The Unapologetic Mathematician

Mathematics for the interested outsider

“Everything” about D-Modules

Over at The Everything Seminar, Greg Muller has a great introductory post about D-modules, which is what representation theory and category theory have to do with partial differential equations. Along the way he touches representable functors, Yoneda’s Lemma, and a bunch of other topics you should be able to follow if you’ve been reading here for a while.

September 7, 2007 Posted by | Uncategorized | Leave a comment

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:
\int_{C\in\mathcal{C}}T(C,C)\rightarrow\prod_{A\in\mathcal{C}}T(A,A)\rightrightarrows\prod_{A,B\in\mathcal{C}}T(A,B)^{\hom_\mathcal{C}(A,B)}
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

Enriched Adjunctions

When I started in on adjoint functors, I gave the definition in terms of a bijection of hom-sets. Then I showed that we can also specify it in terms of its unit and counit. Both approaches (and their relationship) generalize to the enriched setting.

Given a functor F:\mathcal{C}\rightarrow\mathcal{D} and another G:\mathcal{D}\rightarrow\mathcal{C}, an adjunction is given by natural transformations \eta:1_\mathcal{C}\rightarrow G\circ F and \epsilon:F\circ G\rightarrow1_\mathcal{D}. These transformations must satisfy the equations (1_G\circ\epsilon)\cdot(\eta\circ1_G)=1_G and (\epsilon\circ1_F)\cdot(1_F\circ\eta)=1_F. By the weak Yoneda Lemma, this is equivalent to giving a \mathcal{V}-natural isomorphism \phi_{C,D}:\hom_\mathcal{D}(F(C),D)\rightarrow\hom_\mathcal{C}(C,G(D)).

Indeed, a \mathcal{V}-natural transformation in this direction must be of the form \phi_{C,D}=\hom_\mathcal{C}(\eta_C,1_{G(D)})\circ G_{F(C),D}, and one in the other direction must be of the form \varphi_{C,D}=\hom_\mathcal{D}(1_{F(C)},\epsilon_D)\circ F_{C,G(D)}. The equations \phi\circ\varphi=1 and \varphi\circ\phi=1 are equivalent, by the weak Yoneda Lemma, to the equations satisfied by the unit and counit of an adjunction.

The 2-functor \mathcal{V}\mathbf{-Cat}\rightarrow\mathbf{Cat} that sends an enriched category to its underlying ordinary category sends an enriched adjunction to an ordinary adjunction. The function underlying the \mathcal{V}-natural isomorphism \phi is the bijection of this underlying adjunction.

As we saw before, a \mathcal{V}-functor G has a left adjoint F if and only if \hom_\mathcal{D}(D,F(\underline{\hphantom{X}})) is representable for each D\in\mathcal{D}. Also, an enriched equivalence is an enriched adjunction whose unit and counit are both \mathcal{V}-natural isomorphisms. Just as for ordinary adjunctions, we have transformations between enriched adjunctions, a category of enriched adjunctions between two enriched categories, enriched adjunctions with parameters, and so on.

September 4, 2007 Posted by | Category theory | Leave a comment

A little aside on linear algebra

Lacking an open thread, a commenter asks…

I am studying inner product spaces. I have noticed that the inner product along with the norm and the concept of angle are defined without any reference to basis. So, I would think that in an inner product space the inner product and magnitude and direction of a vector are independent of basis.

However, I have noticed that when you introduce a basis the values of the components of vectors may change and their inner products will change.

So, is the inner product, norm and angle independent of basis?

I’ve been waiting to get back around to linear algebra, but I’d rather answer questions than ignore them, so:

Actually, inner products are not basis independent. In fact, in a certain sense, an inner product (let’s assume it’s positive definite, which is probably what you’re considering anyhow) is equivalent to a choice of a basis, up to a certain kind of equivalence. Basically, if we pick a basis we get an inner product in which each basis vector has length one and is perpendicular to every other. On the other hand, if we have a positive-definite inner product we can find such an “orthonormal” basis for it. So the two go hand in hand, and there’s generally many different inner products to put on a given vector space.

Angle is pretty much identified with the inverse cosine of an inner product, so there’s nothing new there.

Norm, however, is even more general than inner product. Every inner product gives rise to a norm (as you’ve probably seen), but there exist norms that are not given by any inner product. This shows up a lot in infinite-dimensional linear algebra, which mathematicians like to call “functional analysis”. In particular, a vector space equipped with a norm (that satisfies a technical condition called “completeness” under this norm) is called a Banach space. If the norm comes from an inner product it’s called a Hilbert space. That there are separate terms speaks to the fact that there are Banach spaces which are not Hilbert spaces. And thus there are normed vector spaces which are not inner product spaces.

September 4, 2007 Posted by | Uncategorized | 3 Comments

The Weak Yoneda Lemma

The Yoneda Lemma is so intimately tied in with such fundamental concepts as representability, universality, limits, and so on, that it’s only natural for us to want to enrich it. Unfortunately, we’re only ready to talk about bijections of sets, not about isomorphisms of \mathcal{V}-objects. So this will give back Yoneda when we consider \mathbf{Set}-categories, but in general it won’t yet have the right feel.

So let’s say we’ve got a \mathcal{V}-functor F:\mathcal{C}\rightarrow\mathcal{V}, an object K\in\mathcal{C}, and a natural transformation \eta:\hom_\mathcal{C}(K,\underline{\hphantom{X}})\rightarrow F. We can construct the composite \mathbf{1}\rightarrow\hom_\mathcal{C}(K,K)\rightarrow F(K), giving an element of the underlying set of F(K). The weak Yoneda Lemma states that this construction gives a bijection between \mathcal{V}\mathrm{-nat}(\hom_\mathcal{C}(K,\underline{\hphantom{X}}),F) — the set of \mathcal{V}-natural transformations from the \mathcal{V}-functor represented by K and the \mathcal{V}-functor F — and the underlying set of the \mathcal{V}-object F(K).

We have the function going one way. We must now take an “element” \xi:\mathbf{1}\rightarrow F(K) and build from it a natural transformation with components \eta_C:\hom_\mathcal{C}(K,C)\rightarrow F(C). And we must also show that it inverts the previous function.

First off, since F is a functor we have an arrow \hom_\mathcal{C}(K,C)\rightarrow\hom_\mathcal{V}(F(K),F(C)), which is the same as F(C)^{F(K)}. Now we can use the arrow \xi to get an arrow F(C)^\mathbf{1}, which is isomorphic to F(C). Every step here is natural in each variable by the litany of natural maps we laid down.

Now, if we compose this natural isomorphism with the identity arrow, it’s not hard to see that we get back \xi. In fact, the identity arrow i_K:\mathbf{1}\rightarrow\hom_\mathcal{C}(K,K) followed by the application of F gives the identity arrow i_{F(K)}:\hom_\mathcal{V}(F(K),F(K)). But then the exponential \hom_\mathcal{V}(\xi,1_{F(K)}) just says to compose \xi with the identity on F(K), and we’re left with \xi.

For the other direction — that starting with an isomorphism, constructing an element, and then constructing another isomorphism gives us back the isomorphism we started with — I refer you to this diagram:
Inverse for the Weak Yoneda Lemma
We start with the isomorphism \eta and construct the isomorphism along the lower-left of the diagram. The top row of the diagram is the identity (show it), and so the upper-right of the diagram is the original isomorphism \eta. I leave it to you to show that each of the three squares commute, and this our two constructions invert each other.

September 3, 2007 Posted by | Category theory | 1 Comment

Quantum Computation

Oh, I almost forgot. I’m evidently going to be advising a senior through a project about quantum computation. Basically, he reads this survey paper by Stan Gudder from the Monthly, and then writes a paper of his own (expository, not original), and talks about it to other undergrads. Should look good, if nothing else.

September 1, 2007 Posted by | Uncategorized | Leave a comment

Category Theory Seminar?

There are vanishingly few topics courses available this semester here. I don’t know if this is standard or not, but I think I might go buggy if I don’t find something to do other than calculus any my own work. You guys are great, yeah, but there’s only so much of the old give-and-take that makes mathematical discussions so exciting.

So I’m sounding out interest in an informal seminar on category theory here in the department. I suppose it’d be sort of like the Secret Russian Seminar, in that it’s not really on the books. I don’t know if I’m qualified to start up an official class or anything, given my rank.

Basically I’ll give a rough sketch of the basics of categories (not even as detailed as what I’ve done here), aiming at getting to applications to logic, semantics, physics, and (of course) topology as soon as possible. I may even try to make it accessible to good undergraduate students, if only to convert that many more people to the dark side benefits of categories.

September 1, 2007 Posted by | Uncategorized | 7 Comments

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