Coend computation

Question

Let

$F:A^{\mbox{op}} \to \mbox{Set}$

and define

$G_a:A\times A^{\mbox{op}} \to \mbox{Set}$

$G_a(b,c) = \mbox{hom}(a,b) \times F(c)$.

I think the coend of $G_a$,

$\int^AG_a$,

ought to be $F(a)$--it's certainly true when A is discrete, since then hom is a delta function. But my colimit-fu isn't good enough to actually compute the thing and verify it's true. Can someone walk me through the computation, please?

Answer 1

Hi Mike. This is what's often called the Density Formula, or (at the n-Lab) the coYoneda Lemma (I think), or (by Australian ninja category theorists) simply the Yoneda Lemma. (But Australian ninja category theorists call everything the Yoneda Lemma.) In any case, it's a kind of dual to the ordinary Yoneda Lemma.

But you asked to be walked through it. First: yes, it is $F(a)$. Another way of writing your coend $$ \int^A G_a $$ is as $$ \int^{b \in A} G_a(b, b) = \int^b \mathrm{hom}(a,b) \times F(b). $$ I claim this is canonically isomorphic to $F(a)$. I'll prove this by showing that for an arbitrary set $S$, the homset $\mathrm{hom}(\mathrm{this}, S)$ is canonically isomorphic to $\mathrm{hom}(F(a), S)$. The claim will then follow from the ordinary Yoneda Lemma.

So, let $S$ be a set. Then $$ \begin{align} \mathrm{Set}(\int^b \mathrm{hom}(a, b) \times F(b), S) & \cong \int_b \mathrm{Set}(\mathrm{hom}(a, b) \times F(b), S) \\ &\cong \int_b \mathrm{Set}(\mathrm{hom}(a, b), \mathrm{Set}(F(b), S)) \\ &\cong \mathrm{Nat}(\hom(a, -), \mathrm{Set}(F(-), S)) \\ &\cong \mathrm{Set}(F(a), S) \end{align} $$ I don't know how much of this you'll want explaining, so I'll just say it briefly for now. If you want further explanation, just ask. The first isomorphism is kinda the definition of colimit. The second is the usual exponential transpose/currying operation. The third is maybe the most important: it's a fundamental fact about ends that if $F, G: C \to D$ are functors then $$ \mathrm{Nat}(F, G) = \int_c D(F(c), G(c)). $$ The fourth and final isomorphism is the ordinary Yoneda Lemma applied to the functor $\mathrm{Set}(F(-), S)$.