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.

May 19, 2008 at 3:51 pm

Yesterday evening, while watching the Angels trounce the Dodgers, I was talking with Dr. George Hockney about how I used what he agreed was a “meta-organizing principle” in a middle-school math class (where it was too high a level of abstraction) and on a midterm exam essay in my College of Ed class EDFN 440: Schooling for a Diverse, Urban Society.

George suggested that my students don’t need to know what my “meta-organizing principle” might be, yet could benefit from my using it to structure my lesson plans.

Hence this is a follow-up to Jeffrey Morton’s summary of Jim Dolan on:

http://arxiv.org/abs/math.QA/0601458

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.

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The midterm exam in my College of Ed class EDFN 440: Schooling for a Diverse, Urban Society, consisted of picking 5 out of 10 paper titles of papers discussed after oral presentations in class, from the anthology:

Fred Schultz et al, ed, Annual Editions, Education, 2008/2009, McGraw-Hill.

We were to summarize in one page essays for each of our 5 selections the major arguments and counterarguments, plus our comments. This was an open-notes exam, but I could not locate my notes at 7 a.m. when I got up and had to rush to get to my full day of high school substitute teaching, from which I went straight to Cal State L.A. and took the exam, using just my memory of the papers in question. I still scored 100% (as I have on every assignment of each of the 3 grad courses I’m taking this quarter), but I added the following after my 5 essays.

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Voluntary Addendum to the 5 Pages — Unifying Principles

6 May 08

Jonathan Vos Post

I know that you, Dr. Stephanie Evans [one of the 19 members of the advisory board to the editor of the anthology] have taken a great deal of time to read my 5 prior midterm pages, and those of the other students in this and the other section. But let me take a very short time to suggest a unifying perspective.

ALL of the papers I’ve discussed deal with 3 different metaphysical kinds of ideas and systems:

(1) stuff; (2) structure; (3) properties.

In “Affecting Public Change” [by William C. Sewell, Educational Foundations, Fall 2005] the “stuff” was the people; the “structure” was educational models (economic or intellectual, bureaucratic or free, intrinsic or extrinsic in values); and the “properties” included prosperity, decency, nonviolence, and individual fulfilment.

In “Creating Moral Schools” [by Bruce Law, Educational Horizons, Fall 2005] the stuff was intentionality, the structure was individual, collective, or coordinated; the properties were “the good”, morality, overlapping consensus, freedom from prejudice, and equality of opportunity.

In “Promoting Altruism in the Classroom” [by E. H. Mike Robinson III and Jennifer R. Curry, Childhood Education, Winter, 2005/2006] the stuff was behavior (acts) and cognition; the structure was biological, social interactive, or mental developmental; and the properties were caring, selflessness, noncontingency to reward, and the pro-social.

In “What Colleges Forget to Teach” [by Robert P. George, City Journal, Winter 2006] the stuff is civic understanding; the structure is the U.S. Constitution; and the properties include explicit versus implicit, originalism versus “living constitution”, and rigid versus flexible.

In “The Agony of American Education” [Lisa Snell, Reason magazine, April 2006] the stuff was choice; the structure was a set of constraints; and the properties were a thousand options either feared or deeply desired.

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Besides my 100% score on the 5 essays (20 of 20 points on each, this last, extra, page was marked: “20. Jonathan — your good work would have scored a 20. Thanks for your synthesis.”

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Do you have any comments?

May 19, 2008 at 6:53 pm

Jonathan:

I’d never thought much about applying the stuff/structure/properties distinction to areas outside math and physics, but it seems potentially interesting.

One comment occurs to me. It begins because I find some of the examples a bit hard to understand as examples of this principle, in this new less domain-specific form. That’s because I don’t quite get what they are, which may be at least partly a result of not having read the original papers. For example: intentionality as “stuff” I get, and “the good” as a “property” may be hard to quantify, but metaphysically I see the point. Saying the “structure” is “individual, collective, or coordinated” is confusing to me – those sound like properties the structure could have. Presumably somewhere in the background is an explanation of what it actually is.

But there is one good thing comes out of this confusion. I had been intending to say that “properties are adjectives” like the above, and so “structure is described by nouns”. This clarifies an interesting problem, because “stuff” is also described by nouns. So when you try to use this trichotomy (or, more generally, -chotomy, if there is such a term) in ordinary language, you run into this trouble. Structure really should live at a grammatical level somewhere between noun and adjective.

More to the point, English grammar subtly favours the two-level dichotomy of “stuff” with “properties” denoted by nouns and adjectives respectively. (Probably most languages do, at least Indo-European ones, but I’m no linguist). This deficiency of ordinary language would get to be a greater problem the more levels of “eka-stuff” you want to distinguish: all but the top level will tend to be denoted by nouns. On the whole, I find English (and French, and the smatterings I know of Russian, German, Spanish and Hindi) are very noun-centric languages.

So, while on this point – are the three (or more) levels actually “metaphysical” levels, as you suggest, or are they more “grammatical” or “semantic”? This is a general question I always want to ask when someone says “metaphysics”.

Thanks for the prodding to think about this.

May 27, 2008 at 3:30 pm

I agree that natural languages distort the stuff-structure-properties meta-view.

In general, we have the Sapir-Whorf hypothesis.

Language forces Physics intuition into nonproductive avenues sometimes, Mathematical cases less so (due to rigor in notation).

For example, once Maxwell gave (and Hertz experimentally observed) electromagnetic radiation as such, moving at the speed of light, English- and German- and Russian- and French-speakers naturally asked “what is waving?”

Thus, the Luminiferous Aether was imagined, to provide a noun/structure to match the properties of Maxwell’s (and, in Vector form, Heaviside’s) equations.

Failure to detect Aether wind from Earth’s orbiting then sun led to a crisis in Physics, purportedly solved (for a while) by Special Relativity.

But SR threw out Classical Physics’ notion of space and time, yet kept the classical notion of Observer. Quantum Mechanics kept the Classical Physics’ notion of space and time, but re-thought the Observer. GR and QM have not yet been reconciled.

Both are massively confused by students and the general public. What (noun) waves in EM? Is space-time the (noun) Aether? Does the universe split (verb) into alternate histories in a multiverse at each quantum event?

Sometimes the words support what the Math and Physics say, through poetic phrasing:

The central ideas of general relativity have been neatly summarized by the American physicist John Archibald Wheeler. In a now famous phrase Wheeler said:

‘Matter tells space how to curve.

Space tells matter how to move.’

Purists might quibble over whether Wheeler should have said ‘space-time’ rather than ‘space’, but as a two-line summary of general relativity this is hard to beat.

http://www.physicalworld.org/restless_universe/html/ru_4_24.html

But note how much the statement reflects English syntax and semantics:

“Matter” as stuff.

“Space” normally structure, but used as eka-stuff?

“curve” normally structure, but used as verb, structure?

“move” verb as property (trajectory or geodesic)?

And note the linguistic anthropomorphism:

‘Matter TELLS space how to curve.

Space TELLS matter how to move.’

It is as if Physics is about magic spells uttered in spoken language, at least metaphorically.

There’s a famous science fiction story in which a Navaho has a different take on Relativity, as his language does not distinguish space from time. There’s an award-winning more recent story about humans communicating with visiting extraterrestrials who have a 2-D space-time language which describes the universe dually to our languages, with Least Action as the focus. The human linguist who learns the alien language changes to the alien view of Time, and can see her (and her daughter’s) future, but cannot speak it without paradox.

Although I am a quite published Science Fiction author/editor, I am no expert on Linguistics. But I did take advanced courses in Computational Linguistics at Caltech and in grad school (where I was initially offered a research assistantship dually in Computer Science and Linguistics; but when I got to UMass/Amherst it seemed that the Computer Science chairman had failed to nail this down with the Linguistics department, so I did my M.S. thesis on Parallel Automated Theorem Proving instead, and my PhD dissertation in “Molecular Cybernetics” — what is now (1/3 century later) called Nanotechnology and Artificial Life.

Thank you for your thoughts on stuff-structure-property applied to Social Science. The papers that I picked are not especially potent; they were chosen from a grad school mid-term, rather than for their depth. Yet I do feel that the approach of Dolan and yourself is more widely applicable.

I’d love to continue this discussion.

— Prof. Jonathan Vos Post