# April 2, 1999 {#week132}

Today I want to talk about $n$-categories and quantum gravity again. For
starters let me quote from a paper of mine about this stuff:

1) John Baez, "Higher-dimensional algebra and Planck-scale physics", to appear in _Physics Meets Philosophy at the Planck Scale_, eds. Craig Callender and Nick Huggett, Cambridge U. Press. Preprint available as [`gr-qc/9902017`](https://arxiv.org/abs/gr-qc/9902017).

By the way, this book should be pretty fun to read --- it'll contain
papers by both philosophers and physicists, including a bunch who have
already graced the pages of This Week's Finds, like Barbour, Isham,
Rovelli, Unruh, and Witten. I'll say more about it when it comes out.

Okay, here are some snippets from this paper. It starts out talking
about the meaning of the Planck length, why it may be important in
quantum gravity, and what a theory of quantum gravity should be like:

> Two constants appear throughout general relativity: the speed of light
> $c$ and Newton's gravitational constant $G$. This should be no surprise,
> since Einstein created general relativity to reconcile the success of
> Newton's theory of gravity, based on instantaneous action at a
> distance, with his new theory of special relativity, in which no
> influence travels faster than light. The constant $c$ also appears in
> quantum field theory, but paired with a different partner: Planck's
> constant $\hbar$. The reason is that quantum field theory takes into account
> special relativity and quantum theory, in which $\hbar$ sets the scale at
> which the uncertainty principle becomes important.
>
> It is reasonable to suspect that any theory reconciling general
> relativity and quantum theory will involve all three constants $c$, $G$,
> and $\hbar$. Planck noted that apart from numerical factors there is a
> unique way to use these constants to define units of length, time, and
> mass. For example, we can define the unit of length now called the
> 'Planck length' as follows:
> $$L = \sqrt{\frac{\hbar G}{c^3}}$$
> This is extremely small: about $1.6\times 10^{-35}$ meters. Physicists have
> long suspected that quantum gravity will become important for
> understanding physics at about this scale. The reason is very simple:
> any calculation that predicts a length using only the constants $c$, $G$
> and $\hbar$ must give the Planck length, possibly multiplied by an
> unimportant numerical factor like $2\pi$.
>
> For example, quantum field theory says that associated to any mass $m$
> there is a length called its Compton wavelength, $L_C$, such that
> determining the position of a particle of mass $m$ to within one Compton
> wavelength requires enough energy to create another particle of that
> mass. Particle creation is a quintessentially quantum-field-theoretic
> phenomenon. Thus we may say that the Compton wavelength sets the
> distance scale at which quantum field theory becomes crucial for
> understanding the behavior of a particle of a given mass. On the other
> hand, general relativity says that associated to any mass $m$ there is a
> length called the Schwarzschild radius, $L_S$, such that compressing an
> object of mass $m$ to a size smaller than this results in the formation
> of a black hole. The Schwarzschild radius is roughly the distance
> scale at which general relativity becomes crucial for understanding
> the behavior of an object of a given mass. Now, ignoring some
> numerical factors, we have
> $$L_C = \frac{\hbar}{mc}$$
> and
> $$L_S = \frac{Gm}{c^2}.$$
> These two lengths become equal when $m$ is the Planck mass. And when
> this happens, they both equal the Planck length!
>
> At least naively, we thus expect that both general relativity and
> quantum field theory would be needed to understand the behavior of an
> object whose mass is about the Planck mass and whose radius is about
> the Planck length. This not only explains some of the importance of
> the Planck scale, but also some of the difficulties in obtaining
> experimental evidence about physics at this scale. Most of our
> information about general relativity comes from observing heavy
> objects like planets and stars, for which $L_S\gg L_C$. Most of our
> information about quantum field theory comes from observing light
> objects like electrons and protons, for which $L_C\gg L_S$. The Planck
> mass is intermediate between these: about the mass of a largish cell.
> But the Planck length is about $10^{-20}$ times the radius of a proton!
> To study a situation where both general relativity and quantum field
> theory are important, we could try to compress a cell to a size
> $10^{-20}$ times that of a proton. We know no reason why this is
> impossible in principle. But we have no idea how to actually
> accomplish such a feat.
>
> There are some well-known loopholes in the above argument. The
> 'unimportant numerical factor' I mentioned above might actually be
> very large, or very small. A theory of quantum gravity might make
> testable predictions of dimensionless quantities like the ratio of the
> muon and electron masses. For that matter, a theory of quantum gravity
> might involve physical constants other than $c$, $G$, and $\hbar$. The latter
> two alternatives are especially plausible if we study quantum gravity
> as part of a larger theory describing other forces and particles.
> However, even though we cannot prove that the Planck length is
> significant for quantum gravity, I think we can glean some wisdom from
> pondering the constants $c$, $G$, and $\hbar$ --- and more importantly, the
> physical insights that lead us to regard these constants as important.
>
> What is the importance of the constant $c$? In special relativity, what
> matters is the appearance of this constant in the Minkowski metric
> $$ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2$$
> which defines the geometry of spacetime, and in particular the
> lightcone through each point. Stepping back from the specific
> formalism here, we can see several ideas at work. First, space and
> time form a unified whole which can be thought of geometrically.
> Second, the quantities whose values we seek to predict are localized.
> That is, we can measure them in small regions of spacetime (sometimes
> idealized as points). Physicists call such quantities 'local degrees
> of freedom'. And third, to predict the value of a quantity that can
> be measured in some region $R$, we only need to use values of quantities
> measured in regions that stand in a certain geometrical relation to $R$.
> This relation is called the 'causal structure' of spacetime. For
> example, in a relativistic field theory, to predict the value of the
> fields in some region $R$, it suffices to use their values in any other
> region that intersects every timelike path passing through $R$. The
> common way of summarizing this idea is to say that nothing travels
> faster than light. I prefer to say that a good theory of physics
> should have *local degrees of freedom propagating causally*.
>
> In Newtonian gravity, $G$ is simply the strength of the gravitational
> field. It takes on a deeper significance in general relativity, where
> the gravitational field is described in terms of the curvature of the
> spacetime metric. Unlike in special relativity, where the Minkowski
> metric is a 'background structure' given a priori, in general
> relativity the metric is treated as a field which not only affects,
> but also is affected by, the other fields present. In other words, the
> geometry of spacetime becomes a local degree of freedom of the theory.
> Quantitatively, the interaction of the metric and other fields is
> described by Einstein's equation
> $$G_{ab} = 8 \pi G T_{ab}$$
> where the Einstein tensor $G_{ab}$ depends on the curvature of the
> metric, while the stress-energy tensor $T_{ab}$ describes the flow of
> energy and momentum due to all the other fields. The role of the
> constant $G$ is thus simply to quantify how much the geometry of
> spacetime is affected by other fields. Over the years, people have
> realized that the great lesson of general relativity is that a good
> theory of physics should contain no geometrical structures that affect
> local degrees of freedom while remaining unaffected by them. Instead,
> all geometrical structures --- and in particular the causal structure ---
> should themselves be local degrees of freedom. For short, one says
> that the theory should be background-free.
>
> The struggle to free ourselves from background structures began long
> before Einstein developed general relativity, and is still not
> complete. The conflict between Ptolemaic and Copernican cosmologies,
> the dispute between Newton and Leibniz concerning absolute and
> relative motion, and the modern arguments concerning the 'problem of
> time' in quantum gravity --- all are but chapters in the story of this
> struggle. I do not have room to sketch this story here, nor even to
> make more precise the all-important notion of 'geometrical
> structure'. I can only point the reader towards the literature,
> starting perhaps with the books by Barbour and Earman, various papers
> by Rovelli, and the many references therein.
>
> Finally, what of $\hbar$? In quantum theory, this appears most prominently
> in the commutation relation between the momentum $p$ and position $q$ of a
> particle:
> $$pq - qp = -i \hbar,$$
> together with similar commutation relations involving other pairs of
> measurable quantities. Because our ability to measure two quantities
> simultaneously with complete precision is limited by their failure to
> commute, $\hbar$ quantifies our inability to simultaneously know everything
> one might choose to know about the world. But there is far more to
> quantum theory than the uncertainty principle. In practice, $\hbar$ comes
> along with the whole formalism of complex Hilbert spaces and linear
> operators.
>
> There is a widespread sense that the principles behind quantum theory
> are poorly understood compared to those of general relativity. This
> has led to many discussions about interpretational issues. However, I
> do not think that quantum theory will lose its mystery through such
> discussions. I believe the real challenge is to better understand why
> the mathematical formalism of quantum theory is precisely what it is.
> Research in quantum logic has done a wonderful job of understanding
> the field of candidates from which the particular formalism we use has
> been chosen. But what is so special about this particular choice? Why,
> for example, do we use complex Hilbert spaces rather than real or
> quaternionic ones? Is this decision made solely to fit the
> experimental data, or is there a deeper reason? Since questions like
> this do not yet have clear answers, I shall summarize the physical
> insight behind $\hbar$ by saying simply that a good theory of the physical
> universe should be a *quantum theory* --- leaving open the possibility
> of eventually saying something more illuminating.
>
> Having attempted to extract the ideas lying behind the constants $c$, $G$,
> and $\hbar$, we are in a better position to understand the task of
> constructing a theory of quantum gravity. General relativity
> acknowledges the importance of $c$ and $G$ but idealizes reality by
> treating $\hbar$ as negligibly small. From our discussion above, we see that
> this is because general relativity is a background-free classical
> theory with local degrees of freedom propagating causally. On the
> other hand, quantum field theory as normally practiced acknowledges $c$
> and $\hbar$ but treats $G$ as negligible, because it is a background-dependent
> quantum theory with local degrees of freedom propagating causally.
>
> The most conservative approach to quantum gravity is to seek a theory
> that combines the best features of general relativity and quantum
> field theory. To do this, we must try to find a *background-free
> quantum theory with local degrees of freedom propagating causally*.
> While this approach may not succeed, it is definitely worth pursuing.
> Given the lack of experimental evidence that would point us towards
> fundamentally new principles, we should do our best to understand the
> full implications of the principles we already have!
>
> From my description of the goal one can perhaps see some of the
> difficulties. Since quantum gravity should be background-free, the
> geometrical structures defining the causal structure of spacetime
> should themselves be local degrees of freedom propagating causally.
> This much is already true in general relativity. But because quantum
> gravity should be a quantum theory, these degrees of freedom should be
> treated quantum-mechanically. So at the very least, we should develop
> a quantum theory of some sort of geometrical structure that can define
> a causal structure on spacetime.

Then I talk about topological quantum field theories, which are
background-free quantum theories *without* local degrees of freedom, and
what we have learned from them. Basically what we've learned is that
there's a deep analogy between the mathematics of spacetime (e.g.
differential topology) and the mathematics of quantum theory. This is
interesting because in background-free quantum theories we expect that
spacetime, instead of serving as a "stage" on which events play out,
actually becomes part of the play of events itself --- and must itself be
described using quantum theory. So it's very interesting to try to
connect the concepts of spacetime and quantum theory. The analogy goes
like this:

| Differential topology | Quantum theory |
| :-------------------- | :------------- |
| $(n-1)$-dimensional manifold (space) | Hilbert space (states) |
| cobordism between $(n-1)$-dimensional manifolds (spacetime) | operator (process) |
| composition of cobordisms | composition of operators |
| identity cobordism | identity operator |

And if you know a little category theory, you'll see what we have here
are two categories: the category of cobordisms and the category of
Hilbert spaces. A topological quantum field theory is a functor from the
first to the second....

Okay, now for some other papers:

2) Geraldine Brady and Todd H. Trimble, "A string diagram calculus for predicate logic", and C. S. Peirce's system Beta, available at `http://people.cs.uchicago.edu/~brady`

    Geraldine Brady and Todd H. Trimble, "A categorical interpretation of Peirce's propositional logic Alpha", _Jour. Pure and Appl. Alg._ **149** (2000), 213--239.

    Geraldine Brady and Todd H. Trimble, "The topology of relational calculus".

Charles Peirce is a famously underappreciated American philosopher who
worked in the late 1800s. Among other things, like being the father of
pragmatism, he is also one of the fathers of higher-dimensional algebra.
As you surely know if you've read me often enough, part of the point of
higher-dimensional algebra is to break out of "linear thinking". By
"linear thinking" I mean the tendency to do mathematics in ways that
are easily expressed in terms of $1$-dimensional strings of symbols. In
his work on logic, Peirce burst free into higher dimensions. He
developed a way of reasoning using diagrams that he called "existential
graphs". Unfortunately this work by Peirce was never published! One
reason is that existential graphs were difficult and expensive to print.
As a result, his ideas languished in obscurity.

By now it's clear that higher-dimensional algebra is useful in physics:
examples include Feynman diagrams and the spin networks of Penrose. The
theory of $n$-categories is beginning to provide a systematic language for
all these techniques. So it's worth re-evaluating Peirce's work and
seeing how it fits into the picture. And this is what the papers by
Brady and Trimble do!

3) J. Scott Carter, Louis H. Kauffman, and Masahico Saito, "Structures and diagrammatics of four dimensional topological lattice field theories", to appear in _Adv. Math._, preprint available as [`math.GT/9806023`](https://arxiv.org/abs/math.GT/9806023).

We can get $3$-dimensional topological quantum field theories from certain
Hopf algebras. As I described in ["Week 38"](#week38), Crane and
Frenkel made the suggestion that by categorifying this construction we
should get $4$-dimensional TQFTs from certain Hopf categories. This paper
makes the suggestion precise in a certain class of examples! Basically
these are categorified versions of the Dijkgraaf-Witten theory.

4) J. Scott Carter, Daniel Jelsovsky, Selichi Kamada, Laurel Langford and Masahico Saito, "Quandle cohomology and state-sum invariants of knotted curves and surfaces", preprint available as [`math.GT/9903135`](https://arxiv.org/abs/math.GT/9903135).

Yet another attack on higher dimensions! This one gets invariants of
2-links --- surfaces embedded in $\mathbb{R}^4$ --- from the cohomology groups of
"quandles". I don't really understand how this fits into the overall
scheme of higher-dimensional algebra yet. They show their invariant
distinguishes between the 2-twist spun trefoil (a certain sphere knotted
in R^4) and the same sphere with the reversed orientation.

5) Tom Leinster, "Structures in higher-dimensional category theory", preprint available at `http://www.dpmms.cam.ac.uk/~leinster`

This is a nice tour of ideas in higher-dimensional algebra. Right now
one big problem with the subject is that there are lots of approaches
and not a clear enough picture of how they fit together. Leinster's
paper is an attempt to start seeing how things fit together.

6) Claudio Hermida, "Higher-dimensional multicategories", slides of a lecture given in 1997, available at `http://www.math.mcgill.ca/~hermida`

This talk presents some of the work by Makkai, Power and Hermida on
their definition of $n$-categories. For more on their work see
["Week 107"](#week107).

7) Carlos Simpson, "On the Breen-Baez-Dolan stabilization hypothesis for Tamsamani's weak $n$-categories", preprint available as
[`math.CT/9810058`](https://arxiv.org/abs/math.CT/9810058).

For quite a while now James Dolan and I have been talking about
something we call the "stabilization hypothesis". I gave an
explanation of this in ["Week 121"](#week121), but briefly, it says
that the $n$th column of the following chart (which extends to infinity in
both directions) stabilizes after $2n+2$ rows:

|       | $n=0$ | $n=1$ | $n=2$ |
| ----- | :--- | :--- | :--- |
| $k=0$ | sets | categories | $2$-categories |
| | | | |
| $k=1$ | monoids | monoidal categories | monoidal $2$-categories |
| | | | |
| $k=2$ | commutative monoids | braided monoidal categories | braided monoidal $2$-categories |
| | | | |
| $k=3$ | " " | symmetric monoidal categories | weakly involutory monoidal $2$-categories |
| | | | |
| $k=4$ | " " | " " | strongly involutory monoidal $2$-categories |
| | | | |
| $k=5$ | " " | " " | " " |

:$k$-tuply monoidal $n$-categories

Carlos Simpson has now made this hypothesis precise and proved it using
Tamsamani's definition of $n$-categories! And he did it using the same
techniques that Graeme Segal used to study $k$-fold loop spaces...
exploiting the relation between $n$-categories and homotopy theory. This
makes me really happy.

8) Mark Hovey, _Model Categories_, American Mathematical Society Mathematical Surveys and Monographs, vol. **63**, Providence, Rhode Island, 1999. Preprint available as `http://www.math.uiuc.edu/K-theory/0278/index.html`

Speaking of that kind of thing, the technique of model categories is
really important for homotopy theory and $n$-categories, and this book is
a really great place to learn about it.

9) Frank Quinn, _Group-categories_, preprint available as [`math.GT/9811047`](https://arxiv.org/abs/math.GT/9811047).

This one is about the algebra behind certain topological quantum field
theories. I'll just quote the abstract:

> A group-category is an additively semisimple category with a monoidal
> product structure in which the simple objects are invertible. For
> example in the category of representations of a group, $1$-dimensional
> representations are the invertible simple objects. This paper gives a
> detailed exploration of "topological quantum field theories" for
> group-categories, in hopes of finding clues to a better understanding
> of the general situation. Group-categories are classified in several
> ways extending results of Frohlich and Kerler. Topological field
> theories based on homology and cohomology are constructed, and these
> are shown to include theories obtained from group-categories by
> Reshetikhin-Turaev constructions. Braided-commutative categories most
> naturally give theories on 4-manifold thickenings of 2-complexes; the
> usual 3-manifold theories are obtained from these by normalizing them
> (using results of Kirby) to depend mostly on the boundary of the
> thickening. This is worked out for group-categories, and in particular
> we determine when the normalization is possible and when it is not.

10) Sjoerd Crans, "A tensor product for Gray-categories", _Theory and Applications of Categories_, Vol. **5**, 1999, No. 2, pp 12--69, available at `http://www.tac.mta.ca/tac/volumes/1999/n2/abstract.html`

A Gray-category is what some people call a semistrict $3$-category: not as
general as a weak $3$-category, but general enough. Technically,
Gray-categories are defined as categories enriched over the category of
$2$-categories equipped with a tensor product invented by John Gray. To
define semistrict $4$-categories one might similarly try to equip
Gray-categories with a suitable tensor product. And this is what Crans
is studying. Let me quote the abstract:

> In this paper I extend Gray's tensor product of $2$-categories to a new
> tensor product of Gray-categories. I give a description in terms of
> generators and relations, one of the relations being an
> ''interchange'' relation, and a description similar to Gray's
> description of his tensor product of $2$-categories. I show that this
> tensor product of Gray-categories satisfies a universal property with
> respect to quasi-functors of two variables, which are defined in terms
> of lax-natural transformations between Gray-categories. The main
> result is that this tensor product is part of a monoidal structure on
> Gray-Cat, the proof requiring interchange in an essential way.
> However, this does not give a monoidal {(bi)closed} structure,
> precisely because of interchange And although I define composition of
> lax-natural transformations, this composite need not be a lax-natural
> transformation again, making Gray-Cat only a partial
> Gray-Cat-cateegory.