Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 22 / 2

Fix the chosen categorical limit object $LA$ and its categorical cone $p_{A,j}:LA\to (DA{)}_j$. For $a:A\to A^{\prime }$, the familyis a categorical cone, by naturality of $Da$. The universal property of $L{A}^{\prime }$ supplies a unique morphismThe identity morphism of $LA$ satisfies the equations for $a=1_A$, so $L{1}_A=1_{LA}$. For composable $a,a^{\prime }$, the equations for $L{a}^{\prime }La$ agree with those for $L(a^{\prime }a)$; uniqueness gives $L(a^{\prime }a)=L{a}^{\prime }La$. Hence the chosen limits define a functor $L:\mathcal{A}\to \mathcal{C}$. This argument also handles an empty indexing category, when each chosen categorical limit is a terminal object.

Equivalently, $LA$ represents the categorical cone functor $C\mapsto \mathrm{Nat}({\Delta }_{J}C,DA)$, and functoriality of chosen representations supplies the same $L$.

Assume $F$ preserves existing categorical limits. For a diagram in a category $E:I\to \mathcal{B}$ with categorical limit cone ${\ell }_i:L\to E_i$, its image $F{\ell }_i$ is a categorical limit cone of $FE$. For any $C\in \mathcal{C}$, a morphism $C\to FL$ is therefore uniquely equivalent to a compatible family of morphisms $C\to F{E}_i$. In the Category of sets, compatible families are exactly the categorical limit of the resulting hom-sets. ThusThe maps are induced by $F{\ell }_i$, so $\mathcal{C}(C,F-)$ preserves the given limit. Applying this to every existing categorical limit proves the implication, with no completeness hypothesis on $\mathcal{B}$.

Conversely, assume that every hom-set functor $\mathcal{C}(C,F-)$ preserves existing categorical limits. For a categorical limit cone ${\ell }_i:L\to E_i$ in $\mathcal{B}$, a categorical cone $q_i:C\to F{E}_i$ determines an element of ${\lim }_i\mathcal{C}(C,F{E}_i)$. The assumed bijectiongives one and only one such $u$ for every $C$ and every categorical cone. That is the universal property of the image cone itself. Therefore the hom-set tests detect limit preservation:This proof works for all diagram sizes for which the relevant categorical limits and compatible-family sets are under consideration.

The representing object in the functor category $[J,\mathcal{C}]$ is the constant diagram in a category ${\Delta }_{J}C$. If $p_j:{\lim }_j{E}_j\to E_j$ is the chosen categorical limit cone, the natural bijection isA natural transformation from the constant diagram in a category is precisely a categorical cone with vertex $C$, and its inverse is the unique mediating morphism from the universal property of the categorical limit. For $\alpha :E\Rightarrow E^{\prime }$, the defining equations for $\lim \alpha$ show that postcomposition by $\lim \alpha$ corresponds to postcomposition by $\alpha$ on the right. This establishes naturality in $E$ and proves representability by the constant diagram.

For every $C$, the preceding part identifies the composite $\mathcal{C}(C,{\lim }_J-)$ with the covariant representable functor $[J,\mathcal{C}]({\Delta }_{J}C,-)$. A covariant representable functor preserves every existing categorical limit, since a morphism into a categorical limit is the same as a compatible family of morphisms into the diagram objects. The limit functor preserves limits. The hom-set detection of categorical limits givesIn particular, it preserves all small categorical limits. They exist in $[J,\mathcal{C}]$: compute each one at every $j$ using completeness of $\mathcal{C}$, and use the uniqueness of the pointwise mediating morphisms to obtain its functor structure and universal natural transformations. Another expression of the same result is the adjunction ${\Delta }_J⊣{\lim }_J$.

Regard the bifunctor as $D^{(2)}:K\to [J,\mathcal{C}]$. Its pointwise categorical limit $E={\lim }_k{D}^{(2)}k$ satisfieswith its action on morphisms obtained by the construction in part (a). Applying the limit-preservation result from part (d) givesThe comparison isomorphism is canonical relative to the chosen categorical limit cones: it is the unique morphism identifying all the projections to $D(j,k)$. This proves commutation of iterated categorical limits, including an empty $J$ or $K$. Equivalently, both iterated constructions have the universal property of the categorical limit of $D$ on $J\times K$.