# The Unapologetic Mathematician

## Ends I

So far in our treatment of enriched categories we’ve been working over a monoidal category $\mathcal{V}$, and we latter added the assumption that $\mathcal{V}$ is symmetric and closed. From here, we’ll also assume that the underlying category $\mathcal{V}_0$ is complete — it has all small limits.

Now let’s consider a functor $T:\mathcal{C}^\mathrm{op}\otimes\mathcal{C}\rightarrow\mathcal{V}$. An “end” for $T$ is a universal $\mathcal{V}$natural transformation $\lambda_C:K\rightarrow T(C,C)$. Universality here means that if $\alpha_C:X\rightarrow T(C,C)$ is another \mathcal{V}\$ natural transformation then there is a unique arrow $f:X\rightarrow K$ so that $\alpha_C=\lambda_C\circ f$. As usual, it’s unique up to isomorphism. We denote the object $K$ by $\int_{C\in\mathcal{C}}T(C,C)$, and abuse the language a bit by calling this object the end. Then we call the $\mathcal{V}$-natural transformation the “counit” of the end.

Because $\mathcal{V}$ is symmetric and closed we have an adjunction $\hom_{\mathcal{V}_0}(X,Z^Y)\cong\hom_{\mathcal{V}_0}(X\otimes Y,Z)\cong\hom_{\mathcal{V}_0}(Y,Z^X)$. Under this adjunction, the two natural transformations

• $T(1_A,\underline{\hphantom{X}})_{A,B}:\hom_\mathcal{C}(A,B)\rightarrow T(A,B)^{T(A,A)}$
• $T(\underline{\hphantom{X}},1_B)_{B,A}:\hom_\mathcal{C}(A,B)\rightarrow T(A,B)^{T(B,B)}$

become transformations

• $\rho_{A,B}:T(A,A)\rightarrow T(A,B)^{\hom_\mathcal{C}(A,B)}$
• $\sigma_{A,B}:T(B,B)\rightarrow T(A,B)^{\hom_\mathcal{C}(A,B)}$

Then the naturality condition states that $\rho_{A,B}\circ\lambda_A=\sigma_{A,B}\circ\lambda_B$.

Therefore when $\mathcal{C}$ is small we can define the end as an equalizer:
$\int_{C\in\mathcal{C}}T(C,C)\rightarrow\prod_{A\in\mathcal{C}}T(A,A)\rightrightarrows\prod_{A,B\in\mathcal{C}}T(A,B)^{\hom_\mathcal{C}(A,B)}$
where one of the arrows on the right is built from $\rho$ and the other is built from $\sigma$. This limit exists by the completeness of $\mathcal{V}_0$. In fact, a very similar argument can push the result a little further, to cover categories which are not small themselves, but which are equivalent to small categories.

We can write the universal property in yet another way. First, note that the set of $\mathcal{V}$-natural $\alpha_C:X\rightarrow T(C,C)$ is in bijection with the set $\hom_{\mathcal{V}_0}(X,\int_C T(C,C))$.

Now, we can write ${\lambda_A}^X:\left(\int_C T(C,C)\right)^X\rightarrow T(A,A)^X$, and it turns out that this is also an end. Indeed, $Y\rightarrow T(C,C)^X$ is $\mathcal{V}$-natural if and only if $Y\otimes X\rightarrow T(A,A)$. Then there exists a unique $g:Y\otimes X\rightarrow\int_C T(C,C)$, which corresponds under the closure adjunction to a unique $f:Y\rightarrow\left(\int_C T(C,C)\right)^X$. Thus $\int_C\left(T(A,A)^X\right)\cong\left(\int_C T(C,C)\right)^X$, and thus the bijection of sets above gets promoted to an isomorphism in $\mathcal{V}_0$