Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 18 / 4

Let $T:{\mathcal{C}}^{\mathrm{op}}\times \mathcal{C}\to \mathcal{D}$. An end of a functor is an object $E$ with maps $e_C:E\to T(C,C)$ satisfying $T(1_C,f)e_C=T(f,1_D)e_D$ for every $f:C\to D$, universal among all such families. Thus every compatible family with vertex $A$ factors uniquely through $E$. Write it as ${\int }_{C}T(C,C)$. These are the wedge equations for a dinatural transformation from a constant functor to $T$.

Dually, a coend of a functor is an object $Q$ with maps $i_C:T(C,C)\to Q$ satisfying $i_{C}T(f,1_C)=i_{D}T(1_D,f)$ as maps from $T(D,C)$, universal among compatible families with a common target. Write it as ${\int }^{C}T(C,C)$. These are cowedge equations, and every cowedge to $A$ factors uniquely through $Q$.

For small $\mathcal{C}$, if the needed products and equalizers exist, the end is the equalizer of ${\prod }_{C}T(C,C)\rightrightarrows {\prod }_{f:C\to D}T(C,D)$. If the needed coproducts and coequalizers exist, the coend is the coequalizer of ${\coprod }_{f:C\to D}T(D,C)\rightrightarrows {\coprod }_{C}T(C,C)$. The two arrows in each construction are precisely the two sides of the compatibility equation. This explains both the mixed variance and the universal property, rather than treating the integral notation as an ordinary numerical integral.

For fixed $U$, the categorical coend on the right is the quotient of ${\coprod }_W\mathcal{C}(U,W)\times X(W)$ by the identificationsDefine ${\Phi }_U([h,x])=X(h)x$. This is well defined because the contravariant functor law gives $X(fh)x=X(h)X(f)x$. Its proposed inverse sends $y\in X(U)$ to $[1_U,y]$. The composite to $X(U)$ is the identity. In the other direction the coend relation, applied with $f=h$ and initial map $1_U$, gives $[h,x]=[1_U,X(h)x]$. Thus the inverse is genuine andFor $a:U^{\prime }\to U$, the coend map replaces $h$ by $ha$; its image is $X(ha)x=X(a)X(h)x$, proving naturality in $U$. A morphism of presheaves acts on $x$ and commutes with every $X(h)$ by naturality, proving naturality in $X$. This is the density formula for presheaves.

The category of elements of $X$ has objects $(W,x)$ with $x\in X(W)$ and morphisms $f:(W,x)\to (V,y)$ satisfying $X(f)y=x$. Send $(W,x)$ to the representable presheaf $H_W$ and a morphism to $H_f$. The Yoneda lemma gives a canonical cocone to $X$, whose leg at $(W,x)$ corresponds to $x$.

For any target presheaf $Z$, a cocone from this diagram to $Z$ is a family of maps $H_W\to Z$, or equivalently elements $z_{W,x}\in Z(W)$, with $z_{W,x}=Z(f)z_{V,y}$ whenever $X(f)y=x$. Such a family is exactly a natural transformation $X\to Z$, via $x\mapsto z_{W,x}$. This bijection respects the canonical cocone, so it proves its colimit universal property. Consequently the canonical colimit presentation of a presheaf isEquivalently, the density coend identifies the disjoint copies of the representables by exactly these element-category arrows. Smallness of $\mathcal{C}$ and the set-valued fibers make the indexing category small.