Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 20 / 5

Let $T:{\mathcal{C}}^{\mathrm{op}}\times \mathcal{C}\to \mathcal{D}$. A categorical end is an object $E$ with morphisms $e_A:E\to T(A,A)$ satisfying, for every $f:A\to B$,It is universal among such families: any $x_A:X\to T(A,A)$ satisfying the same equations is uniquely $e_{A}h$ for one $h:X\to E$. This is a universal dinatural transformation from the constant value $E$, and is written $E={\int }_{A}T(A,A)$.

Dually a categorical coend is $Q$ with morphisms $j_A:T(A,A)\to Q$ satisfyingAny family $x_A:T(A,A)\to X$ satisfying these equations is uniquely $h{j}_A$ for one $h:Q\to X$. Write $Q={\int }^{A}T(A,A)$. For small $\mathcal{C}$, the categorical end can be constructed as the equalizer of ${\prod }_{A}T(A,A)\rightrightarrows {\prod }_{f:A\to B}T(A,B)$ when these objects exist. The categorical coend is the coequalizer of ${\coprod }_{f:A\to B}T(B,A)\rightrightarrows {\coprod }_{A}T(A,A)$ when those objects exist. The two morphisms impose exactly the displayed equations for a dinatural transformation.

Working with a small category $\mathcal{C}$ (or in a fixed ambient universe), for a categorical presheaf $X$ and fixed $U$, apply the categorical coend construction to $T(A,B)=\mathcal{C}(U,B)\times X(A)$. It is the quotient of the disjoint union of pairs $(h:U\to W,x\in X(W))$ by the relationsDefine ${\Phi }_U([h,x])=X(h)x$. It respects every relation because $X(fh)x=X(h)X(f)x$. Define ${\Psi }_U(y)=[1_U,y]$. Then ${\Phi }_U{\Psi }_U(y)=y$. Conversely the relation with $f=h:U\to W$ gives $[h,x]=[1_U,X(h)x]$, so ${\Psi }_U{\Phi }_U$ is the identity morphism. ThereforeFor $k:V\to U$, restriction on the categorical coend sends $[h,x]$ to $[hk,x]$, and ${\Phi }_V([hk,x])=X(k){\Phi }_U([h,x])$. A natural transformation $X\to Y$ likewise acts on the second coordinate and commutes with $\Phi$. Thus the density formula for presheaves is a natural isomorphism, not just a family of unrelated bijections.

The category of elements $\int X$ has objects $(W,x)$ for $x\in X(W)$, and morphisms $f:(W,x)\to (V,y)$ with $X(f)y=x$. Send $(W,x)$ to the representable presheaf $H_W$, and a morphism $f$ to $H_{\bullet }(f)$. The element $x$ supplies a natural transformation $H_W\to X$ with component $h\mapsto X(h)x$, by the Yoneda lemma. These transformations form a cocone.

Pointwise at $U$, the colimit of this diagram in a category has representatives $(W,x,h:U\to W)$, with relation $(W,X(f)y,h)\sim (V,y,fh)$. These are exactly the relations of the preceding categorical coend. Its proved natural isomorphism with $X(U)$ therefore identifies the cocone with the universal colimit cocone. More directly, every representative equals $(U,X(h)x,1_U)$ through the morphism $h:(U,X(h)x)\to (W,x)$; two representatives with the same image therefore agree. As $\mathcal{C}$ is small, $\int X$ is small, so this is an ordinary small colimit. Consequently