Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 20 / 4

For $D:\mathcal{I}\to \mathcal{C}$, define the functor $\mathrm{Cone}(-,D):{\mathcal{C}}^{\mathrm{op}}\to \mathbf{Set}$ bywhere $\Delta X$ is the constant diagram functor; a morphism into $X$ acts by precomposition on every categorical cone leg. A categorical limit is a representation of a functor of the formThe image of $1_L$ is the universal categorical cone $\lambda :\Delta L\to D$. By naturality of the representing isomorphism, the image of $h:X\to L$ is $\lambda \circ \Delta h$. Thus this definition is exactly existence and uniqueness of factorization of every categorical cone through $\lambda$. Conversely that universal property gives the representing natural bijection.

Choose for each $j\in \mathcal{J}$ a categorical limit $L_j$ of $F(-,j)$, with projections ${\lambda }_{i,j}:L_j\to F(i,j)$. For $v:j\to k$, the morphisms $F(1_i,v){\lambda }_{i,j}$ form a categorical cone over $F(-,k)$, since the two actions of the product functor $F$ commute. Define $L(v):L_j\to L_k$ by the equationsExistence and uniqueness follow from the universal property. The identity morphism of $L_j$ satisfies the equations for $v=1_j$, and $L(w)L(v)$ satisfies those for $wv$, so uniqueness gives $L(1_j)=1_{L_j}$ and $L(wv)=L(w)L(v)$. Hence $L:\mathcal{J}\to \mathcal{D}$ is a functor.

If $L_j^{\prime }$ with projections ${\lambda }_{i,j}^{\prime }$ are other choices, there is a unique isomorphism $u_j:L_j\to L_j^{\prime }$ carrying every projection to the corresponding projection. The same equations show $u_{k}L(v)=L^{\prime }(v)u_j$, because their composites with all ${\lambda }_{i,k}^{\prime }$ agree. Thus the choices give a unique natural isomorphism compatible with the chosen categorical limit categorical cones. Without the compatibility condition, uniqueness of an arbitrary natural isomorphism is not asserted.

Let $M$ be a categorical limit of $L:\mathcal{J}\to \mathcal{D}$, with projections $p_j:M\to L_j$. Set ${\gamma }_{i,j}={\lambda }_{i,j}{p}_j$. The equations for the categorical cones $\lambda$ and $p$, and the defining equations for $L(v)$, show that this family is a categorical cone over $F:\mathcal{I}\times \mathcal{J}\to \mathcal{D}$.

For any categorical cone $x_{i,j}:X\to F(i,j)$, fix $j$. Its $\mathcal{I}$-legs give a unique $h_j:X\to L_j$. For $v:j\to k$, both $L(v)h_j$ and $h_k$ have composite $x_{i,k}$ with every ${\lambda }_{i,k}$, so they are equal. Thus the $h_j$ form a categorical cone over $L$, which factors uniquely through $h:X\to M$. The equalities ${\gamma }_{i,j}h=x_{i,j}$ follow. Any other such $h$ has the same composites with every $p_j$, because the projections ${\lambda }_{i,j}$ detect equality; uniqueness for $M$ then makes it equal to $h$. Therefore $M$ is a categorical limit of the whole product diagram in a category.

A precise commutation of iterated categorical limits statement is the following. For small categories $\mathcal{I},\mathcal{J}$ and $F:\mathcal{I}\times \mathcal{J}\to \mathcal{D}$, if every column categorical limit and the resulting $\mathcal{J}$-limit exist, thenIf every row categorical limit and the resulting $\mathcal{I}$-limit also exist, thenEach isomorphism is the unique one identifying all projections to $F(i,j)$. Indeed, the preceding part proves that each iterated categorical limit has the universal property of the whole product diagram in a category. Its uniqueness gives the canonical isomorphism. Equivalently, categorical cones with vertex $X$ over the product diagram in a category are exactly compatible families, indexed by $j$, of categorical cones over $F(-,j)$; regrouping the same compatibility equations cannot change their representing object. In a complete category all the stated existence hypotheses hold for small $\mathcal{I},\mathcal{J}$. The theorem is about categorical limits with categorical limits, and makes no unconditional claim about commuting a categorical limit with a colimit.