Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 18 / 3

For a small diagram $F:I\to \mathcal{D}$ in a locally small category, define its cone functor on ${\mathcal{D}}^{\mathrm{op}}$ bywhere $\Delta A$ is the constant diagram. Precomposition acts on cone vertices. A categorical limit is a representation of this functor: an object $L$ with natural bijections $\mathcal{D}(A,L)\cong {\mathrm{Cone}}_F(A)$. The image of $1_L$ is the universal cone, and the inverse bijection supplies the unique mediating arrow for every cone. Thus the representable functor formulation and the usual terminal-cone universal property coincide.

Choose a limit $L_j$ for each diagram $F(-,j)$, with legs $p_{i,j}:L_j\to F(i,j)$. For an arrow $v:j\to j^{\prime }$, the maps $F(1_i,v)p_{i,j}$ form a cone over $F(-,j^{\prime })$, since the two-variable functor respects the commuting arrows in $I\times J$. The limit property therefore gives a unique arrow $L(v):L_j\to L_{j^{\prime }}$ satisfyingThe arrows $L(1_j)$ and $1_{L_j}$ have identical composites with all projections, so they are equal. The same uniqueness shows $L(v^{\prime }v)=L(v^{\prime })L(v)$. Hence the assignment extends to a functor $L:J\to \mathcal{D}$.

For the chosen limit objects and cones it is uniquely determined, literally, by these equations. Changing the choices gives a unique family of isomorphisms between the limits compatible with their projections; the equations above show that this family is a natural isomorphism of the resulting functors. Thus it is unique up to a unique cone-compatible natural isomorphism, not independent of choices as an equality of objects.

Let $M={\lim }_{j\in J}{L}_j$ with projections $q_j:M\to L_j$. The composites $p_{i,j}{q}_j$ form a cone over $F:I\times J\to \mathcal{D}$: the $I$ equations hold in each inner limit, and the $J$ equations follow from the naturality of $p$ and the outer cone.

Given any cone $a_{i,j}:A\to F(i,j)$, the inner limit property gives unique $b_j:A\to L_j$ with $p_{i,j}{b}_j=a_{i,j}$. The $J$ cone equations for $a$ force $L(v)b_j=b_{j^{\prime }}$ by uniqueness of inner mediating arrows. Thus the $b_j$ form a cone to $L$, and there is a unique $b:A\to M$ with $q_{j}b=b_j$. This factors all $a_{i,j}$, and the two stages of uniqueness prove uniqueness for the whole product diagram. Consequently

The categorical commutation of iterated categorical limits theorem is the following. For small categories $I,J$ and a two-variable functor $F:I\times J\to \mathcal{D}$, if the two collections of inner limits and their outer limits exist, then there are canonical isomorphismsIn particular all these limits exist if $\mathcal{D}$ has all $I$-shaped and all $J$-shaped limits. The isomorphisms are natural in $F$ and uniquely characterized by preserving every projection to $F(i,j)$.

Part (iii) proves that each iterated limit represents the functor of cones on the product diagram, with the same flattened projections. Two such representations have unique inverse cone-compatible arrows, proving the isomorphisms. For a natural transformation $F\to F^{\prime }$, the two composites in any naturality square have the same product-diagram projections, hence agree by limit uniqueness. This proves naturality as well. Completeness of $\mathcal{D}$ is a convenient sufficient hypothesis, rather than a necessary one for this individual diagram.