The Unapologetic Mathematician

Mathematics for the interested outsider

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:
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

September 6, 2007 Posted by | Category theory | 1 Comment

Category Theory Seminar!

It looks like not only is this seminar going to go off, but one of the tenured faculty is going to set it up as a seminar associated to the department’s VIGRE grant, partly because his grad student needs a VIGRE seminar on his transcript this semester and this would be the closest to his actual research interests. Here’s the rough description I sent off to the professor, which will be more or less the outline on the books:

As I’ve envisioned it, the seminar will start with 2-3 weeks (maybe 4, depending on how smoothly it goes) of basic information, to cover:

1) categories, functors, and natural transformations
2) limits, adjunctions, representable functors, universals (Yoneda lemma?)
3) monoidal and monoidal closed categories

At that point most of the basic language is in place to do applications. We can then branch out to survey various applications in 1-3 weeks each, including (but not limited to):

1) “quantum topology” (applications to knot theory and related fields)
2) elementary topoi (applications to foundations)
3) lambda calculi (applications to functional programming semantics)
4) Lawvere’s “sketches” (applications to universal algebra)
5) n-categories (applications to mathematical physics)

These branches could be dealt with by interested seminar participants as time permits.

I’m hoping that the material could be made accessible to even a good senior undergraduate, but if the VIGRE seminar format is such that it’s more appropriate to restrict to graduate students and up, that can work too.

I’m surely not going to measure up to John Baez’ quantum gravity seminar, especially not on my first time out, but I’m really excited about it anyhow. When we start up I’ll try to get some regular attendee take notes so I can follow Baez’ lead in scanning/posting them.

September 6, 2007 Posted by | Uncategorized | 1 Comment