The Unapologetic Mathematician

Mathematics for the interested outsider

Enriched Naturality Revisited

Let’s look back at the enriched versions of representable functors. If we fix an object C\in\mathcal{C} we have a \mathcal{V}-natural transformation \hom_\mathcal{C}(C,\underline{\hphantom{X}})_{B,C}:\hom_\mathcal{C}(A,B)\rightarrow\hom_\mathcal{C}(C,B)^{\hom_\mathcal{C}(C,A)}. This corresponds under the closure adjunction to \circ:\hom_\mathcal{C}(A,B)\otimes\hom_\mathcal{C}(C,A)\rightarrow\hom_\mathcal{C}(C,B). There’s a similar transformation for composition on the other side.

Now remember that the closest thing we have to a “morphism” in an enriched category is an element of the underlying set of a hom-object. That is, we can talk about an arrow f:\mathbf{1}\rightarrow\hom_\mathcal{C}(A,B). We often abuse the language and say that this is a morphism from A to B, which in fact it’s a morphism in the underlying category.

Now, even though this f isn’t really a morphism in our enriched category, we can still come up with a morphism sensibly called \hom_\mathcal{C}(C,f):\hom_\mathcal{C}(C,A)\rightarrow\hom_\mathcal{C}(C,B). Here’s how it goes:

  • We start with \hom_\mathcal{C}(C,A) and use the left unit isomotphism to move to \mathbf{1}\otimes\hom_\mathcal{C}(C,A).
  • We now hit \mathbf{1} with our morphism f to land in \hom_\mathcal{C}(A,B)\otimes\hom_\mathcal{C}(C,A).
  • Finally, we compose to end up in \hom_\mathcal{C}(C,B).

We can similarly take g:\mathbf{1}\rightarrow\hom_\mathcal{C}(A,B) and construct a morphism \hom_\mathcal{C}(g,C):\hom_\mathcal{C}(B,C)\rightarrow\hom_\mathcal{C}(A,C).

These composites should look familiar from the definition of enriched naturality for a transformation \eta:S\rightarrow T. In fact, we have a more compact diagram to replace that big hexagon:
Enriched Naturality Diagram (revised)
Notice here that the right and bottom arrows in this square expand out to become the top and bottom of a hexagon, and we slip the functors S and T into place.


August 29, 2007 - Posted by | Category theory


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