The n-Category Café

A comment on the notion of higher vector spaces: the KV definition is not really on par with the BC definition:

• the KV notion is a very restricted notion of (2,2)-vector spaces,

• the BC notion is the general notion of (2,1)-vector space

(both over the given ground field $k$).

The general situation is this:

• pick some commutative $\infty$-ring $k$ (can be modeled by an $E_\infty$-ring);

• write $Mod_k$ for the $(\infty,1)$-category of $\infty$-modules over $k$ (can be modeled by $E_\infty$-modules);

  the objects in here are “$(\infty,1)$-vector spaces” over $k$;

• write $2 Mod_k$ for the $(\infty,2)$-category whose

  • objects are algebra objects $A$ in $Mod_k$ (to be thought of as placeholders for their $(\infty,1)$-category $Mod_A$ of modules);

  • morphisms are bimodule objects $N$ in $Mod_k$ (to be thought of as placeholders for $k$-linear functors $(-) \otimes_A N : Mod_A \to Mod_B$);

• generally for $n \gt 1$, by induction, write $n Mod_k$ for the $(\infty,n)$-category whose

  • objects are algebra objects $A$ in $(n-1)Mod_k$;

  • morphisms are bimodule objects in $(n-1)Mod_k$;

In this general pattern, the cases discussed here are the following:

• the ground $\infty$-ring $k$ is an ordinary field.

• $Mod_k$ is the $(\infty,1)$-category of chain complexes of $k$-modules. A general element in here is a $(\infty,1)$-vector space over $k$. In particular a BC-vector space is a 1-truncated such object, hence a $(2,1)$-vector space.

• $2 Mod_k$ is the $(\infty,2)$-category whose objects are given by dg-algebras, and whose morphisms are given by dg-bimodules.

• consider the ordinary algebras $k^{\oplus n}$, regarded trivially as a dg-algebra. The corresponding object in $2 Mod_k$ is the KV-2-vector space of dimension $n$.

nLab: (∞,n)-vector space

Oh, and presumably “Monte Python” is a uniform random sampling from a population of snakes?

Thanks, I’ll fix that typo. Did you know that as you increase the size of a random sample until it converges to the actual distribution its chosen from, you get the ‘full Monte’?

You may be just discussing the difference between Fermi-Dirac statistics which apply to distinguishable fermions and Bose-Einstein statistics which apply to indistinguishable bosons.

Actually Stuart is discussing the difference between Bose–Einstein statistics and Maxwell–Boltzmann statistics.

Bose–Einstein statistics apply to bosons: indistinguishable particles that transform according to the trivial representation of the symmetric group $S_n$ on $S^n H$, the $n$th symmetric tensor power of the single-particle Hilbert space $H$. Maxwell–Boltzmann statistics apply to boltzons: distinguishable particles, which transform according to the permutation representation of $S_n$ on $H^{\otimes n}$, the $n$th tensor power of $H$.

People don’t talk about boltzons very much because we don’t know any particles that act exactly like boltzons when we take quantum mechanics into account. But in certain ‘classical limits’ Maxwell–Boltzmann statistics work well. For classical balls in a box, they’re just right.

To answer Stuart: I understand your puzzlement! I was completely shocked when I realized that we can use the mathematics of Fock space, including the commutation relations

$$a a^\dagger - a^\dagger a = 1$$

for boltzons as well as bosons. The big difference is that we normalize states a different way, because the states describe probability distributions rather than wavefunctions. This is what I’ve been working on for the last year, and I’m writing a little book on it with Jacob Biamonte, tentatively called Quantum Techniques for Stochastic Processes.

I agree this is weird. The sense I have, after playing with this for a few years on and off, is that this analogy isn’t physically meaningless, but it’s not obvious how literally to take it.

So it might possibly make this clearer to understand that not only do the commutation relations for balls-in-boxes and for the harmonic oscillator look similar formally, they arise for very similar reasons. This isn’t in our current paper, by the way, but it will be an important part of the second installment, tentatively subtitled “Models on Free Structures”, and is mentioned briefly in this talk I gave at the Higher Structures workshop last year. It’s also the part of this project which really uses the formal analysis of the harmonic oscillator which Jamie Vicary did in the paper John linked to above, A categorical framework for the quantum harmonic oscillator.

So, to recap how one gets the Fock space for the harmonic oscillator… Start with the Hilbert space for a particle with no particular features - namely just $\mathbb{C}$. There is only one state here, in the sense of a ray in the Hilbert space, so this system is a “particle” of which the only thing to say about it is that it’s there.

The bosonic Fock space is then $\oplus_n \mathbb{C}^{\otimes_s n}$, the direct sum of all symmetric tensor products of some number of copies of this space. One way to say this is that the symmetric tensor product of a space $V$ with itself is the equalizer of two maps $V \otimes V \rightarrow V \otimes V$, namely the identity and the swap map. Likewise, $V^{\otimes_s n}$ is the equalizer of all the permutation automorphisms that appear because $V^{\otimes n}$ is automatically a representation of $S_n$. So the symmetric product is the trivial representation.

Since $\mathbb{C}^{\otimes_s n} \cong \mathbb{C}$, this is just a sum of a bunch of 1-dimensional spaces, each of which describes an $n$-particle system, which again has only one state. The only thing to say about this state is that it has $n$ particles in it. Jamie’s original paper explains this by means of a monad on $Hilb$, which is essentially the “free commutative monoid” monad: the Fock space is the free commutative monoid on $\mathbb{C}$. This fact gives a bunch of special maps, including a bialgebra structure on the Fock space, and the raising and lowering operators can be constructed out of this. The commutation relations are a consequence of that.

Now, groupoidifying this is a categorification, so this description has to be weakened. To start with, we take a groupoid describing a system with only one configuration (the “it’s there” state for our particle). This will be the trivial groupoid $\mathbf{1}$, with one object and only the identity morphism. Then we want to take the “groupoidified Fock space”.

Since groupoids live in a 2-category, the equivalent of the “free commutative monoid” monad turns out to be a bit weaker, namely the “free symmetric monoidal category” 2-monad. We get a “direct sum” (i.e. in $Span(Gpd)$, the disjoint union) of a bunch of objects which show up as certain 2-limits. In particular, we freely generate a bunch of objects like $(\mathbb{1} \otimes ... \otimes \mathbb{1})$, and we must get not EQUATIONS, but ISOMORPHISMS corresponding to all the switch maps. This is essentially where the groupoid of finite sets and bijections come from: think of $\mathbb{1}$ as the groupoid which contains exactly the 1-element set - the free symmetric monoidal category this generates is the groupoid which contains all finite sets and their bijections.

All the structure which appeared in the construction of Fock space also appears here - and the things which correspond to raising and lowering operators are exactly what we intuitively describe as “put a ball in” and “take a ball out” (i.e. functors which add or remove elements of any given set). The categorified commutation relations then hold for the same sort of reasons as before.

The picture in $Hilb$ is then related to the picture in $Span(Gpd)$ by the degroupoidification functor, which gets along with all the structures involved. The fact that degroupoidification assigns a vector space to a groupoid which consists of invariant functions on the groupoid necessarily means that it produces the trivial representation of all those symmetric groups. One could probably get a fermionic Fock space by forcibly turning the groupoid of finite sets into a super-groupoid, where the odd permutations have a negative sign, and changing the degroupoidification functor accordingly.

In fact, the 2-vector space which we assign to the groupoid actually consists of all representations - so in some sense the weakening of the symmetry in the construction of Fock space makes it indeterminate what statistics the particles have. The bosonic representations are all in there, though.

(Note also that there’s no reason that we have to start with the trivial groupoid for this process to make sense - it’s just the choice that gives the harmonic oscillator and the Heisenberg algebra we’re looking for. The end result will then involve also the representation theory of whatever groupoid one starts with, as well as the symmetric groups.)

So the fact that this all works is not merely a coincidence - but this sort of formal description obviously can’t settle whether or not the non-coincidence is also physically relevant.

Short version: yes, that’s weird. I don’t know how seriously to take it.