20th Century Space and 10th Century Space

A couple of posts ago, I mentioned Max Jammer’s book “Concepts of Space” as a nice genealogy of that concept, with one shortcoming from my point of view - namely, as the subtitle suggests, it’s a “History of Theories of Space in Physics”, and since physics tends to use concepts out of mathematics, it lags a bit - at least as regards fundamental concepts. Riemannian geometry predates Einstein’s use of it in General Relativity by fifty some years, for example. Heisenberg reinvented matrices and matrix multiplication (which eventually led to wholesale importation of group theory and representation theory into physics). More examples no doubt could be found (String Theory purports to be a counterexample, though opinions differ as to whether it is real physics, or “merely” important mathematics; until it starts interacting with experiments, I’m inclined to the latter, though of course contra Hardy, all important mathematics eventually becomes useful for something).

What I said was that it would be nice to see further investigation of concepts of space within mathematics, in particular Grothendieck’s and Connes’. Well, in a different context I was referred to this survey paper by Pierre Cartier from a few years back, “A Mad Day’s Work: From Grothendieck To Connes And Kontsevich, The Evolution Of Concepts Of Space And Symmetry”, which does at least some of that - it’s a fairly big-picture review that touches on the relationship between these new ideas of space. It follows that stream of the story of space up to the end of the 20th century or so.

There’s also a little historical/biographical note on Alexander Grothendieck - the historical context is nice to see (one of the appealing things about Jammer’s book). In this case, much of the interesting detail is more relevant if you find recent European political history interesting - but I do, so that’s okay. In fact, I think it’s helpful - maybe not mathematically, but in other ways - to understand the development of mathematical ideas in the context of history. This view seems to be better received the more ancient the history in question.

On the scientific end, Cartier tries to explain Grothendieck’s point of view of space - in particular what we now call  topos theory - and how it developed, as well as how it relates to Connes’.  Pleasantly enough, a key link between them turns out to be groupoids!  However, I’ll pass on commenting on that at the moment.

…

Instead, let me take a bit of a tangent and jump back to Jammer’s book.  I’ll tell you something from his chapter “Emancipation from Aristotelianism” which I found intriguing.  This would be an atomistic theory of space - an idea that’s now beginning to make something of a comeback, in the guise of some of the efforts toward a quantum theory of gravity (EDIT: but see comments below).  Loop quantum gravity, for example, deals with space in terms of observables, which happen to take the form of holonomies of connections around loops.  Some of these observables have interpretations in terms of lengths, areas, and volumes.  It’s a prediction of LQG that these measurements should have “quantized”, which is to say integer, values: states of LQG are “spin networks”, which is to say graphs with (quantized) labels on the edges, interpreted as areas (in a dual cell complex).  (Notice this is yet again another, different, view of space, different from Grothendieck’s or Connes’, but shares with Connes especially the idea of probing space in some empirical way.  Grothendieck “probes” space mainly via cohomology - how “empirical” that is depends on your point of view.)

The atomistic theory of space Jammer talks about is very different, but it does also come from trying to reconcile a discrete “quantum” theory of matter with a theory linking matter to space.  In particular, the medieval Muslim philosophical school known as al Kalam tried to reconcile the Koran and Islamic theology with Greek philosophy (most of the “Hellenistic” world conquered by Alexander the Great, not least Egypt, is inside Dar al Islam, which is why many important Greek texts came into Europe via Arabic translations).  Though they were, as Jammer says, “Emancipating” themselves from Aristotle, they did share some of his ideas about space.

For Aristotle, space meant “place” - the answer to the questions “where is it?” and “what is its shape and size?”. In particular, it was first and foremost an attribute of some substance.  All “where?” questions are about some THING.  The answer is defined in terms of other things: my cat is on the ground, under the tree, beside the house.  The “place” of an object was literally the inner shell of the containing body that held it (which was contained by some other body, and so on - there being no vacuum in Aristotle).  So my “place” is defined by (depending how you look at it) my skin, my clothes, or the walls of the room I’m in.  This is a relational view of space, though more hard-headed than, say, Leibniz’s.

The philosophers of the Kalam had a similar relational view of space, but they didn’t accept Aristotle’s view of “substances”, where each thing has its own essential identity, on which attributes are hung like hats.  Instead, they believed in atomism, following Democritus and Leucippus: bodies were made out of little indivisible nuggets called “atoms”.  Macroscopic things were composites of atoms, and their attributes resulted from how the atoms were put together.  Here’s Jammer’s description:

The atoms of the Kalam are indivisible particles, equal to each other and devoid of all extension.  Spatial magnitude can be attributed only to a combination of atoms forming a body.  Although a definite position (hayyiz) belongs to each individual atom, it does not occupy space (makan).  It is rather the set of these positions - one is almost tempted to say, the system of relations - that constitutes spatial extension….

In the Kalam, these rather complicated and surprisingly abstract ideas were deemed necessary in order to meet Aristotle’s objections against atomism on the ground that a spatial continuum cannot be constituted by, or resolved into, indivisibles nor can two points be continuous or contiguous with one another.

So like people who prefer a “background independent” quantum theory of gravity, they wanted to believe that space (geometry) derives from matter, and that matter is discrete, but space was commonly held to be continuous.  Also alike, they resolved the problem by discarding the assumption of continuous space, and, by consideration of motion, to discrete time.

There are some differences, though.  The most obvious is that the nodes of the graph in a spin network state don’t represent units of matter, or “atoms”.  For that matter, quantum field theory doesn’t really have “atoms” in the sense of indivisible units which don’t break apart or interact.  Everything interacts in QFT.  (In some sense, interactions are more fundamental units in QFT than “particles” are - particles only (sic!) serve to connect one interaction with another.)

Another key difference is how space relates to matter.  In Aristotle, and in the Kalam, space is defined directly by matter: two bits of matter “define” the space between them.  In General Relativity (the modern theory with the “relational” view of space), there’s still room for space as an actor in its own right, like Newton’s absolute space-as-independent-variable - in other words, room for a vacuum, which Aristotle categorically denied could even conceivably exist.  In GR, what matter determines is the curvature of space (more precisely the Einstein tensor of the curvature).

Well, so the differences are probably more informative than the similarities,

(Edit: To emphasize a key difference glossed over before…  It was coupling to quantum matter which suggested quantizing the picture of space.  Discreteness of the spectrum of various observables is a logically separate prediction in each case.  Either matter or space(time) could have had continuous spectrum for the relevant observables and still been quantized - discrete matter would have given discreteness for some observed quantities, but not area, length, and so on.  So in the modern setting, the link is much less direct.)

 but the fact that theories of related discreteness in matter, space, and time, have been around for a thousand years or more is intriguing.  The idea of empty space as an independent entity - in the modern form only about three hundred years old - appears to be the real novel part.  One of the nice intuitions in Carlo Rovelli’s book on Quantum Gravity, for me at least, was to say that, rather than there being a separate “space”, we have a theory of fields defined on other fields as background - one of which, the “gravitational field” has customarily been taken for “space”.  So spatial geometry is a field, and it has some propagating (through space!) degrees of freedom - the particle associated to this field is a graviton.  Nobody’s ever seen one, mind you - but supposing they exist makes many of things easier.

To re-state a previous point: I think this is a nice aspect of categorification for dealing with space.  Extending the “stuff/structure/properties” trichotomy to allow space to resemble both “stuff” and relations between stuff leaves room for both points of view.

…

I mention this because tomorrow I leave London (Ontario) for London (England), and thence to Nottingham, for the Quantum Gravity and Quantum Geometry Conference.  It’s been a while since I worked much on quantum gravity, per se, but this conference should be interesting because it seems to be a confluence of mathematically and physically inclined people, as the name suggests.  I read on the program, for example, that Jerzy Lewandowski is speaking on QFT in Quantum Curved Spacetime, and suddenly remember that, oh yes, I did a Masters thesis (viz) on QFT in curved (classical) spacetime… but that was back in the 20th century!

It’s been a while, and I only made a small start at it before, but that whole area of physics is quite pretty.  Anyway, it should be interesting, and there are a number of people I’m looking forward to talking to.

 

Two Kinds of Categorification And Other Stuff

I’d just like to post something about a conceptual clarification that came up recently. Last week I gave the first of a couple of talks in the Algebra seminar in our department, about the ideas of structure types and stuff types, more or less as outlined in this paper which I put out a couple of years ago. It summarizes and traipses a little way beyond the matter of the 2003/2004 quantum gravity seminar at UCR, whence on this paper by John Baez and Jim Dolan, and even further back on work by André Joyal, particularly in the paper “Foncteurs analytiques et espèces de structures“, which regrettably doesn’t seem to be available either online. (I gave a blackboard version of the talk, but it was an expanded form of this one hour version.)

(Semantic side note: these espèces de structures are often referred to as “combinatorial species” in English. This is the more common translation than “structure type”, but unfortunately, it doesn’t capture the modifier “de structures“, instead choosing the more generic “combinatorial”, which makes it hard to distinguish “structure types” from “stuff types” in the Baez-Dolan sense. Also, “species” is probably over-specific as a translation of “espèces” in a way that “type” isn’t. The generic sense of “species” as “a kind of” in English is a bit recherché.)

In any case, what I’m interested in this post is the sense in which stuff types give a “categorification” of a vector space. In a nutshell, a stuff type is a groupoid over (the groupoid whose objects are finite sets, and whose morphisms are bijections). That is, it’s really a functor , which we call the “underlying set” functor. For example, consider the groupoid of all binary trees, where the underlying set is the set of nodes (or, a different example, the set of leaves). Any isomorphism between two such trees gives a bijection between the underlying sets, so this actually is a functor. Or one could take the functor , where the “underlying set” of a pair of sets is just , and likewise for morphisms. (Notice that different bijections “up above” in the bundle may give the same bijection “below” - in cases where this doesn’t happen, we have one of Joyal’s “structure types”). In some ways, it’s better to think of it as a bundle of groupoids - one fibre over each object in

The thing is, that map gives an invariant for objects in the category of groupoids, but not a complete invariant. Unlike, say, finite sets and the natural numbers. Natural numbers correspond exactly to isomorphism classes of sets - not so with groupoid cardinalities. So there’s an equivalence relation, and reducing the object set modulo that equivalence relation gives a structure - but it’s not the minimal throwing-away of information about objects that taking isomorphism classes would be.

But in any case, it’s the whole category of groupoids (over ) which gets “decategorified” down to a vector space, in that world. There is a concept of groupoid cardinality, which is given by Baez and Dolan in the paper above, and which is also linked to Tom Leinster’s definition of the Euler characteristic of a category. This adds up, over all the isomorphism classes of objects, , the reciprocals of the sizes of automorphism groups. Reasons why this is the nicest concept of cardinality are described in some of those references, but all that really matters here is that groupoid cardinality gets along with disjoint unions of groupoids (corresponding to sums of cardinalitys), and products of groupoids (which get the product of the two cardinalities). That is, the categorical coproduct and product, respectively, define operations on the set of cardinalities!

In particular, taking stuff types - groupoids over , we can take the cardinalities of the fibres over sets of each size giving the coordinate in a vector. So then is, the slice category has this “cardinality” on objects into a set, and the structure of the category gives well-defined operations on this set, turning it into a vector space. In fact, there’s an operation (weak pullback) which makes it an inner product space. (To make this work in complex cardinalities takes some fudging with phases in , but it can be done.)

The details are interesting, and I’m coming back to looking at some of this again, but what I want to point out at the moment is a more fundamental point, which has to do with the offhanded use of the handy, but imprecise, term “categorify”. With the category of (-) stuff types, we have a category with a “decategorification” map that compresses it into a vector space. This sure sounds like a “categorified vector space”. In fact, this seems to be what people who hear the term “categorification” often want it to mean: I look for a categorification of mathematical object X by finding a category which, secretly, looks like X.

The problem is, there’s another concept attached to the phrase “categorified vector space”, namely that of 2-vector space in the sense of Kapranov and Voevodski, as discussed, say, here. There’s a different level of abstraction at work here. The specific category of stuff types provides a categorification (if that indeed is the right word to use) of a specific vector space. The concept of a KV 2-vector space categorifies the concept of a regular vector space in a particular way: putting “additive” structure on objects, and “C-linear” structure on morphisms. (The Baez-Crans version does the same job in a different way).

You don’t think of a specific KV 2-vector space “decategorifying to” a specific vector space. Indeed, just taking the “minimal” equivalence relation - isomorphism classes of objects - what we get from a KV 2-vector space is more like an -module (over a rig, not a ring). Basically, 2-vectors have components which are vector spaces, and therefore classified by their dimension. The relationship between THIS kind of 2-vector space and the non-categorified concept is that real vector spaces show up as the -sets in a KV 2-vector space.

Elucidating exactly what’s going on with these two forms of categorification would be nice - perhaps somebody’s done it, but if so, I don’t know who. I also don’t know any nice conditions that tell you when you have a “category that can be mistaken for a vector space”, like stuff types: a good characterization of these things would be nice. Or again: both versions of “categorification” of vector space have special relationships to groupoids - but of two very different natures (in one, the groupoids can be interpreted as 2-vectors - in the other, there are whole 2-vector spaces associated to groupoids). Just a coincidence?

Another possibility that comes to mind would be to form some kind of hybrid structure - where the “vector spaces” which show up in the -sets in a KV 2-v.s. are secretly this fake-vector space type of category. Since both types seem to have physics-y ambitions, such a setup that combines both approaches is appealing, rather than a muddled and confusing competition for the term “categorification”.

I don’t have a good ending to this story, which is why this is a blog, not a book.

 

Correspondences and Spans

In the past couple of weeks, Masoud Khalkhali and I have been reading and discussing this paper by Marcolli and Al-Yasry. Along the way, I’ve been explaining some things I know about bicategories, spans, cospans and cobordisms, and so on, while Masoud has been explaining to me some of the basic ideas of noncommutative geometry, and (today) K-theory and cyclic cohomology. I find the paper pretty interesting, especially with a bit of that background help to identify and understand the main points. Noncommutative geometry is fairly new to me, but a lot of the material that goes into it turns out to be familiar stuff bearing unfamiliar names, or looked at in a somewhat different way than the one I’m accustomed to. For example, as I mentioned when I went to the Groupoidfest conference, there’s a theme in NCG involving groupoids, and algebras of -linear combinations of “elements” in a groupoid. But these “elements” are actually morphisms, and this picture is commonly drawn without objects at all. I’ve mentioned before some ideas for how to deal with this (roughly: is easy to confuse with the algebra of matrices over ), but anything special I have to say about that is something I’ll hide under my hat for the moment.

I must say that, though some aspects of how people talk about it, like the one I just mentioned, seem a bit off, to my mind, I like NCG in many respects. One is the way it ties in to ideas I know a bit about from the physics end of things, such as algebras of operators on Hilbert spaces. People talk about Hamiltonians, concepts of time-evolution, creation and annihilation operators, and so on in the algebras that are supposed to represent spaces. I don’t yet understand how this all fits together, but it’s definitely appealing.

Another good thing about NCG is the clever elegance of Connes’ original idea of yet another way to generalize the concept “space”. Namely, there was already a duality between spaces (in the usual sense) and commutative algebras (of functions on spaces), so generalizing to noncommutative algebras should give corresponding concepts of “spaces” which are different from all the usual ones in fairly profound ways. I’m assured, though I don’t really know how it all works, that one can do all sorts of things with these “spaces”, such as finding their volumes, defining derivatives of functions on them, and so on. They do lack some qualities traditionally associated with space - for instance, many of them don’t have many, or in some cases any, points. But then, “point” is a dubious concept to begin with, if you want a framework for physics - nobody’s ever seen one, physically, and it’s not clear to me what seeing one would consist of…

(As an aside - this is different from other versions of “pointless” topology, such as the passage from ordinary topologies to, sites in the sense of Grothendieck. The notion of “space” went through some fairly serious mutations during the 20th century: from Einstein’s two theories of relativity, to these and other mathematicians’ generalizations, the concept of “space” has turned out to be either very problematic, or wonderfully flexible. A neat book is Max Jammer’s “Concepts of Space“: though it focuses on physics and stops in the 1930’s, you get to appreciate how this concept gradually came together out of folk concepts, went through several very different stages, and in the 20th century started to be warped out of all recognition. It’s as if - to adapt Dan Dennett - “their word for milk became our word for health”.I would like to see a comparable history of mathematicians’ more various concepts, covering more of the 20th century. Plus, one could probably write a less Eurocentric genealogy nowadays than Jammer did in 1954.)

Anyway, what I’d like to say about the Marcolli and Al-Yasry paper at the moment has to do with the setup, rather than the later parts, which are also interesting. This has to do with the idea of a correspondence between noncommutative spaces. Masoud explained to me that, related to the matter of not having many points, such “spaces” also tend to be short on honest-to-goodness maps between them. Instead, it seems that people often use correspondences. Using that duality to replace spaces with algebras, a recurring idea is to think of a category where morphism from algebra to algebra is not a map, but a left-right -bimodule, . This is similar to the business of making categories of spans.

Let me describe briefly what Marcolli and Al-Yasry describe in the paper. They actually have a 2-category. It has:

Objects: An object is a copy of the 3-sphere with an embedded graph .

Morphisms: A morphism is a span of branched covers of 3-manifolds over :

such that each of the maps is branched over a graph containing (perhaps strictly). In fact, as they point out, there’s a theorem (due to Alexander) proving that ANY 3-manifold can be realized as a branched cover over the 3-sphere, branched at some graph (though perhaps not including a given , and certainly not uniquely).

2-Morphisms: A 2-morphism between morphisms and (together with their maps) is a cobordism , in a way that’s compatible with the structure of the $lateux M_i$ as branched covers of the 3-sphere. The are being included as components of the boundary - I’m writing it this way to emphasize that a cobordism is a kind of cospan. Here, it’s a cospan between spans.

This is somewhat familiar to me, though I’d been thinking mostly about examples of cospans between cospans - in fact, thinking of both as cobordisms. From a categorical point of view, this is very similar, except that with spans you compose not by gluing along a shared boundary, but taking a fibred product over one of the objects (in this case, one of the spheres). Abstractly, these are dual - one is a pushout, and the other is a pullback - but in practice, they look quite different.

However, this higher-categorical stuff can be put aside temporarily - they get back to it later, but to start with, they just collapse all the -categories into -sets by taking morphisms to be connected components of the categories. That is, they think about taking morphisms to be cobordism classes of manifolds (in a setting where both manifolds and cobordisms have some branched-covering information hanging around that needs to be respected - they’re supposed to be morphisms, after all).

So the result is a category. Because they’re writing for noncommutative geometry people, who are happy with the word “groupoid” but not “category”, they actually call it a “semigroupoid” - but as they point out, “semigroupoid” is essentially a synonym for (small) “category”.

Apparently it’s quite common in NCG to do certain things with groupoids - like taking the groupoid algebra of -linear combinations of morphisms, with a product that comes from multiplying coefficients and composing morphisms whenever possible. The corresponding general thing is a categorical algebra. There are several quantum-mechanical-flavoured things that can be done with it. One is to let it act as an algebra of operators on a Hilbert space.

This is, again, a fairly standard business. The way it works is to define a Hilbert space at each object of the category, which has a basis consisting of all morphisms whose source is . Then the algebra acts on this, since any morphism which can be post-composed with one starting at acts (by composition) to give a new morphism starting at - that is, it acts on basis elements of to give new ones. Extending linearly, algebra elements (combinations of morphisms) also act on .

So this gives, at each object , an algebra of operators acting on a Hilbert space - the main components of a noncommutative space (actually, these need to be defined by a spectral triple: the missing ingredient in this description is a special Dirac operator). Furthermore, the morphisms (which in this case are, remember, given by those spans of branched covers) give correspondences between these.

Anyway, I don’t really grasp the big picture this fits into, but reading this paper with Masoud is interesting. It ties into a number of things I’ve already thought about, but also suggests all sorts of connections with other topics and opportunities to learn some new ideas. That’s nice, because although I still have plenty of work to do getting papers written up on work already done, I was starting to feel a little bit narrowly focused.

 

Ottawa

I recently got back to London, Ontario from a trip to Ottawa, the first purpose of which was to attend the Ottawa Mathematics Conference. The other purpose was to visit family and friends, many of whom happen to be located there, which is one reason it’s taken me a week or so to get around to writing about the trip. Now, the OMC was a general-purpose conference, mainly for grad students, and some postdocs, to give short talks (plus a couple of invited faculty from Ottawa’s two universities - the University of Ottawa, and Carleton University - who gave lengthier talks in the mornings). This is not a type of conference I’ve been to before, so I wasn’t sure what to expect.

From one, fairly goal-oriented, point of view, the style of the conference seemed a little scattered. There was no particular topic of focus, for instance. On the other hand, for someone just starting out in mathematical research, this type of thing has some up sides. It gives a chance to talk about new work, see what’s being done across a range of subjects, and meet people in the region (in this case, mainly Ottawa, but also elsewhere across Eastern and Southern Ontario). The only other general-purpose mathematics conference I’ve been to so far was the joint meeting of the AMS in New Orleans in 2007, which had 5000 people and anyone attending talks would pick special sessions suiting their interests. I do think it’s worthwhile to find ways of circumventing the various pressures toward specialization in research - it may be useful in some ways, but balance is also good. Particularly for Ph.D. students, for whom specialization is the name of the game.

One useful thing - again, particularly for students - is the reminder that the world of mathematics is broader than just one’s own department, which almost certainly has its own specialties and peculiarities. For example, whereas here at UWO “Applied” mathematics (mostly involving computer modelling) is done in a separate department, this isn’t so everywhere. Or, again, while my interactions in the UWO department focus a lot on geometry and topology (there are active groups in homotopy theory and noncommutative geometry, for example), it’s been a while since I saw anyone talk about combinatorics, or differential equations. Since I actually did a major in combinatorics at U of Waterloo, it was kind of refreshing to see some of that material again.

There were a couple of invited talks by faculty. Monica Nevins from U of Ottawa gave a broad and enthusiastic survey of representation theory for graduate students. Brett Stevens from Carleton talked about “software testing”, which surprised me by actually being about combinatorial designs. Basically, it’s about the problem of how, if you have many variables with many possible values each, to design a minimal collection of “settings” for those variables which tests all possible combinations of, say, two variables (or three, etc.). One imagines the variables representing circumstances software might have to cope with - combinations of inputs, peripherals, and so on - so the combinatorial problem is if there are 10 variables with 10 possible values each, you can’t possibly test all 10 billion combinations - but you might be able to test all possible settings of any given PAIR of variables, and much more efficiently than just an exhaustive search, by combining some tests together.

Among the other talks were several combinatorial ones - error correcting codes using groups, path ideals in simplicial trees (which I understand to be a sort of generalization to simplicial sets of what trees are for graphs), heuristic algorithms for finding minimal cost collections of edges in weighted graphs that leave the graph with at least a given connectivity, and so on. Charles Starling from U of O gave an interesting talk about how to associate a topological space to an aperiodic tiling (roughly, any finite-size region in an aperiodic tiling is repeated infinitely many times - so the points of the space are translations, and two translations are within of one another if they produce matching regions about the origin of size - then the thing is to study cohomology of such spaces, and so forth).

The talk immediately following mine was by Mehmetcik Pamuk about homotopy self-equivalences of 4-manifolds, which used a certain braid of exact sequences of groups of automorphisms (among other things). I expected this to be very interesting, and it was certainly intriguing, but I can’t adequately summarize it - whatever he was saying, it proved to be hard to pick up from just a 25 minute talk. I did like something he said in his introduction, though: nowadays, if a topologist says they’re doing “low-dimensional” topology, they mean dimension 3, and “high-dimensional” means dimension 4. This is a glib but indicative way to point out that topology of manifolds in dimensions 1 and 2 is well understood (the connected components are, respectively, circles and n-holed tori), and in dimension 5 and above have been straightened out more recently thanks to Smale.

There were some quite applied talks which I missed, though I did catch one on “gravity waves”, which turn out not to be gravitational waves, but the kind of waves produced in fluids of varying density acted on by gravity. (In particular, due to layers of temperature and pressure in the atmosphere, sometimes denser air sits above less dense air, and gravity is trying to reverse this, producing waves. This produces those long rippling patterns you sometimes see in high-altitude clouds. Lidia Nikitina told us about some work modelling these in situations where the ground topography matters, such as near mountains - and had some really nice pictures to illustrate both the theory and the practice.)

On the second day there were quite a few talks of an algebraic or algebra-geometric flavour - about rings of algebraic invariants, about enumerating lines in special “blow-up” varieties, function fields associated to hyperelliptic curves, and so on - but although this is interesting, I had a harder time extracting informative things to say about these, so I’ll gloss over them glibly. However, I did appreciate the chance to gradually absorb a little more of this area of math by osmosis.

The flip side of seeing what many other people are doing was getting a chance to see what other people had to say about my own talk - about groupoids, spans, and 2-vector spaces. One of the things I find is that, while here at UWO the language of category theory is widely used (at least by the homotopy theorists and noncommutative geometry people I’ve been talking to), it’s not as familiar in other places. This seems to have been going on for some time - since the 1970’s if I understand the stories correctly. After MacLane and Eilenberg introduced categories in the 1940’s, the concept had significant effects in algebraic geometry/topology, homological algebra, and spread out from there. There was some deep enthusiasm - possibly well-founded, though I won’t claim so - that category theory was a viable replacement for set theory as a “foundation” for mathematics. True or not, that idea seemed to be one of those which was picked up by mathematicans who didn’t otherwise know much about category theory, and it seems to be one that’s still remembered. So maybe it had something to do with the apparent fall from fashion of category theory. I’ve heard that theory suggested before: roughly, that many mathematicians thought category theory was supposed to be a new foundation for mathematics, couldn’t see the point, and lost interest.

Now, my view of foundations is roughly suggested in my explanation of the title of this blog. I tend to think that our understanding of the world comes in bits and pieces, which we refine, then try to stick together into larger and more inclusive bits and pieces - the “Atlas” of charts of the title. This isn’t really just about the physical world, but the mathematical world as well (in fact I’m not really a Platonist who believes in a separate “world” of mathematical objects - though that’s a different conversation). This is really just a view of epistemology - namely, empirical methods work best because we don’t know things for sure, not being infinitely smart. So the “idealist”-style program of coming up with some foundational axioms (say, for set theory), and deriving all of mathematics from them without further reference to the outside doesn’t seem like the end of the story. It’s useful as a way of generating predictions in physics, but not of testing them. In mathematics, it generates many correct theorems, but doesn’t help identify interesting, or useful, ones.

So could category theory be used in foundations of mathematics? Maybe - but you could also say that mathematics consists of manipulating strings in a formal language, and strings are just words in a free monoid, so actually all of mathematics is the theory of monoids with some extra structure (giving rules of inference in the formal language). Yet monoid theory - indeed, algebra generally - is not mainly interesting as foundations, and probably neither is category theory.

On the whole, it was an interesting step out of the usual routine.

 

Categorification and Matter

First, the obligatory excuse found in most sporadic blogs: I haven’t taken the time to write anything here recently. I was busy for a while, between the trip to UC Davis to speak (giving a form of this talk) at the “Strings and Gravity” seminar there, and then catching up on teaching - the end of the term is coming up. There: now that’s out of the way.

Right now I want to say something a bit broader than I have been doing - somewhere between “intuitive justification” and “philosophy”. The motivation is that whenever I talk about ETQFT’s and how to see them as introducing matter into quantum gravity, there’s always some puzzlement about this “categorification” business. To people who think a lot about category theory, it may seem natural, but many of those interested in physical questions don’t fall in this category, and the whole idea of “categorifying” a theory seems like a weird, arbitrary imposition.

So talking to these different audiences has forced me to think about how to give an intuitive account of why this might be a good idea. Ideally this will not be so precise as to be incomprehensible, or so vague as to be useless. In reality, this will be at best a rough sketch of such a justification.

Stuff, Structure, and Properties

One aspect of the relationship which I wanted to comment on, one that almost seems like a pun, is the trichotomy which John Baez and Jim Dolan like to use in describing mathematical, um, widgets (I would use the more standard term “objects”, or maybe “structures”, but both of these words have technical meanings in the following) in categorical terms. This is the distinction between “stuff”, “structure”, and “properties”. (More details here and via subsequent links - some of which shows up in my first paper). Almost any usual mathematical widget can be broken down this way: (1) they consist of some “stuff”, often in the form of some sets; (2) the stuff is equipped with “structure”, often described by some functions; (3) the structure satisfies some “properties”, often expressed as equations.

For example: a group is (1) a set of elements, equipped with (2) a group operation (expressed as a function ), and a special identity element (picked out by a function from the one-element set, ), and an inverse for each element (given by an inverse function . These satisfy (3) the group axioms, which are some equations involving expressing some properties - associativity, the properties of and inverses.

In this case, the structure live inside the category of sets and functions - but similar things could be said in any other category. For instance, in the category of topological spaces and continuous functions, the same setup gives the definition of a topological group, likewise divided into “stuff” (objects, in this case topological spaces), “structure” (some morphisms), and “properties” (equations between morphisms).

Widgets which live in an -category of some kind have more of these layers - such a widget will be specified by one or more objects, equipped with specified morphisms and 2-morphisms, satisfying some equations. A monoidal category, for instance, is this kind of widget: it has a category worth of “elements”, equipped with a monoidal operation given as a functor, equipped in turn with specified 2-isomorphisms such as the “associator”, which satisfies some equations such as the Pentagon identity. There are now FOUR levels to specify. I think it was Jim Dolan who came up with the following way of extending the “stuff/structure/properties” terminology (his explanation).

The highest level - equations - always deserves the name “properties”, since they either hold, or don’t (at least, there’s a truth value associated to them - but let’s not worry about multiple-valued logics). By analogy, this suggests the data for our widget given by the -morphisms in the -category where it lives should be called “structure”. The -morphisms (which are the objects in a 1-category) should be called “stuff”.

For the , , and generally -morphisms, Jim introduces the prefix “eka”, as in “eka-stuff”, which follows Mendeleev’s nomenclature for elements predicted by his form of the periodic table of elements which were heavier than known ones. This nomenclature in turn comes from the Sanskrit “eka”, meaning “one” - the new elements were one level lower on the periodic table.

So specifying a widget in a 2-category involves “eka-stuff/stuff/structure/properties”. This is suggestive, in that it seems as if categorification - adding a new level - is like digging out a new sub-basement beneath a house. First “eka-stuff”, then “eka-eka-stuff”, and so on, to “eka^k-stuff”. Since, in many versions of -category, given two objects and , the totality of morphisms form an -category, this is somewhat correct: there is an -categorical structure describing each .

(The periodic-table analogy, I suppose, is meant to imply that the best-understood layer is the layer of equations - which describe properties. This opposes what is probably the more common intuition people have when first encountering higher categories, that we know what “objects” are, but find “higher morphisms” confusing. But when writing things concretely, it’s the highest-level morphisms which look most familiar, like functions.)

A key point here is that “stuff having structure satisfying properties” is a fairly intuitive framework for talking about things. Categorification gives us a more nuanced layering. It may seem odd to speak of “eka-stuff equipped with stuff equipped with structure satisfying properties” (even worse if you want to be consistent, and say “equipped with” instead of “satisfying”). But now the second layer - stuff, refers to 1-morphisms. Here is a layer which has some aspects we associate with “structure”: it describes relations between the eka-stuff (objects). On the other hand, it also has aspects we associate with “stuff” (it can be equipped with its own structure). When would one want something that is on the one hand something like a relational attribute between things (structure), and on the other hand something like an object in its own right (stuff).

One answer: to describe space. As a good Leibnizian, I prefer to think of space relationally: it describes how objects are situated in terms of structural relationships. On the other hand, General Relativity tells us that if we think about space, rather than spacetime, we need to describe it as having dynamics which satisfy some property. From this point of view, space is like material stuff that changes over time, according to some differential equation (classically, at least).

Matter = Stuff?

Now, part of the point of applying extended TQFT ideas to gravity is that the categorification introduces matter into the formerly empty background of topological gravity - in particular, the state of a bit matter is described by looking at the boundary conditions on a codimension-2 surface in spacetime (or codimension-1 surface in space) surrounding it. The “pun” I alluded to above is the idea that introducing matter amounts to introducing a new layer of “stuff”. Adding matter means adding “stuff”…

The pun isn’t quite dead on, however, because in the ETQFT setup, adding matter is actually adding “eka-stuff”: digging out a sub-basement on which the “stuff” of geometrized space and its dynamics can rest.

So how does the periodic table of stuff/structure/properties relate to an extended TQFT? To start with, consider the case of an ordinary TQFT in 2 dimensions. It’s well known that such TQFT’s correspond to commutative Frobenius algebras (though see e.g. this paper by Aaron Lauda and Hendryk Pfeiffer, where they explain this, and a generalization of it). That is, a TQFT defines an object with (1) Stuff: a vector space, equipped with (2) Structure: unit, counit, multiplication, and comultiplication maps, satisfying (3) Properties: a bunch of axioms, including the Frobenius relation, commutativity, and algebra axioms like associativity.

The key thing is that this correspondence comes from the fact that a 2D TQFT is a functor into from the category , which happens to be a symmetric monoidal category freely generated by one object (the circle), and some morphisms (corresponding to four cobordisms: the cap, cup, “pair of pants”, and “inverted pair of pants”), subject to just the topological relations making the circle with these maps into a “Frobenius object”. (Since the cobordisms are only defined up to diffeomorphism).

Then any actual “physical” setting will look like: a bunch of circles, say of them, connected to another bunch of circles, say of them, by some cobordism. We could call this a “string world sheet” (although not in the sense of string theory, exactly, since over there one typically has conformal structure on the cobordisms too, and talks about a CFT, not a TQFT, living on the sheet). In general, the cobordism will be an -punctured, genus- torus (with orientations that distinguish the inputs from the outputs). So if the dynamics of the “physical” world are described by a TQFT corresponding to Frobenius algebra , this topology will mean the space of states of the world is given by at the beginning and at the end (this is “stuff”). A state evolves through “time” by the morphism (”structure”) corresponding to the cobordism - a particular combination of multiplication and comultiplication maps for the

In a theory of gravity without matter, we can see three levels as well - “slices” of space with some geometric information, connected by spacetimes with geometric information, which satisfy some equations. In particular, the geometric information on spacetime has to satisfy Einstein’s equation, if we’re talking about the classical world, or some sort of Hamiltonian constraint in (some approaches to) quantum gravity. In any case, it must have some property to be admissible. So this suggests the classifications: “space geometry” - stuff; “spacetime geometry” - structure; “dynamical laws” - properties.

Categorification suggests adding to this list: “matter/boundary conditions” - eka-stuff. That is, the eka-stuff in a specific physical setting will be a “2-space of states” for matter as measured at a particular boundary. In a 3D ETQFT, for instance, the boundaries to space will be unions of circles (just as in a 2D TQFT), so this will be generated by a 2-space of states for a circle. The circle could be thought of as the boundary around a single excised particle, but in fact that only covers the irreducible 2-states: in general, it’s a boundary around some region containing a system. Space geometry relates such boundaries to each other: it is “stuff” relating the “eka-stuff”. That stuff (space geometry), in turn, can be equipped with structure - maps associated to a spacetime topology, which describe how it evolves in “time” (though a-priori there’s no special time direction - the “stuff” could equally well describe the world-sheet of the system boundary, and the structure describing how that evolution extends outward spatially).

It seems to me there’s a lot here, but to really say it properly would require being much more technically precise than I’m up to at the moment. So that’s about all I have to say about that.

 

Visiting PI - Colloquium by Robert Spekkens

One of the first things I did after arriving at PI on Wednesday (and having lunch) was to attend the colloquium talk which was being given by Robert Spekkens. It was called “Why the Quantum?”, but as he described it, the real point of the talk was to take a close look at the features of quantum physics that are commonly considered “weird” or “mysterious” and see what’s really innovative in the departure from classical physics. For the most part, “physics” here means “mechanics”, but he also touched on optics, theory of computation, and briefly on electromagnetism and gravity in a more speculative way.

The main message of his talk is that very few of the things about quantum physics which seem strange are really all that innovative. He showed this by describing a kind of classical theory that has many of them - interference, noncommuting observables, entanglement, “wavefunction collapse”, wave-particle duality, teleportation and a no-cloning theorem, superposition of states, and so forth. All of these, he told us, will show up in a model based on a classical mechanical system, where the “quantum” theory is a theory of probability distributions (or, equivalently, of the knowledge of observers about a classical system) subject to a restriction about what distributions are allowed.

The point is to start with some classical system: let’s say it’s a mechanical system of some moving particles. Then there’s a configuration space of all the possible (classical) configurations of the system - one point in this space for each configuration. Classical mechanics is then about defining a “flow” on this space, which tells you where a point will move over time (how the system will go from one configuration to another). Then Liouville mechanics is about probability distributions in this space: you might not know exactly which configuration the system is in, but you have a way of estimating the probabilities. Then you impose the restriction that the only allowed probability distributions are ones for which the products of the variances for conjugate variables are at least Planck’s constant. (Actually, I think Spekkens formulated this differently, but that’s about what it amounts to, as I understand it.) The result is equivalent to “Gaussian quantum mechanics” - one where probability distributions are all Gaussians.

This also puts limits on what the rule for evolving states can be: any rule for how individual states evolve over time also gives a result for how probability distributions evolve over time. (Picture a cloud of ink, with varying density, flowing along in moving water - knowing the flow lines tells you where the cloud goes.) If there are restrictions on what kind of probability distributions can be set up, these have to be preserved over time - otherwise, you could set up an allowed distribution, and then wait until it evolves into a disallowed one. In particular, for Gaussian quantum mechanics, he told us that systems with a quadratic Hamiltonian will satisfy this condition.

The important fact here is that this is a “realist” interpretation. It says the quantum mechanical uncertainty reflects that QM is a theory about your knowledge of the state of the system, which, however, really exists. Often in quantum mechanics, one defines a “wave function” as a function living on configuration space (complex-valued, not real-valued like a probability density, but a function nonetheless). However, it’s now pretty standard to think of this wave function as the “real” state of the system - the view that it represents a state of knowledge was popular for a while, but ran into various problems in the form of experiments that are hard to account for, such as Bell inequality violations. The point of the talk was to see just how many of the “strange” features of quantum mechanics are genuine problems for this view, and to show the answer is “not many”.

The features he claimed are really mysterious from this point of view are fairly few: Bell inequality violations, some no-go theorems for models of physics involving local hidden variables such as the Kochen-Specker Theorem, and a few others. So Spekkens’ suggestion was that this concept of quantum mechanics as a theory of probability with an “epistemic” restriction (i.e. limits on what’s knowable) might be salvaged if the underlying classical theory were non-local - and perhaps had some other odd features yet to be precisely delineated - to begin with. However, it might not have to be terribly strange apart from that, since quantum mechanical features like interference and superposition of states all show up in the restricted statistical picture.

The gist of his argument then seemed to be that to really straighten out some foundational issues in quantum physics, one approach would be: (a) come up with a well-founded justification for the assumption about restrictions on possible probability distributions, and (b) come up with at least one (and as few as possible) other principles to account for the remaining mysterious things - he also suggested they all seem to have something to do with “contextuality”. As I understand it, this last is the idea that an observable might have definite, but multiple, values - and that which values are seen depend on which groups of observables are measured together. I don’t know what, if anything, to make of that oddball-sounding idea.

However, he did argue that in some cases at least, the restriction can be justified by the observer effect: you have to look at a system using some apparatus, whose state you don’t know completely, and which interferes with the system in order to observe it (for instance, measuring the position of a particle by scattering it off another one, whose state is partly unknown, and imparts an unknown momentum).

My overall reaction to the talk is that it’s interesting to know that realist interpretations of quantum physics (where the “reality” is more or less classical, and quantum effects some kind of afterthought, or epistemic effect) aren’t as dead as they might have seemed. However, the view that says classical physics emerges as some kind of limiting case of quantum effects seems better developed, at least mathematically, than the reverse. As for his claim that we “understand” the classical picture “physically”, whereas it’s not so for the quantum picture - I personally can only agree that’s true for me, but I don’t entirely see what you can conclude from that.

The bottom line seems to be that there are still problems in epistemology. I suspected as much already - though I’m not sure if I “knew” it, whatever that means.

 

Visiting Perimeter Institute

Once again, I keep meaning to write some less math-heavy posts, if for no other reason than to keep in the habit of thinking up things to write in here. Now is a good occasion to do this, since I’m visiting at the Perimeter Institute in Waterloo to give a talk called “Extended Topological Quantum Field Theories and Quantum Gravity” at the quantum gravity seminar on Thursday (the 28th). This is basically an updated and refined version of the talk I gave for my thesis defense, in which I’ve tried to make more of the link to physics - in particular, to BF theory, and to 3D quantum gravity. This turns out to be hard to do in an hour-long talk and still cover things adequately. Still, I find it worthwhile to get the point of view of real physicists on these apparently physics-related ideas, after thinking about them as a mathematician for some time.

After I arrived, I had lunch with a bunch of the quantum gravity people here. The conversation ranged from hunting for jobs, through cultural differences between Europe, Canada, and the US (a standard conversation to be had anywhere in Canada at the drop of a hat), all the way over to “Why is spacetime 4-dimensional?” Lee Smolin put this last one to me when I was describing how categorification is related to considering higher co-dimensions of spacetime/space/surfaces in space. It’s a reasonable question, though not one I have any answer to. But when you cook up a theory - like this ETQFT stuff - which in principle works in any number of dimensions, and you want it to be physical, you’re left wondering “why so few dimensions?”

Okay - it’s not the main point of what I’m doing here, but it’s a nice light question to blog about, since I don’t pretend to have even a good guess at the answer.

It takes a certain mentality to think that 4 dimensions is astonishingly few - however, I have that mentality, as do many mathematicians. You can work with infinite-dimensional spaces in mathematics - why should “real”, “physical” space only have four? Actually, the segue into this had to do with the question of why all the Lie groups that turn up in physical gauge theories are so tiny - , , - rather than, say, , which describes rotations in a 745 (complex) dimensional space. Again: gauge theory makes just as much sense with big gauge groups as small ones - so what’s special about the low dimensions?

Well, I don’t know the answer - but it’s the kind of question mathematicians probably should be asked more often. We’re perfectly happy to deal with a 745 dimensional space and not worry about the fact that it’s non-physical. But if mathematics really underlies physics in any deep way, there should be some good mathematics in the answer.

There were some possibilities tossed around: what if the exceptional group really does turn out to be important in fundamental physics, and the real gauge group of the right physical theory has to lie inside it somewhere? Then there’s an upper bound on how many dimensions you can have - though, unfortunately, is 248-dimensional, so the upper bound is a bit high. (Mind you, the symmetries of 4D space is, in itself, a 10-dimensional group, so things are not quite as bad as they appear - but still worse than they should be). There’s also no obvious reason why should have such a special role.

A more physics-y answer is that in 5D and higher, you don’t get confinement - quarks and gluons just fly around like a dilute gas, and there would be no matter in the sense we know it. This is a great concise description of why we should be happy to live in a 4D spacetime. The objection to this is that it’s basically an appeal to the anthropic principle: “If space weren’t 4D, we wouldn’t be here to wonder why.” If you’ve read Lee Smolin’s most recent book, you’ll know he doesn’t care for appeals to the anthropic principle. Neither do I, for that matter. If you assume that every possible universe actually exists (which is at least metaphysically parsimonious - no need for two separate categories of “possible” and “actual”), the anthropic principle is undeniable. The problem is, it doesn’t predict very much until you work out enough about what universes are possible that you might as well just try to answer the question for its own sake. Still, maybe it’s just true that there are a huge number of actual universes, and some of them are no good for intelligent life. But that just means the question has no answer, so you might as well give up. It doesn’t take you anywhere. So suppose there’s a reason: what could it be?

In 3 and 4 dimensions, there are regular polyhedra - or, equivalently, discrete subgroups of the rotation group - that don’t correspond to the series which always exists. In 2D, there are infinitely many regular polygons, and in all dimensons, there are simplexes, cubes, and duals of cubes… but in 3 and 4D there are some extras, all of which boil down to the icosahedron, its dual, or things you can construct from it in 4D. Why this should make any difference, I have no idea.

And there are a couple of other special things in low dimensions, which are no more obviously relevant, but seem compelling to me, perhaps because I’m a mathematician…

In 4 dimensions, but no other dimensionality, there are “exotic” which are homeomorphic but not diffeomorphic to the usual . The heuristic explanation for why (which is as much as I really grasp) is that 4D is “big enough” for complicated twisty things to exist, but “too small” for there to always be room to untangle them - so only in 4D can “things be complicated”. Which is suggestive, but hardly a full answer.

4 dimensions is the only case where the classification of manifolds is not understood (now that the Poincaré conjecture has been settled - there were still some lingering doubts last I heard, but they seem to be evaporating day by day). in 2D, manifolds are basically just toruses with some genus; in 3D manifolds can be cut up into pieces each of which can be geometrized (a la Thurston). In 5D and higher, you can classify (in principle) manifolds by constructing them via surgeries. The reason this doesn’t work in 4D is that surgeries building new manifolds correspond to cobordisms between the input and output manifolds, and in 5 or more dimensions, cobordisms are rather trivial (actually, this only refers to cobordisms where the inclusions of the source and target manifolds are homotopy equivalences, which isn’t totally general).

This last bit seems the most intriguing to me, since I’ve been thinking about TQFT’s and ETQFT’s, which are field theories living on cobordisms. But that still doesn’t add up to an answer to the physical question. It would be nice to understand, for instance, whether the above fact means anything helpful in terms of the physics of such a theory.

Anyway, I’ll try to write up something about those theories from a physical point of view after I’ve had a chance to chit-chat about them with some physicists after my talk. It probably won’t answer this rather vague and (perhaps?) unanswerable question, but there seem to be some interesting things to say. Maybe before then (but after I’ve had a chance to give my talk, no doubt!) I’ll also give a little write-up of the colloquium talk by Robert Spekkens I attended today about foundations of quantum mechanics.

 

2-Hilbert Spaces - pt 2

So I gave a little talk shortly before leaving London for Christmas. I had mostly written it up, but then I’ve been on the road for a while in Montreal, Ottawa, and Calgary, without consistent net access. However, now I have a moment to put this up.

The talk carried on from the previous one I described last post. It began to move in the direction of representation theory of 2-groups on 2-vector spaces and 2-Hilbert spaces, but didn’t get that far. This was partly because I had to finish describing what 2-linear maps and 2-maps look like for such spaces, and then because I had to explain about 2-groups and give some examples. I’ll say more about the representation-theory stuff in January. But here I’ll just summarize at least the rest of the description of the category , and also by describing 2-linear maps and so forth. Then I’ll comment a little more philosophically about what these are about.

So I explain how there’s a 2-vector space (in some suitable sense, not the KV sense) of measurable fields of Hilbert spaces on a space , analogous to the vector space of complex functions on a space. Also similarly, given a measure on , we get an inner product. Then there’s a (2-)Hilbert space where this inner product is always well-defined (as a complex scalar, or a genuine Hilbert space - which is the equivalent of a scalar at the next level up).

Well, then Crane and Yetter’s paper describes constructively how to get 2-linear maps (additive, linear functors) between such 2-vector spaces. They don’t as far as I can see, show that all functors arise this way, but it seems likely. The way is to say you get a functor from:

1) A measurable field of Hilbert spaces (this is similar to the linear maps between KV 2-vector spaces, which are like matrices of vector spaces)

2) A -indexed family of measures on - these give you the measures you need to do the “inner product” involved in “matrix multiplication” at each (note that this stuff is only well-defined up to sets of measure 0, as usual). So we have, on objects:

and a related expression for morphisms, using the identity on .

It’s probably worth pointing out that the measures on are used in the direct integral here, and so their only real role is to define the inner product on - the underlying vector space at each point in the new field would be the same no matter what these measures were (up to the fact that if the resulting inner product is degenerate, we need a quotient space where it’s not).

So this gives 2-linear maps, which are functors. Natural transformation between these functors come from the fact that is itself a category, and in fact a 2-vector space in the sense we’re using here ( is “enriched over itself”). So morphisms between these fields of Hilbert spaces basically amount to fields of bounded operators as usual. This is actually not quite right, because we need to account for the different measures: basically, you use a measure which is the geometric mean of those associated to source and target - check out Crane and Yetter’s paper if you want the details.

That finishes up a summary of how 2-Hilbert spaces work. The next thing I’ll be talking to our group about is how to use these for a categorified form of representation theory.

…

But first, what is the point of all this stuff? Not yet asking about representation theory in this setting - why is it interesting enough to bother? It’s worth thinking about what a categorification of a Hilbert space is supposed to be. In particular, let’s try locating them in the world of quantum mechanics.

A quantum system is usally portrayed as having states represented by vectors in a Hilbert space. The only things you can “do” to states involve applying operators to the whole space: project them into subspaces, “rotating” them by some unitary evolution operator, and so on. In a 2-Hilbert space, states, or “2-vectors” are objects in a category, which means there are not only these “macro” operations on the whole space, but also morphisms between any two states you pick. In fact, this is the source of the inner product on a 2-Hilbert space - there is a Hilbert space (in the usual sense) of morphisms between any two states, and in the world of 2-Hilbert spaces, this is the equivalent of a scalar.

In QM, the inner product is telling you an amplitude to observe a system in state if it was set up in state - this is saying something about “how related” is to . The categorifed picture saying this is just makes more explicit what kind of relationship this is.

Now, if you happen to pick the same vector to start and end with, considering , what this is saying is that there’s some bunch of “symmetry operations” on a state. (Taking just the invertible ones gives an actual symmetry group for a given state.) This is saying that “state 2-vectors” have some internal degrees of freedom. Their amplitudes give a measure of how many such degrees of freedom there are.

The fact that a 2-Hilbert space is described as an enriched category means that the usual picture of a quantum system returns when you look in individual components of a state 2-vector. In particular, the coefficients of a 2-state vector can be thought of as Hilbert spaces representing a system in that particular component. So, for instance, part of the big project I’m describing in these notes is to depict quantum gravity (at least in 3 dimensions) as an extended TQFT, which represents a physical system with these 2-Hilbert spaces. A 2-state vector here describes the situation on a boundary of space - matrix elements of a 2-linear map are Hilbert spaces of connections on a given manifold interpolating between chosen boundary states. Natural transformations between 2-linear maps are what give amplitudes for spacetimes joining such slices of space.

So what is a state 2-vector? All these properties should fit together into some nice scheme: classical configurations can exist in a “2-state” in some kind of superposition, where each configuration gets its own internal degrees of freedom. The inner product emerges naturally from this, considering morphisms between 2-states. Every morphism between 2-states has to respect the classical configurations, giving for each one a map between the internal spaces associated to it in the two 2-states. Is there a more elegant way to sum this up? Probably so, but at the moment I don’t quite see how to put it.

However: next time, I’ll carry on with some representation theory.

 

Musing, and a Tale of Two Kauffmans

Writing sizeable chunks of math blog takes longer than I expected. Here are a few non-intensive things that occurred to me.

…

While I was walking home from the UWO campus, I was reminded of the nature of Canada in late November: everything, from sky to plantlife to earth, is in shades of grey, brown, ochre, and the occasional desaturated greenish-whatever. Autumn leaves have pretty much stopped falling, and are on the ground turning greyish versions of whatever colours they were before. There are whole vistas of bare branches, dead underbrush, and so on.

Which seems dreary for a while, until you’re immersed in it, as I am on the particular route I walk home, along London, Ontario’s Thames River (not to be confused with the River Thames in London, England), which is lined with parks. Then, after a while, all the subtle differences in shading and texture start to jump out at you more and more, until brownish moss on a tree under overcast late-afternoon light is vibrant green, a patch of snow is glowing bluish white, the occasional flicker of sunset through the cloud cover is warm pumpkin-orange, that one particular bush’s leaves look startlingly red… and then you see something artificial, like someone’s nylon jacket, or a kid’s plastic play-structure, and their colours look implausibly oversaturated, like a badly photoshopped picture.

Which got me to thinking about fine distinctions that seem drab outside their context - the way these colours look at first. Or nitpicky, like having 30 different words for “cold” and the different qualities it can have, or recognizing 15 different types of snowflake from a distance. Coming back to Canada after several years in California, I noticed all this specialized knowledge I’d forgotten about, and seems terribly arcane outside its native habitat. It occurred to me that this is how mathematics probably seems to outsiders - like physicists, or statisticians… (I jest)

For instance: I often have the experience of using the term “categorification” in describing something I’m doing - often in scare-quotes, followed by some kind of explanation - only to have it echoed back as “categorization”, and wonder whether to risk pedantry and explain that they’re not the same thing at all. “Categorification, not to be confused with…”

…

On another note, I went looking for this paper by Carter, Kauffman and Saito, on a kind of invariant of 4-manifolds which generalizes 3D Dijkgraaf-Witten invariants, on the supposition that it would be closely related to some things I’ve been thinking about, from a diagrammatic point of view I’ve not paid much attention to in the last year or so. As I was looking through seach results, I noticed a paper from about 10 years ago by Kauffman and Smolin with an interesting sounding title, A Possible Solution to the Problem of Time in Quantum Cosmology. Since Lee Smolin has written on linking topological field theory and quantum gravity, I guessed it would also be interesting to look at. Only after reading the first few pages did I notice that the first listed author was not Louis Kauffman, who studies knot theory (and things tangent thereto), but Stuart Kauffman, who studies biocomplexity and complex systems.

I happen to be interested in the work of both Kauffmans - more immediately and professionally that of Louis, but I also read a couple of Stuart’s more accessible books, “At Home in the Universe”, and “Investigations” - and since the paper was short, I finished reading it. The basic premise is that the configuration space for 4D quantum gravity may not be constructible by any finite procedure (classifying spin networks, they say, might present a problem; doing path integrals over all 4-manifold topologies certainly does). So the “problem of time”, that there’s no role for time in describing dynamics in terms of paths through a configuration space, wouldn’t make sense - at least for a constructivist. (Or indeed a constructivist, though of course they shouldn’t be confused.) One thing that threw me off in noticing which Kauffman was involved was that part of this portion of the argument was about classifying knots.

That cleared itself up when they got to the part proposing a solution - that the total space of possible states isn’t a-priori given, but time re-enters the situation as the universe evolves, at each time step having some amplitude to move into each configuration in a (newly defined!) space called the adjacent possible. Having read Stuart K.’s books, this was when I realized my mistake - he describes this concept in “Investigations” in the context of a biosphere, or an economy, where a theorist also doesn’t have an explicit description of all possible future states given in advance.

It seems like this idea has a lot in common with type theory as a solution to Russell’s paradox: the collection of all sets isn’t a set, and so to get at it, sets are generated starting with nothing in successive stages. Whether this also doubles as a solution to the problem of time, I don’t know. In any case, it’s an interesting idea. It definitely would be a problem to have to do path integrals over a space of all topologies for 4-manifolds, when these can’t be classified, so some sort of suggestions are definitely a good thing here.