The n-Category Café

As far as I can tell, the amount of ejected material, tidal debris, etc, will depend on the collision scenario (orbital energy of the galaxies) as well as on their mass distribution (baryonic and non-baryonic) and possibly on other finer details such as the amount of diffuse mass of the Local Group (these latter are important for taking into account dynamical friction effects).

I have never simulated such a detailed collision between the Milky Way and Andromeda, but Cox and Loeb did.

(My previous work with N-body simulations focused on merging elliptical-like objects in the context of the origin of the “fundamental plane” relation, and were intended as to study the overall features of the mergers, not their detailed interactions. If memory serves, however, I could have about ~ 15% of mass ejections in some cases, or even more, depending on the “violence” of the merger, that is, how strongly the gravitational potential would fluctuate during the collision).

Christine

“Thanks! And this is close to one of my naive questions: “why everything is abelian?”. I mean that the large part of mathematics is built on the notion of abelian (additive) group: rings, fields, bodies, modules, vector spaces, etc.”

I think the answer is: for historical reasons only.

There are mathematical fields (like analysis in metric spaces, see the work of Gromov) where the natural notion replacing a vector space is a simply connected Lie group whose Lie algebra admits a positive graduation (aka Carnot group). A particular example is the Heisenberg group.
See http://xxx.arxiv.org/abs/0705.1440
for a definition of linearity in this context.

200 years ago there was a single geometry:
euclidean. Is it risky to say that in 100
years from now we shall have non-euclidean analysis, which is not based on abelian groups?

Osman said, “why everything is abelian?”

Some people have argued that any upper bounds on the amount of non-abelianness around us should be pretty weak. But let me try to actually give some upper bounds.

The basic idea is that objects which are non-abelian often fail to admit higher operations. For example, abelian groups can admit ring structures, but a bilinear map on a general group factors through its abelianization.

Bergman and Hausknecht’s book “Cogroups and co-rings in categories of associative rings” is a pretty extensive reference for things like this. Part of it is based on the work of Tall–Wraith Let me try to say a few words about what they do, as far as I understand it.

In what sense is a (possibly non-commutative) ring naturally the kind of object that knows how to act on an abelian group? The answer is that is represents a comonad on abelian groups. More precisely, we say an endofunctor on the category of abelian groups is representable if its composition with the forgetful functor to sets is representable. If $R$ is an abelian group, then the functor $Hom(R,-)$ naturally takes values in the category of abelian groups, and a ring structure on $R$ is the same thing as a comonad structure on the functor.

The interesting thing is that Bergman and Hausknecht show that it’s hard to do similar things in non-abelian situations. For example, there is a theorem of Kan that says that the category of representable endofunctors on the category of groups is equivalent to the category of pointed sets. It follows that the category of representable comonads on the category of groups is the same thing as a monoid in the category of pointed sets, which is the same thing as a monoid. In other words, the only thing that naturally knows how to act on groups is just a monoid. Whereas in the category of abelian groups, we went up to the next level and got rings.

A similar thing happens at the next level. What are the representable comonads on the category of rings? What about the category of commutative rings? Here things are a little more interesting, because there are objects that know how to act on rings, but not by ring endomorphisms! For instance, a Lie algebra knows how to act on a commutative ring. Such a thing is just a homomorphism from the given Lie algebra to the Lie algebra of derivations of the ring. In fact, more generally, any cocommutative bialgebra knows how to act on a commutative ring. In the case of Lie algebras, you get the commutative ring representing the functor by taking the symmetric algebra of the enveloping algebra. In the case of a group, you take the symmetric algebra of its group algebra.

The great thing is that in the case of commutative rings, there are even more examples, ones that don’t come from bialgebras. For example, in positive characteristic, the Frobenius map knows how to act on a commutative ring. There is a representable comonad $T$ on the category of commutative $Z/pZ$-algebras such that an action of $T$ on $R$ is nothing more than saying that the Frobenius map on $R$ is bijective. We could make another comonad that requires that the Frobenius map be the identity. Even more interesting, over $Z$, we could ask for lifts of the Frobenius map from characteristic $p$ to characteristic $0$.

But all these wonderful things can’t happen with noncommutative rings. Of course there is no non-commutative Frobenius map, at least in the straightforward sense. In fact, Bergman and Hausknecht prove that, at least when $F$ is $Q$ or $Z/pZ$, the only representable comonads on the category of $F$-algebras are ones coming from bialgebras combined with ones coming from anti-endomorphisms, ie maps from an algebra to its opposite. (Admittedly, they don’t address the situation over $Z$, which is much more important.)

Now that I’ve said all that, they also give some evidence that goes against this point of view. For example, even though the only representable comonads on the category of groups come from boring monoids, there are actually representable comonads on the category of monoids that don’t come from monoids. For example, it’s possible to endow a monoid $M$ with a “right-inverse” operator. This is an anti-endomorphism $r: M\to M$ satisfying the identity $m\cdot r(m)=1$ and such that $r$ composed with itself is the identity. You can then use this structure to make other representable comonads. In fact, they give a complete description of all the representable comonads, and they have a similar flavor involving left and right inverse operators. It would be pretty interesting to study the next level of this hierarchy.

Categorifying negative numbers is actually tricky. John Baez talked about negative sets here, and gave a number of good references.

Another thing that people have tried is to use superalgebra; for example, instead of considering linear combinatorial species $F_0$ (i.e., species valued in vector spaces), consider “virtual” species valued in $\mathbb{Z}_2$-graded vector spaces. A virtual species is given by a pair of linear species

$(F_0, F_1)$

which is to be thought of as a formal difference $F_0 - F_1$ (hence, at the decategorified level, this involves passage to a Grothendieck group). In good cases, a virtual species may also come equipped with differential graded structure, and one may consider the derived category of DG-species as another method for dealing with categorified negatives.

But none of these methods is perfect.

Expressing the concept of “positive energy representation” in the language of groupoids sounds tricky, since it’s rather closely wedded to the complex numbers. For example, there’s no obvious notion of a positive-energy real representation of a Lie group.

To see this, consider the primordial example: the real line! A strongly continuous unitary (complex) representation $U$ of $\mathbb{R}$ has a skew-adjoint generator

$$A = {d \over d t} U(t) |_{t=0}$$

which then has

$$U(t) = exp(t A) .$$

The spectrum of $A$ lies on the imaginary line, so it makes no sense to say it’s positive.

Using the magic of $i$, we can then define a self-adjoint generator

$$H = i A$$

which has

$$U(t) = exp(-i t H)$$

Since the spectrum of $H$ lies on the real axis, it makes sense to declare $H$ positive when its spectrum lies in $[0,+\infty)$. We then say $U$ is a positive-energy representation.

Note how the ghost of Galois hovers over the proceedings, snickering quietly. If we’d picked $-i$ instead of $i$ as our square root of $-1$, what used to be a ‘positive-energy representation’ would now be declared ‘negative-energy’, and vice versa!

If we were working over the real numbers, we wouldn’t have positive-energy representations.

See how everything changes:

A strongly continuous orthogonal (real) representation $U$ of $\mathbb{R}$ has a skew-adjoint generator

$$A = {d \over d t} U(t) |_{t=0}$$

which then has

$$U(t) = exp(t A) .$$

The spectrum of $A$ lies on the imaginary line, so it makes no sense to say it’s positive. In fact, if any number $a$ lies in the spectrum, so must its complex conjugate $\overline{a}$!

So far, groupoidification seems to work most easily for the most ‘field-independent’ aspects of linear algebra. The concept of positive energy is very special to the complex numbers. So, if we want to groupoidify the theory of positive-energy representations, we have our work cut out for us.

So far, Jim Todd and I are much closer to groupoidifying the whole theory of unitary representations of finite-dimensional compact simple Lie groups — or equivalently, finite-dimensional holomomorphic representations of complex simple Lie groups. The concept of ‘positive energy representation’ doesn’t apply here.

In short: there’s a lot of work to do, and I’ve got to tell you about it before you can see where the frontier is… but ‘positive energy representations’ are a bit beyond the frontier, right now.

By the way, I mentioned this business about skew-adjoint versus self-adjoint generators in my discussion of real, complex and quaternionic quantum mechanics in week251:

Another special way in which $\mathbb{C}$ is better than $\mathbb{H}$ or $\mathbb{R}$ is that only for a complex Hilbert space is there a correspondence between continuous 1-parameter groups of unitary operators and self-adjoint operators. We always get a skew-adjoint operator, but only in the complex case can we convert this into a self-adjoint operator by dividing by $i$.

In the case of $\mathbb{R}$ there aren’t enough square roots of -1. In the case of the quaternions, $\mathbb{H}$, there turn out to be ‘too many’ — and the problem is, they don’t commute. So, if $A$ is linear, $H = i A$ is not!

You know the following, but I feel like repeating it to the rest of the world:

Jeffrey Morton showed how to develop a lot of Feynman diagram theory for perturbed harmonic oscillators purely combinatorially,

At the beginning I had the impression that Jeffrey’s work was intended as a categorification of quantum mechanics. But around that comment of mine which I mentioned, I began wondering if maybe the natural home of all these nice constructions and observations he gives is less so a categorified quantum mechanics – something we would imagine applying to stuff more general than quantum point particles – but rather something like an unravelling of the true structure underlying ordinary quantum mechanics. Something which we should really be applying to ordinary quantum point particles if we were doing it right, instead of that linear algebra which we instead do use.

What I have grasped of the “tale of groupoidification” in fact seems to strengthen this feeling.

But, I’m not sure I understand what you mean. I’ll need to reread your comments about this. Or maybe you could summarize the essence.

The idea was this: quantization of a point particle wants to be a pushforward operation: the parallel transport functor which describes its classical coupling to an electromagnetic field, say, is “pushed forward to a point”.

This woks nice on objects (and in fact on $(k \lt n)$-morphisms when we do $n$-particles). But for it to work on ($n$-)morphisms, we have to intervene by hand: choose a measure and do this and that to get the path integral we want.

That seems to indicate that something is wrong. Especially when we work with arbitrary $n$-particles: here push-forward quantization is nice and canonical for a fraction of $n/(n+1)$ of the system, but becomes awkward and “man made” for the last (and most important) $\frac{1}{n+1}$st part of the journey.

And part of the reason is clearly: at top level we are pushing forward 0-functors (functions). And there is no canonical way to do that. So if these 0-functors secretly were 1-functors, with everything else accordinly shifted by one degree, then maybe we’d have a better chance to grasp all of quantization as just one nice natural and canonical push-forward operation.

So what I tried to play around with beginning at that comment I mentioned is to see what can be improved if we assume (quite following Jeffrey Morton, but thinking about ordinary QM all along) that all the numbers we encounter are actually (to be thought of as) sets or the like.

At that point I was thinking of replacing $\mathbb{C}$-modules (vector spaces) with $\mathbb{C}\mathrm{Set}$-module categories.

It seems to me that this is pretty close to groupoidification, and maybe what I should really have considered is the latter. I need to think more about this and try to better understand this.

But in any case, it seemed to me that if we assumed that our “wave functions” are actually not 0-functors but 1-functors, which assigns sets (certainly with some extra structure, but maybe consider just plain finite sets for the moment) to points, then suddenly it does make sense to consider the path integral as an honest push-forward to a point at all levels.

I tried to indicate how this does seem to work in The Canonical 1-Particle.

The crucial information there is the following:

if we do the push-forward quantization for a 1-dimensional target space which is modelled by a graph which has from every vertex precisely two edges emanating to the two neighbour vertices, the as we compute the quantum propagator for a single time step, we have to compute the colimit of our set-valued wave function over categories of the shape

$$(x-1) \leftarrow (x) \rightarrow (x+1)$$

This is actually one of the simple toy examples discussed in Tom Leinster’s paper. Its Leinster measure is $$1 \leftarrow -1 \rightarrow 1 \,.$$ If you think about it, you see the exponentiated lattice Laplace operator here, to first order.

First one might think that this is just a coincidence. But I don’t think so. In that entry I went one step further to the path integral over two time steps by push-forward. And it does come from a colimit over a category whose Leinster-measure gives the second power of the exponentiated lattice Laplace operator, i.e. indeed the right path integral result in that order.

So for that reason I was hoping/thinking that we will find that if only we look close enough we find that all these numbers in ordinary quantum mechanics are relly sets of some sort, and all these vector spaces really modules for these kinds of sets.

Now, what you keep saying in the tale of groupoidification seems to be quite consistent with this hope. Roughly at least. Maybe I should replace modules for sets by groupoids, instead. Or something like this.

Urs wrote:

At the beginning I had the impression that Jeffrey’s work was intended as a categorification of quantum mechanics. But around that comment of mine which I mentioned, I began wondering if maybe the natural home of all these nice constructions and observations he gives is less so a categorified quantum mechanics — something we would imagine applying to stuff more general than quantum point particles — but rather something like an unravelling of the true structure underlying ordinary quantum mechanics.

Right, that’s what his work is supposed to be. And yet it is a categorification, technically.

So, this raises an interesting general point. We’re always running around trying to categorify everything, but there are at least two things categorification can do. One is to take a mathematical structure and boost its dimension by one: for example, going from the math of particle physics to the math of string physics. Another is to take a structure that was sort of misunderstood and ‘squashed’, and understand it better by unsquashing it: for example, taking combinatorics phrased in terms of natural numbers, and phrasing it in terms of finite sets. These seem like quite different things to do — at least they seem that way to me right now! — but categorification can be good for both.

Joyal’s work on ‘species’ is a classic case of the latter sort of categorification. Joyal took the generating functions combinatorists like to use for counting things, and realized these are secretly just squashed versions of species, namely functors

$$F: FinSet_0 \to Set.$$

This realization then let us unsquash some of the math behind the quantum harmonic oscillator and Feynman diagrams.

But, it turns out that everything works better — the combinatorics, the harmonic oscillator and the Feynman diagrams! — if we go half a step further and consider stuff types, namely weak 2-functors

$$F: FinSet_0 \to Gpd$$

Note that I say ‘half a step’ further, which is related to some of your later points. Why just ‘half a step’? Isn’t going from $Set$ to $Gpd$ a full step upwards on the $n$-categorical ladder?

Well, in a way not! Species are secretly the same thing as faithful functors

$$\Psi = \hat{F} : X \to FinSet_0$$

where $X$ is a groupoid, while stuff types are the same as arbitrary functors

$$\Psi = \hat{F} : X \to FinSet_0$$

where $X$ is a groupoid.

Maybe this is a full step up the $n$-categorical ladder in some sense, but it usually feels more like ‘doing things right’. It’s sort of like the difference between standing on a ladder with one foot higher than the other, and standing on it with both feet at the same level. That’s why it feels like going up a ‘half step’.

Now I think this really is related to your idea:

The idea was this: quantization of a point particle wants to be a pushforward operation: the parallel transport functor which describes its classical coupling to an electromagnetic field, say, is “pushed forward to a point”.

This woks nice on objects (and in fact on $(k \lt n)$-morphisms when we do $n$-particles). But for it to work on ($n$-)morphisms, we have to intervene by hand: choose a measure and do this and that to get the path integral we want.

That seems to indicate that something is wrong. Especially when we work with arbitrary $n$-particles: here push-forward quantization is nice and canonical for a fraction of $n/(n+1)$ of the system, but becomes awkward and “man made” for the last (and most important) $\frac{1}{n+1}$st part of the journey.

Thanks for re-explaining this. I see much better what you mean, now.

In Jeff’s work, states of the quantum harmonic oscillator are replaced by stuff types

$$\Psi : X \to FinSet_0$$

and operators are replaced by stuff operators, namely spans of groupoids:

                 T
                / \
               /   \
              /     \
             /       \
            /         \
           /           \
          v             v
      FinSet_0        FinSet_0

But, all this generalizes to situations where we replace $FinSet_0$ by other groupoids… so we’re really trying to do quantum mechanics in a new way, using spans of groupoids as a replacement for operators.

(In reality, so far we can do only a small fragment of quantum mechanics this way. But, we’re allowed to dream…)

In particular, as you note, the push-pull (or pull-push) operations involved in path integrals become nicer in this new framework, since we don’t need to specify a measure. Given a ‘state’ thought of as a functor between groupoids:

$$\Psi : X \to A$$

and given an ‘operator’ thought of as a span of groupoids:

                 T
                / \
               /   \
              /     \
             /       \
            /         \
           /           \
          v             v
          A             B

we can pull back $\Psi$ to $T$ and then push it forwards to $B$, getting a state we could call

$$T \Psi: T X \to B$$

(In this formalism, the tricky part is the pulling back, not the pushing forwards! That’s because I’m thinking of states as functors $\Psi = \hat{F}: X \to A$ instead of weak 2-functors $F: A \to Gpd$. In the latter approach, which is equivalent, the tricky part is the pushing forwards!)

So, yeah — I think I roughly understand how your remarks connect to the Tale of Groupoidification.

Let me try to summarize in one sentence. When you see a measure space, maybe you should try to understand it better by seeing it as a squashed version of a groupoid, or category.

On the page 11 they categorify ring of integers through FinSet 2. I do not understand how they obtain their form of cartesian product

I think most category theorists would shy away from calling it a Cartesian product, and prefer to call it a tensor product instead. In any case, it’s obtained via a beautiful piece of theory due to Brian Day (I think originally in his PhD thesis).

What Day showed is that a tensor product structure on a category $\mathbb{C}$ induces a corresponding tensor product on the functor category $[\mathbb{C}, \mathrm{Set}]$.

The general definition is very easy to understand if (and only if) you know and love coends. But the special case here is easy to understand even if you don’t. If we have a monoid $M$, we can think of $M$ as being a discrete category with a tensor product on it. This then induces a tensor product on $\mathrm{Set}^{|M|}$. Here’s how it works:

An object of $\mathrm{Set}^{|M|}$ is an indexed collection $\langle X_i | i \in M\rangle$, and the tensor is $$\langle X_i\rangle \otimes\langle Y_i\rangle = \langle \sum_{a\cdot b = i}X_a\times Y_b\rangle;$$ the unit object $\langle I_i\rangle$ has $I_1 = 1$ and $I_j = \emptyset$ for $j\neq 1$.

If you want to introduce negativity, the natural monoid to consider is the set $\{+1, -1\}$ under multiplication, which is isomorphic to $(\mathbb{Z}_2, +)$. Let’s calculate the induced tensor on $\mathrm{Set}^2$: an object is a pair $(X_+, X_-)$, the unit is $(1, 0)$, and $$(X_+, X_-)\otimes(Y_+, Y_-) = (X_+\times Y_+ + X_-\times Y_-, X_+\times Y_- + X_-\times Y_+),$$ since $+1 = +1\times +1 = -1\times-1$, and $-1 = +1\times-1 = -1\times+1$.

So $\mathrm{Set}^2$ admits at least four different tensor products: there is the categorical product and categorical coproduct (taken elementwise), and the convolution tensors induced by the two different 2-element monoids $(\mathbb{Z}_2, +)$ and $(\mathbb{Z}_2, \times)$. (Easy exercise: give an explicit form for the tensor induced by the latter.)

How does this compare to a map that respects the G action up to homotopy?

Are the maps φ_g to be coherent in some sense? e.g φ_gh = ?

There are some conditions on $G$, right, where the 2-group $INN(G)$ has $G$’s worth of objects and precisely one morphism between any two objects? Perhaps the case where the center of $G$ is trivial?

Anyway, when $INN(G)$ has the form I just mentioned, it’s equivalent to the trivial 2-group, so its representation theory, when done in a morally correct way, should be completely dull. Here by “representation theory” I mean the study of its (suitably weak) actions on anything, modulo the correct concept of equivalence.