The Unapologetic Mathematician

Enriched Categories II

So we have the basic data of a category $\mathcal{C}$ enriched over a monoidal category $\mathcal{V}$. Of course, what I left out were the relations that have to hold. And they’re just the same as those from categories, but now written in terms of $\mathcal{V}$ instead of $\mathbf{Set}$: associativity and identity relations, as encoded in the following commutative diagrams: Notice how these are very similar to the axioms for a monoidal category or a monoid object. And this shouldn’t be unexpected by now, since we know that a monoid is just a (small) category with only one object. In fact, if we only have one object in a $\mathcal{V}$-enriched category we get back exactly a monoid object in $\mathcal{V}$!

Now, often we’re thinking of our hom-objects as “hom-sets with additional structure”. There should be a nice way to forget that extra structure and recover just a regular category again. To an extent this is true, but for some monoidal categories $\mathcal{V}$ the “underlying set” functor isn’t really an underlying set at all. For now, though, let’s look at a familiar category of “sets with extra structure” and see how we get the underlying set out of the category itself.

Again, the good example to always refer back to for enriched categories is $\mathbf{Ab}$, the category of abelian groups with tensor product as the monoidal structure. We recall that the functor giving the free abelian group on a set is left adjoint to the forgetful functor from abelian groups to sets. That is, $\hom_\mathbf{Ab}(F(S),A)\cong\hom_\mathbf{Set}(S,U(A))$. We also know that we can consider an element of the underlying set $U(A)$ of an abelian group as a function from a one-point set into $U(A)$. That is, $\hom_\mathbf{Set}(\{*\},U(A))\cong U(A)$. Putting these together, we see that $U(A)\cong\hom_\mathbf{Ab}(\mathbb{Z},A)$, since $\mathbb{Z}$ is the free abelian group on one generator.

But $\mathbb{Z}$ is also the identity object for the tensor product! The same sort of argument goes through for all our usual sets-with-structure, telling us that in all these cases the “underlying set” functor is represented by the monoidal identity $\mathbf{1}$, which is the free object on one generator. We take this as our general rule, giving the representable functor $V(\underline{\hphantom{X}})=\hom_{\mathcal{V}_0}(\mathbf{1},\underline{\hphantom{X}}):\mathcal{V}_0\rightarrow\mathbf{Set}$. In many cases (but not all!) this is the usual “underlying set” functor, but now we’ve written it entirely in terms of the monoidal category $\mathcal{V}$!

As time goes by, we’ll use this construction to recover the “underlying category” of an enriched category. The basic idea should be apparent, but before we can really write it down properly we need to enrich the notions of functors and natural transformations.

August 14, 2007 Posted by | Category theory | Leave a comment

Tweaky

So I’m up at 03:00 because of an unexpected nap this evening. When I got back from dinner, there was a blackout that included my building. So I slept a bit, then woke up completely a few hours later. Having little else to do, I finally hunkered down on some tweaks to the site.

I moved the tagline a bit and dropped the “rant” phrase. I wanted to change “outraged” to “outspoken”, but even then it wouldn’t really look nice where it now goes. Actually, not only have I not been that outraged, I haven’t been as ranty as I expected I’d be.

Over the last seven months this place has gone in directions I didn’t really expect, and it’s really taken on a life of its own. As for myself — the nominal author — I find myself taken along for the ride. I’m reminded of the apocryphal story about the university that planted grass everywhere, waited to see where it got trampled from people walking across it, then put the sidewalks where people naturally walked. Making predictions is difficult, especially about the future. Unless you’re an anti-blogger, in which case I suppose it’s difficult to predict the past, but that’s a whole ‘nother story.

In the place of the old tagline, I’ve got a more robust (and appropriate) “about” panel. It should give a better explanation of what the hell is going on here, and how to read the page for newcomers.

I haven’t started tearing through and restructuring the “categories” panel in the sidebar, but I’ll hold off on that until I get the cable modem set up here rather than the slow and buggy “Mu-Fi“.

I’m also going to see what I can do about getting a search panel set up, but without paying for the capability to edit my CSS directly (in both money to WordPress and time invested in learning how), I’m not sure what I’ll be able to cobble together.

Finally, inspired by the adventures this evening, I’m starting a more “bloggy” weblog about my life as a Connecticut Yankee in the Lower Garden District. I’m using a phrase I picked up from Nick Maggio (a grad student here) for the title: Yankee Freak-Out. There’s not much there yet, but I’ll start posting tomorrow afternoon.

Assuming, of course, that I’ve got the power.

August 14, 2007 Posted by | Uncategorized | 2 Comments