi \colon \hspace{-3pt}\text{よ}(X) \to \mathcal{F}) \overset{\pi}{\longmapsto} X$$

which may be interpreted as taking a section down to the object it lies over. Roughly speaking, the density theorem states that every presheaf $\mathcal{F}$ can be recovered as a colimit over $\int_{\mathscr{C}} \mathcal{F}$ of presheaves represented by these projections.

Proof. For each $(X, \phi) \in \int_{\mathscr{C}} \mathcal{F}$, let $\rho_{X, \phi} = \phi$; we show that $(\mathcal{F}, \rho)$ is a universal cocone under $\!\text{よ} \circ \pi$. By the Yoneda lemma, there exists a natural isomorphism of profunctors

$$\Theta \colon \mathrm{Hom}_{[\mathscr{C}^{\mathrm{op}}, \mathbf{Set}]}(\!\text{よ}[-], -) \cong \langle -, - \rangle,$$

where $\langle -, - \rangle \colon \mathscr{C}^{\mathrm{op}} \hspace{-1pt} \times \hspace{-1pt} [\mathscr{C}^\mathrm{op}, \mathbf{Set}] \to \mathbf{Set}$ is the canonical pairing. A cocone $(\mathcal{G}, \sigma)$ under $\!\text{よ} \circ \pi$ defines, for $X\hspace{-1pt} \in \hspace{-1pt} \mathscr{C}$, a map

$$\sigma_X \colon \mathrm{Hom}_{[\mathscr{C}^{\mathrm{op}}, \mathbf{Set}]}(\!\text{よ}(X), \mathcal{F}) \to \mathrm{Hom}_{[\mathscr{C}^{\mathrm{op}}, \mathbf{Set}]}(\!\text{よ}(X), \mathcal{G}); \quad \quad \sigma_X(\phi) = \sigma_{X, \phi}.$$

By definition, a presheaf morphism $\psi \colon \hspace{-0.4pt} \mathcal{F} \to \mathcal{G}$ which induces a morphism of cocones must satisfy $\mathrm{Hom}(\!\text{よ}(X), \hspace{-0.2pt}\psi) \hspace{-0.2pt} = \hspace{-0.2pt} \sigma_X$. Since $\Theta$ is natural, the diagram

commutes, which determines each component of $\psi$, since $\psi_X$ is just the pushforward $\langle X, \psi \rangle$. It follows that

$$\begin{equation*} \psi_X = \Theta_{X, \mathcal{G}}^{-1} \circ \sigma_X \circ \Theta_{X, \mathcal{F}}. \tag{$\star$} \end{equation*}$$

It remains to show that such a presheaf morphism $\psi$ exists. Note that a morphism $f \colon Y \to X$ in $\mathscr{C}$ induces a morphism $f \colon (Y, \phi \circ \!\text{よ}(f)) \to (X, \phi)$ in the comma category, which implies

$$\sigma_{Y}(\phi \circ \!\text{よ}(f)) = \sigma_{X}(\phi) \circ \text{よ}(f)$$

as $(\mathcal{G}, \sigma)$ is a cocone under $\!\text{よ} \circ \pi$. It follows that $\sigma$ is natural when viewed as a map $\mathrm{Hom}(\!\text{よ}[-], \mathcal{F}) \to \mathrm{Hom}(\!\text{よ}[-], \mathcal{G})$. Furthermore, $\Theta$ is already natural in the first argument, so the formula $(\star)$ indeed defines a morphism of presheaves.