The Unapologetic Mathematician

Mathematics for the interested outsider

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.

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June 26, 2007 - Posted by | Category theory

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