Limits and Colimits of Presheaves

A presheaf on $\mathcal{A}$ is a functor ${\mathcal{A}}^{\mathrm{op}}\to \mathbf{Set}$, and presheaves on $\mathcal{A}$ form the category $[{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}]$. Limits and colimits in this category are computed pointwise, so it inherits completeness and cocompleteness from $\mathbf{Set}$. The Yoneda embedding preserves limits but not colimits, and although a colimit of representables need not be representable, every presheaf is a colimit of representables.¹

Hom-functors turn limits into limits.

Representables preserve limits

Fix a locally small category $\mathcal{A}$ and an object $A$. The hom-functor$\mathcal{A}(A,-):\mathcal{A}\to \mathbf{Set}$ sends an object $X$ to the set of maps $A\to X$. For a product, a map into $X\times Y$ is a pair of maps, so

naturally in every argument. The same pattern holds for equalizers: a map into $\mathrm{Eq}(s,t)$ is a map $f:A\to X$ with $sf=tf$, which is exactly an element of the equalizer of $\mathcal{A}(A,s)$ and $\mathcal{A}(A,t)$ in $\mathbf{Set}$. These are two instances of a single statement.²

The lemma is a reading of the explicit limit formula in $\mathbf{Set}$. An element of $\lim \mathcal{A}(A,D)$ is a compatible family $(f_I)$ with $f_I\in \mathcal{A}(A,D(I))$ and $\mathcal{A}(A,Du)(f_I)=f_J$ for every $u:I\to J$. That compatibility is $D(u)\circ f_I=f_J$, which is the cone condition. Combined with the representation of the cone functor, $\mathrm{Cone}(A,D)\cong \mathcal{A}(A,\lim D)$, the lemma gives the headline.

Dualizing replaces $\mathcal{A}$ by ${\mathcal{A}}^{\mathrm{op}}$: the contravariant hom-functor $\mathcal{A}(-,A):{\mathcal{A}}^{\mathrm{op}}\to \mathbf{Set}$ preserves limits, which means it turns colimits in $\mathcal{A}$ into limits in $\mathbf{Set}$,

The covariant hom-functor turns limits into limits and the contravariant one turns colimits into limits; neither produces a colimit. The duality is not symmetric.

Limits in functor categories

Now let $\mathcal{A}$ be small and $\mathcal{S}$ locally small, so $[\mathcal{A},\mathcal{S}]$ is again locally small. For each $A\in \mathcal{A}$ the evaluation functor ${\mathrm{ev}}_A:[\mathcal{A},\mathcal{S}]\to \mathcal{S}$ sends $X$ to $X(A)$. Given a diagram $D:\mathcal{I}\to [\mathcal{A},\mathcal{S}]$ of functors, evaluating at $A$ produces an ordinary diagram $D(-)(A):\mathcal{I}\to \mathcal{S}$. The claim is that the limit of $D$ is computed by taking the limit of each $D(-)(A)$ and reassembling.

In short, limits in a functor category are computed pointwise, the points being the objects of $\mathcal{A}$. The proof assembles the pointwise limits $L(A)=\lim D(-)(A)$ into a functor: a map $f:A\to A^{\prime }$ induces a map $L(f):L(A)\to L(A^{\prime })$ as the unique factorization from the induced-map lemma, the projections then form a cone $L\Rightarrow D(I)$ in $[\mathcal{A},\mathcal{S}]$, and universality is checked one object at a time.³ Colimits are computed pointwise by the dual argument.

Two consequences:

• Completeness is inherited. If $\mathcal{S}$ has all limits (respectively colimits) of shape $\mathcal{I}$, so does $[\mathcal{A},\mathcal{S}]$, and each evaluation functor ${\mathrm{ev}}_A$ preserves them.
• Presheaf categories are complete and cocomplete. Since $\mathbf{Set}$ has all small limits and colimits, $[{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}]$ has them too, computed pointwise, for any small $\mathcal{A}$.

A caution attaches to the first point. When $\mathcal{S}$ lacks some limits, the functor category can still contain a limit of that shape which no evaluation functor preserves; such limits are not pointwise. The theorem only builds pointwise limits from limits already present in $\mathcal{S}$.⁴

Limits commute with limits

Pointwise computation gives a clean proof that the order of taking limits does not matter. For shapes $\mathcal{I}$, $\mathcal{J}$ and a category $\mathcal{S}$ with all limits, a functor $D:\mathcal{I}\times \mathcal{J}\to \mathcal{S}$ can be curried either way, and its three possible limits agree.⁵

The proof runs each side through the representation isomorphism until both name a representing object for $[\mathcal{I}\times \mathcal{J},\mathcal{S}](\Delta -,D)$, then invokes uniqueness. By analogy with swapping the order of integration, the result is sometimes called a Fubini theorem. The analogy is only partial: colimits commute with colimits, but limits do not in general commute with colimits. Taking $\mathcal{S}=\mathbf{Set}$ with four one-element sets,

so a product of sums differs from the corresponding sum of products.

The Yoneda embedding and limits

The Yoneda embedding$H_{\bullet }:\mathcal{A}\to [{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}]$ sends $A$ to the representable $H_A=\mathcal{A}(-,A)$. Since representables preserve limits and limits of presheaves are pointwise, the embedding carries limits in $\mathcal{A}$ to limits in the presheaf category.

Concretely, if $\mathcal{A}$ has binary products then $H_{X\times Y}\cong H_X\times H_Y$, which evaluated at $A$ is the familiar $\mathcal{A}(A,X\times Y)\cong \mathcal{A}(A,X)\times \mathcal{A}(A,Y)$. Viewing $\mathcal{A}$ as sitting inside its presheaf category, the corollary says a limit of representables never leaves $\mathcal{A}$: it is again representable.

Colimits are the opposite story. The embedding does not preserve them. If $\mathcal{A}$ has an initial object $0$, then $H_0(0)=\mathcal{A}(0,0)$ is a one-element set, whereas the initial presheaf is constant at $\varnothing$. So $H_0$ is not initial: the embedding sends the initial object of $\mathcal{A}$ to a non-initial presheaf.⁶ The next section addresses this failure: every presheaf is nonetheless a colimit of representables.

Every presheaf is a colimit of representables

A canonical construction writes every presheaf as a colimit of representables; its indexing category is the category of elements.

The objects of $\mathcal{E}(X)$ are the elements of $X$ gathered across all objects of $\mathcal{A}$; by Yoneda they are just the generalized elements of $X$ of representable shape. Composing the projection with the Yoneda embedding gives a diagram of representables indexed by these elements.

The proof computes cocones. A cocone on $H_{\bullet }\circ P$ with vertex $Y$ is a family of natural transformations $H_A\to Y$, one per element $(A,x)$, compatible across maps of $\mathcal{E}(X)$. By the Yoneda lemma each transformation $H_A\to Y$ is an element of $Y(A)$, and the compatibility condition turns the family into a single natural transformation $X\to Y$. So cocones on $H_{\bullet }\circ P$ with vertex $Y$ correspond naturally to maps $X\to Y$, which is the universal property of the colimit.⁷ The name echoes topology: $\mathcal{A}$ is dense in its presheaf category because every presheaf is a colimit (a limit-of-points, in the topological sense) of objects of $\mathcal{A}$.

The density theorem is dual to the Yoneda lemma; the two are the same statement seen from opposite sides, one about maps out of representables and one about maps into them. Together they say that a presheaf category is generated by its representables under colimits, which makes $[{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}]$ the free cocompletion of $\mathcal{A}$.