The Unapologetic Mathematician

Category Representations

We’ve seen how group representations are special kinds of algebra representations. But even more general than that is the representation of a category.

A group is a special monoid, within which each element is invertible. And a monoid is just a category with a single object. Similarly, an $\mathbb{F}$-algebra is just like a monoid but enriched over the category of vector spaces over $\mathbb{F}$. That is, it’s a one-object category with an $\mathbb{F}$-bilinear composition. It makes sense to regard both of these structures as categories of sorts. A representation will then be a functor from one of these categories.

The clear target category is $\mathbf{Vect}_\mathbb{F}$. So what’s a functor $\rho$ from, say, a group $G$ (considered as a category) to $\mathbf{Vect}_\mathbb{F}$? First the single object of the category $G$ picks out some object $V\in\mathbf{Vect}_\mathbb{F}$. That is, $V$ is a vector space over $\mathbb{F}$. Then for each arrow $g$ in $G$ — each group element — we have an arrow $\rho(g)\in\hom_\mathbb{F}(V,V)$. Since $g$ has to be invertible, this $\rho(g)$ must be invertible — an element of $\mathrm{GL}(V)$.

What about an algebra? Now our source category $A$ and our target category $\mathbf{Vect}_\mathbb{F}$ are both enriched over $\mathbf{Vect}_\mathbb{F}$. It only makes sense, then, for us to consider $\mathbb{F}$-linear functors. Such a functor $F$ again picks out a single vector space $V$ for the single object of $A$ (considered as a category). Every arrow $a$ in $A$ gets sent to an arrow $\alpha(a)\in\hom_\mathbf{F}(V,V)$. This mapping is linear over the field $\mathbb{F}$.

So what do category representations get us? Well, one thing is this: consider a combinatorial graph — a collection of “vertices” with some directed “edges” joining them. A path in the graph is a sequence of directed edges joined tip-to-tail, and the collection of all paths in the graph constitutes the “path category” of the graph (exercise: identify the identity paths). A representation of this path category is what mathematicians call a “quiver representation”, and they’re big business.

More interesting to me is this: the category $\mathcal{T}ang$ of tangles (or $\mathcal{OT}ang$ of oriented tangles, $\mathcal{F}r\mathcal{T}ang$ of framed tangles, or $\mathcal{F}r\mathcal{OT}ang$ of framed, oriented tangles). This is a monoidal category with duals, as is $\mathbf{Vect}_\mathbb{F}$, and so it only makes sense to ask that our functors respect those structures as well. We don’t ask that it send the braiding to the symmetry on $\mathbf{Vect}_\mathbb{F}$, since that would trivialize the structure.

So what is a representation of the category $\mathcal{T}ang$? It is my contention that this is nothing but a knot invariant, viewed in a more natural habitat. A little more generally, knot invariants are the restrictions to knots (and links) of functors defined on the category of tangles, which can often (always?) be decategorified — or otherwise rendered down — into representations of $\mathcal{T}ang$. This is my work: to translate existing knot theoretical ideas into this algebraic language, where I believe they find a better home.

October 27, 2008 -

1. […] of algebras, and the special case of representations of groups, but with an eye to the categorical interpretation. So, representations are functors. And this immediately leads us to the category of such functors. […]

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2. I certainly wish I had an extra hour or two a day to follow this stuff carefully as you produce it, rather than occasionally going over random stretches from the past – I have a sort of suspicion that knot theory might be relevant to linguistics (what I do), based on the fact that Kosta Dosen once made a remark about knot theory being ‘the real reason why proofnets work’, and proofnets having some uses in linguistics, according to some people, but it’s a long way from there to anything publishable… Comment by Avery Andrews | October 28, 2008 | Reply

3. Not knowing what a proofnet is, this is only a guess, but an educated one: proofnets (like much in logic) can be expressed in the language of monoidal categories with extra structure, and the universal braided monoidal category with duals is that of tangles. I have a feeling that’s what the connection is. Comment by John Armstrong | October 28, 2008 | Reply

4. I think that’s basically correct, although I don’t adequately control a lot of the details. Jonathan Cohen’s blog (that logic blog), currently dormant due to thesis-writing, has quite a lot of stuff about them and related topics.

So what a proofnet is is a graph-structure based on a bunch of assumptions and a desired conclusion, which you then complete by drawing in links between the atomic formulas in the assumptions and conclusion, and if some conditions are satisified, the result constitutes a proof of the conclusion from the assumptions.

Intruigingly for me, these things can be made to look very much like certain kinds of diagrams that linguists have tended to produce anyway, and, luckily for me, linguistics so far only seems to need the absolutely simplest kind of proofnet. But that doesn’t make knot theory any easier… Comment by Avery Andrews | October 28, 2008 | Reply

5. Proof nets are essentially so-called Kelly-Mac Lane graphs, which are graphs whose edges link together argument placeholders in functors that are related by extranatural transformations, which can be natural transformations in the ordinary sense, or dinatural transformations. A simple example would be a proof net for an evaluation map $Y^X \otimes X \to Y$

which is natural in the ordinary sense in the argument $Y$ and dinatural in the argument $X$. The two instances of $X$ and the two instances of $Y$ would each be linked by edges in the appropriate KM graph.

The evaluation map is really a generalization of modus ponens in logic, where the tensor plays the role of a generalized conjunction and the exponential plays the role of material implication $X \Rightarrow Y$. The evaluation morphism can be thought of as a particular deduction which witnesses the entailment $(X \Rightarrow Y) \wedge X \to Y$. But a key point in categorical proof theory is that one can and should distinguish between different deductions or proofs of the same entailment, i.e., different ways of defining morphisms lying over the same entailment. KM graphs or proof nets are the chief graphical means of laying bare these distinctions.

A simple example is that even in very minimal fragments of logic, one can point to two distinct ways of inferring an entailment of the form $x \wedge x \to x \wedge x$; at the categorical level these would correspond to an identity map and to a symmetry map $\sigma: x \otimes x \to x \otimes x$. These morphisms have different KM graphs: if you draw the domain over the codomain, then the identity map, being an instance of the transformation $x \otimes y \to x \otimes y$, would have a KM graph or proof net which links up the arguments with two vertical lines, whereas for the second morphism, the KM graph would involve crossing edges. Here we are dealing with a fragment of logic which can be appropriately “categorified” (and yes, I am using the term appropriately!) in the form of symmetric monoidal categories or, since we have brought up knot theory, braided monoidal categories.

The KM graphs or proof nets which pertain to that fragment of logic which can be categorified to the theory of braided monoidal categories with duals (or tortile categories, etc.) are really just the familiar tangle diagrams. The KM graphs or proof nets here would be the appropriate invariants which decide coherence for free braided monoidal categories with duals, free tortile categories, what have you.

Speaking of linguistics: it is no accident that Lambek, who pioneered the application of logical techniques to the study of coherence problems in category (and who showed back in 1968 that KM graphs decide coherence for monoidal biclosed categories without units — what he called “residuated categories”), was and is very interested in the application of these sorts of categorical structures to linguistics. If you haven’t seen his writings on this topic, I’d strongly urge you to have a look!

That said, it does sound to me that Kosta is stretching it just a bit, or at least that kind of pronouncement carries an air of mystery to me. Proof nets “work” without any reference to true “knottiness” — they work fine for many symmetric (as opposed to braided) monoidal structures, as in the cases of symmetric monoidal closed categories, or *-autonomous categories (which is the categorical milieu for linear logic, which is the fragment of logic for which Girard invented the term “proof net”). So I’m not quite sure what Kosta is on about here. I would agree however that there is great potential for much more extensive interactions between low-dimensional topology and categorical logic. Comment by Todd Trimble | October 29, 2008 | Reply

6. I guess ‘the real reason why …’ means ‘obscure oracular pronouncement coming up’! I basically know what KM diagrams etc. are, but don’t have enough facility (at least yet) to follow much of anything done with them.

Chomsky vis a vis Lambek might be an interesting story some day; apparently Lambek was amazed to discover in the late 80s that there was this collection of linguists working on his Categorial Grammar, which he assumed had been completely overwhelmed and buried by the Chomskian onslaught, and things based on or similar to it have continued to propagate steadily. Most current linguistic theories can be regarded as amalgams of both of their ideas. Comment by Avery Andrews | October 29, 2008 | Reply

7. […] in the category of algebras over . But let’s consider this category itself. We also said that an algebra is a category enriched over , but with only one object. So we should really be thinking about the category of […]

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