Commuting Limits & Presheaf Completeness

Proof

Throughout the proof, the product category enters through one computation. For maps \(f : I \to I'\) in \(\mathbf{I}\) and \(g : J \to J'\) in \(\mathbf{J}\), composition in \(\mathbf{I} \times \mathbf{J}\) is componentwise, so \[ (f, g) = (f, 1_{J'}) \circ (1_I, g) = (1_{I'}, g) \circ (f, 1_J). \] We call this the interchange identity.

Applying the functor \(D\) to it gives \[ D(f, g) = D(f, 1_{J'}) \circ D(1_I, g) = D(1_{I'}, g) \circ D(f, 1_J). \]

The functor \(\widehat{D}\). Fix \(I \in \mathbf{I}\). The assignment \(D(I, -)\) sends \(J\) to \(D(I, J)\) and \(g : J \to J'\) to \(D(1_I, g)\). It preserves identities because \(D(1_I, 1_J) = D(1_{(I,J)}) = 1_{D(I,J)}\), and it preserves composition because \((1_I, g') \circ (1_I, g) = (1_I, g' \circ g)\), so applying \(D\) gives \(D(1_I, g') \circ D(1_I, g) = D(1_I, g' \circ g)\). Hence \(\widehat{D}(I) = D(I, -)\) is a functor \(\mathbf{J} \to \mathscr{S}\).

For a map \(f : I \to I'\), define \(\widehat{D}(f) : D(I, -) \to D(I', -)\) by the components \(\widehat{D}(f)_J = D(f, 1_J)\). Naturality is the interchange identity: for \(g : J \to J'\), \[ \begin{align*} D(I', -)(g) \circ \widehat{D}(f)_J &= D(1_{I'}, g) \circ D(f, 1_J) \\\\ &= D(f, g) \\\\ &= D(f, 1_{J'}) \circ D(1_I, g) \\\\ &= \widehat{D}(f)_{J'} \circ D(I, -)(g). \end{align*} \]

Functoriality of \(\widehat{D}\) in \(f\) follows from the same computation in the first coordinate: \((f', 1) \circ (f, 1) = (f' \circ f, 1)\) gives \(\widehat{D}(f')\circ\widehat{D}(f) = \widehat{D}(f' \circ f)\) componentwise, and \(\widehat{D}(1_I) = 1_{\widehat{D}(I)}\) since every component is \(D(1_I, 1_J)\).

The assignment on natural transformations. Let \(\theta : D \to D'\) be a natural transformation. For fixed \(I\), the family \((\theta_{(I, J)})_{J \in \mathbf{J}}\) is natural in \(J\), being the naturality of \(\theta\) at the maps \((1_I, g)\), so it is a map \(\widehat{\theta}_I : \widehat{D}(I) \to \widehat{D'}(I)\) in \([\mathbf{J}, \mathscr{S}]\). The family \((\widehat{\theta}_I)_I\) is natural in \(I\): the required identity \(\widehat{D'}(f) \circ \widehat{\theta}_I = \widehat{\theta}_{I'} \circ \widehat{D}(f)\) reads, at the component \(J\), \(D'(f, 1_J) \circ \theta_{(I, J)} = \theta_{(I', J)} \circ D(f, 1_J)\), which is the naturality of \(\theta\) at \((f, 1_J)\). Since composition and identities of natural transformations are computed componentwise, \(\theta \mapsto \widehat{\theta}\) preserves both, and \(D \mapsto \widehat{D}\), \(\theta \mapsto \widehat{\theta}\) is a functor \([\mathbf{I} \times \mathbf{J}, \mathscr{S}] \to [\mathbf{I}, [\mathbf{J}, \mathscr{S}]]\).

The inverse. Given \(E : \mathbf{I} \to [\mathbf{J}, \mathscr{S}]\), define \(\check{E} : \mathbf{I} \times \mathbf{J} \to \mathscr{S}\) on objects by \(\check{E}(I, J) = E(I)(J)\) and on a map \((f, g) : (I, J) \to (I', J')\) by \[ \check{E}(f, g) = E(f)_{J'} \circ E(I)(g) = E(I')(g) \circ E(f)_J, \] the two composites agreeing by the naturality of \(E(f)\) at \(g\). Identities are preserved: \(E(1_I) = 1_{E(I)}\) by functoriality of \(E\) and \(E(I)(1_J) = 1\) by functoriality of \(E(I)\), so \(\check{E}(1_I, 1_J)\) is a composite of identities. For composition, take \((f', g') : (I', J') \to (I'', J'')\) and compute \[ \begin{align*} \check{E}(f', g') \circ \check{E}(f, g) &= E(f')_{J''} \circ E(I')(g') \circ E(f)_{J'} \circ E(I)(g) \\\\ &= E(f')_{J''} \circ E(f)_{J''} \circ E(I)(g') \circ E(I)(g) \\\\ &= E(f' \circ f)_{J''} \circ E(I)(g' \circ g) = \check{E}\big((f', g') \circ (f, g)\big), \end{align*} \] where the second equality moves \(E(f)_{J'}\) past \(E(I')(g')\) by the naturality of \(E(f)\) at \(g'\), and the third uses functoriality of \(E\) and of \(E(I)\).

On morphisms, take a map \(\alpha : E \to E'\) in \([\mathbf{I}, [\mathbf{J}, \mathscr{S}]]\) and define \(\check{\alpha}\) by \(\check{\alpha}_{(I, J)} = (\alpha_I)_J\). Its naturality at \((f, g)\) follows by composing the naturality of \(\alpha\) at \(f\), read componentwise, with the naturality of \(\alpha_I\) at \(g\). Functoriality of \(E \mapsto \check{E}\) on maps is again componentwise.

Mutually inverse. On objects and maps of either category the two constructions unwind each other by inspection. In one direction, \(\check{\widehat{D}}(I, J) = \widehat{D}(I)(J) = D(I, J)\) and \[ \check{\widehat{D}}(f, g) = \widehat{D}(f)_{J'} \circ \widehat{D}(I)(g) = D(f, 1_{J'}) \circ D(1_I, g) = D(f, g), \] while on natural transformations \(\check{\widehat{\theta}}_{(I, J)} = (\widehat{\theta}_I)_J = \theta_{(I, J)}\). In the other direction, \(\widehat{\check{E}}(I)(J) = \check{E}(I, J) = E(I)(J)\), while \(\widehat{\check{E}}(f)_J = \check{E}(f, 1_J) = E(f)_J\) and \(\widehat{\check{E}}(I)(g) = \check{E}(1_I, g) = E(I)(g)\) by the identity laws for \(E\) and \(E(I)\), so \(\widehat{\check{E}} = E\), and \(\widehat{\check{\alpha}} = \alpha\) componentwise. Both composites equal the identity functors, which proves (a).

Diagonals. The constant functor \(\Delta_{\mathbf{I} \times \mathbf{J}}\, S\) takes the value \(S\) on every object and \(1_S\) on every map. Hence \(\widehat{\Delta_{\mathbf{I} \times \mathbf{J}}\, S}(I)\) is the functor \(\mathbf{J} \to \mathscr{S}\) constant at \(S\), which is \(\Delta_{\mathbf{J}}\, S\), and \(\widehat{\Delta_{\mathbf{I} \times \mathbf{J}}\, S}(f)\) has every component equal to \(1_S\), which is the identity transformation, exactly the value of \(\Delta_{\mathbf{I}}(\Delta_{\mathbf{J}}\, S)\) on \(f\). On a map \(h : S \to S'\), every component of \(\widehat{\Delta_{\mathbf{I} \times \mathbf{J}}\, h}\) at every stage is \(h\), and the same is true of \(\Delta_{\mathbf{I}}(\Delta_{\mathbf{J}}\, h)\). This proves (b).

The mirrored isomorphism. The two coordinates of \(\mathbf{I} \times \mathbf{J}\) play symmetric roles in every step above: the product category entered each verification only through the interchange identity and componentwise composition, both unchanged when the coordinates are exchanged, and every remaining ingredient was the functoriality or naturality of the given data, which makes no reference to the coordinates. Running the same proof with the roles of \(\mathbf{I}\) and \(\mathbf{J}\) exchanged establishes (c).

Proof

We prove the second isomorphism. The objects and maps of \(\mathbf{I} \times \mathbf{J}\) are pairs, so the product of two small categories is small, and every category and diagram in sight satisfies the smallness hypotheses of the results we invoke.

First, the inner limit exists. Since \(\mathscr{S}\) has all limits of shape \(\mathbf{J}\), the functor category \([\mathbf{J}, \mathscr{S}]\) has all limits of shape \(\mathbf{I}\) as soon as \(\mathscr{S}\) does, and \(\mathscr{S}\) has them by hypothesis. So \(\lim_{\mathbf{I}} \widehat{D}\) exists as an object of \([\mathbf{J}, \mathscr{S}]\), and the outer limit \(\lim_{\mathbf{J}} \big( \lim_{\mathbf{I}} \widehat{D} \big)\) exists in \(\mathscr{S}\) because \(\mathscr{S}\) has all limits of shape \(\mathbf{J}\). Abbreviate \(L = \lim_{\mathbf{J}} \big( \lim_{\mathbf{I}} \widehat{D} \big)\).

Now compute, for an arbitrary object \(S \in \mathscr{S}\), \[ \begin{align*} \mathscr{S}(S, L) &\cong [\mathbf{J}, \mathscr{S}]\big(\Delta_{\mathbf{J}} S, \lim_{\mathbf{I}} \widehat{D}\big) \\\\ &\cong [\mathbf{I}, [\mathbf{J}, \mathscr{S}]]\big(\Delta_{\mathbf{I}} \Delta_{\mathbf{J}} S, \widehat{D}\big) \\\\ &\cong [\mathbf{I} \times \mathbf{J}, \mathscr{S}]\big(\Delta_{\mathbf{I} \times \mathbf{J}} S, D\big). \end{align*} \]

The first bijection is the adjunction between the diagonal and the limit for shape \(\mathbf{J}\) in \(\mathscr{S}\), read from right to left. The second is the same adjunction for shape \(\mathbf{I}\) in the locally small category \([\mathbf{J}, \mathscr{S}]\), which has all limits of shape \(\mathbf{I}\) as noted above, applied at the object \(\Delta_{\mathbf{J}} S\) and the diagram \(\widehat{D}\). The third applies the inverse of the currying isomorphism to hom-sets. It carries \(\widehat{D}\) back to \(D\) and, by part (b), carries \(\Delta_{\mathbf{I}} \Delta_{\mathbf{J}} S\) to \(\Delta_{\mathbf{I} \times \mathbf{J}} S\). Bijectivity comes for free, since a functor with a two-sided inverse is bijective on every hom-set.

Each bijection is natural in \(S\). The adjunction bijections are natural in their first variable, and naturality of the first bijection in \(S\), respectively of the second in \(\Delta_{\mathbf{J}} S\), gives naturality in \(S\) after composing with the functors \(\Delta_{\mathbf{J}}\) and \(\Delta_{\mathbf{I}} \Delta_{\mathbf{J}}\). The currying bijection is natural in \(S\) because the inverse isomorphism is a functor and, by part (b) again, sends precomposition by \(\Delta_{\mathbf{I}} \Delta_{\mathbf{J}} h\) to precomposition by \(\Delta_{\mathbf{I} \times \mathbf{J}} h\) for every map \(h\) of \(\mathscr{S}\).

The composite is therefore a bijection \(\mathscr{S}(S, L) \cong [\mathbf{I} \times \mathbf{J}, \mathscr{S}](\Delta_{\mathbf{I} \times \mathbf{J}} S, D)\), natural in \(S\), and the right-hand side is exactly the set of cones on \(D\) with vertex \(S\). A natural isomorphism from \(\mathscr{S}(-, L)\) to \(\operatorname{Cone}(-, D)\) is a representation of the cone functor, so by the correspondence between limits and representations, the diagram \(D\) has a limit and \(L\) is a limit object: \(\lim_{\mathbf{I} \times \mathbf{J}} D\) exists and is isomorphic to \(L\).

For the first isomorphism of the statement, run the same argument with the roles of the two factors exchanged, using the mirrored transpose of part (c). The transpose \(\widetilde{D}\) is a diagram of shape \(\mathbf{J}\) valued in \([\mathbf{I}, \mathscr{S}]\), and its inner limit \(\lim_{\mathbf{J}} \widetilde{D}\) exists there because \(\mathscr{S}\) has all limits of shape \(\mathbf{J}\). The chain of bijections, started from \(\mathscr{S}\big(S, \lim_{\mathbf{I}} \big( \lim_{\mathbf{J}} \widetilde{D} \big)\big)\), terminates in the same set of cones on \(D\). Both iterated limits therefore represent \(\operatorname{Cone}(-, D)\), so both are vertices of limit cones on \(D\), and all three objects of the statement are isomorphic because limits are unique up to isomorphism.

Proof

Write \(D^{\vee} = \Phi \circ D^{\mathrm{op}} : \mathbf{I}^{\mathrm{op}} \to [\mathbf{A}^{\mathrm{op}}, \mathscr{S}^{\mathrm{op}}]\). The categories \(\mathbf{A}^{\mathrm{op}}\) and \(\mathbf{I}^{\mathrm{op}}\) are small, and \(\mathscr{S}^{\mathrm{op}}\) is locally small because its hom-sets are hom-sets of \(\mathscr{S}\), so \(D^{\vee}\) satisfies the standing hypotheses of the previous page.

We translate each notion in the statement. A colimit of \(D\) is by definition a limit of \(D^{\mathrm{op}} : \mathbf{I}^{\mathrm{op}} \to [\mathbf{A}, \mathscr{S}]^{\mathrm{op}}\), and cocones on \(D\) are by definition cones on \(D^{\mathrm{op}}\). Composition with the isomorphism \(\Phi\), by the first dictionary entry, carries cocones on \(D\) bijectively to cones on \(D^{\vee}\), with a cocone of vertex \(X\) corresponding to a cone of vertex \(X^{\mathrm{op}}\), and with colimit cocones corresponding exactly to limit cones.

The pointwise notions translate the same way. Whiskering the identity \(\operatorname{ev}_A \circ\, \Phi = (\operatorname{ev}_A)^{\mathrm{op}}\) with \(D^{\mathrm{op}}\) gives, for every \(A \in \mathbf{A}\), \[ \operatorname{ev}_A \circ\, D^{\vee} = (\operatorname{ev}_A)^{\mathrm{op}} \circ D^{\mathrm{op}} = \big(\operatorname{ev}_A \circ\, D\big)^{\mathrm{op}} = \big(D(-)(A)\big)^{\mathrm{op}} \] as diagrams \(\mathbf{I}^{\mathrm{op}} \to \mathscr{S}^{\mathrm{op}}\). A colimit of \(D(-)(A)\) in \(\mathscr{S}\) is by definition a limit of \(\big(D(-)(A)\big)^{\mathrm{op}}\) in \(\mathscr{S}^{\mathrm{op}}\), so the hypothesis says precisely that every pointwise diagram \(D^{\vee}(-)(A)\) of \(D^{\vee}\) has a limit in \(\mathscr{S}^{\mathrm{op}}\).

Likewise, the \(\operatorname{ev}_A\)-image of the cone corresponding to a cocone \(c\) on \(D\) is the opposite of the \(\operatorname{ev}_A\)-image of \(c\). Hence \(c\) evaluates to colimit cocones at every object exactly when its corresponding cone on \(D^{\vee}\) evaluates to limit cones at every object.

The pointwise limit theorem now applies to \(D^{\vee}\). Its part (a) produces a cone on \(D^{\vee}\) that evaluates to a limit cone at every object, and the corresponding cocone on \(D\) evaluates to a colimit cocone at every object, proving (a). For (b), let \(c\) be a cocone on \(D\) evaluating to colimit cocones everywhere. Its corresponding cone on \(D^{\vee}\) evaluates to limit cones everywhere, hence is a limit cone by part (b) of the cited theorem, and therefore \(c\) is a colimit cocone. The final clause follows from (a) and (b) together.

Proof

Existence is immediate from the theorem. For any diagram \(D : \mathbf{I} \to [\mathbf{A}, \mathscr{S}]\), each pointwise diagram \(D(-)(A)\) has a colimit because \(\mathscr{S}\) has all colimits of shape \(\mathbf{I}\), so \(D\) has a colimit.

For preservation, let \(c\) be a colimit cocone on \(D\), with vertex \(X\), and fix \(A \in \mathbf{A}\). Part (a) of the theorem provides a cocone \(c'\) on \(D\), with vertex \(X'\), that evaluates to a colimit cocone at every object, and \(c'\) is a colimit cocone by part (b).

Colimit cocones on the same diagram factor through one another by a unique isomorphism: there is an isomorphism \(j : X' \to X\) composing the legs of \(c'\) into the legs of \(c\). Applying the functor \(\operatorname{ev}_A\) to these factorizations shows that the evaluated cocone \(\operatorname{ev}_A(c)\) is obtained from the colimit cocone \(\operatorname{ev}_A(c')\) by composing every leg with the isomorphism \(\operatorname{ev}_A(j)\). A cocone obtained from a colimit cocone by composing with an isomorphism of vertices is again a colimit cocone, since factorizations through the two vertices correspond bijectively along the isomorphism. Hence \(\operatorname{ev}_A(c)\) is a colimit cocone on \(D(-)(A)\), which is what preservation of colimits demands.

Proof

The category \(\mathbf{Set}\) is locally small. It has all limits of every small shape by the explicit construction as a subset of a product, and it has all colimits of every small shape by the explicit construction as a quotient of a sum. The category \(\mathbf{A}^{\mathrm{op}}\) is small. For each small shape \(\mathbf{I}\), the limit corollary of the previous page and the colimit corollary above, applied with \(\mathbf{A}^{\mathrm{op}}\) in place of \(\mathbf{A}\), give limits and colimits of shape \(\mathbf{I}\) in \([\mathbf{A}^{\mathrm{op}}, \mathbf{Set}]\) and their preservation by every evaluation functor.

Proof

The family is a cone, since applying the functor \(H_\bullet\) to the cone equations \(Du \circ p_I = p_J\) gives \(H_{Du} \circ H_{p_I} = H_{p_J}\). By part (b) of the pointwise limit theorem, applied with \(\mathbf{A}^{\mathrm{op}}\) as the small domain and \(\mathbf{Set}\) as the locally small target, the cone is a limit cone as soon as its image under \(\operatorname{ev}_A\) is a limit cone for every \(A \in \mathbf{A}\). The pointwise hypothesis of that theorem holds once those evaluated images are shown to be limit cones, since a limit cone in particular provides a limit.

Fix \(A \in \mathbf{A}\) and compute the composite \(\operatorname{ev}_A \circ\, H_\bullet : \mathbf{A} \to \mathbf{Set}\). On objects it sends \(B\) to \(H_B(A) = \mathbf{A}(A, B)\). On a map \(f : B \to B'\) it sends \(f\) to the component \((H_f)_A\), which post-composes with \(f\). This is exactly the covariant hom-functor \(\mathbf{A}(A, -)\).

The \(\operatorname{ev}_A\)-image of the displayed family is therefore \[ \Big( \mathbf{A}(A, \lim D) \xrightarrow{\mathbf{A}(A,\, p_I)} \mathbf{A}(A, D(I)) \Big)_{I \in \mathbf{I}}, \] which is a limit cone on \(\mathbf{A}(A, D)\) because representables preserve limits. Every evaluation of the cone is thus a limit cone, and the cone is a limit cone.