# The Unapologetic Mathematician

## Representations and Universals and Colimits (oh, my!)

Let’s look at universal arrows a bit more closely. If we have a functor $F:\mathcal{D}\rightarrow\mathcal{C}$, an object $C\in\mathcal{C}$, and a universal arrow $u:C\rightarrow F(U)$, then we can take any arrow $f:U\rightarrow D$ and form the composition $F(f)\circ u:C\rightarrow F(D)$. The universal condition says that to every arrow $d:C\rightarrow F(D)$ corresponds a unique $f:U\rightarrow D$ so that $d=F(f)\circ u$. That is, there is a bijection $\eta_D:\hom_\mathcal{C}(C,F(D))\cong\hom_\mathcal{D}(U,D)$ for every $D\in\mathcal{D}$.

It’s even better than that, though: these bijections commute with arrows in $\mathcal{D}$. That is, if we have $f:D_1\rightarrow D_2$ then the naturality condition $f\circ\eta_{D_1}(d)=\eta_{D_2}(F(f)\circ d)$ holds for all $d:C\rightarrow D_1$. This makes the $\eta_D$ into the components of a natural isomorphism $\eta:\hom_\mathcal{C}(C,F(\underline{\hphantom{X}}))\rightarrow\hom_\mathcal{D}(U,\underline{\hphantom{X}})=h_U$. That is, a universal arrow from $C$ to $F$ is a representation of the functor $\hom_\mathcal{C}(C,F(\underline{\hphantom{X}})):\mathcal{D}\rightarrow\mathbf{Set}$.

Conversely, let’s say we have a representable functor $K:\mathcal{D}\rightarrow\mathbf{Set}$, with representation $\eta_D:h_U\rightarrow K(D)$. Yoneda’s lemma tells us that this natural transformation corresponds to a unique element of $K(U)$, which as usual we think of as an arrow $u:*\rightarrow K(U)$. So a representation of a functor is a universal arrow in $\mathbf{Set}$.

So, representations of functors are the same as universal arrows. Colimits are special kinds of universal arrows, and since a universal arrow is an initial object in an apropriate comma category it is a colimit. Representations are universals are colimits.

If we swap the category $\mathcal{C}$ for the dual category $\mathcal{C}^\mathrm{op}$, we change limits into colimits, couniversals into universals, and contravariant representations into covariant ones. All the reasoning above applies just as well to $\mathcal{C}^\mathrm{op}$. Thus, we see that in $\mathcal{C}$ representations of contravariant functors are couniversals are limits.

There’s actually yet another concept yet to work into this story, but I’m going to take a break from it to flesh out another area of category theory for a while.

June 26, 2007 - Posted by | Category theory