I spent most of last week attending four of the five days of the workshop “Categories, Quanta, Concepts”, at the Perimeter Institute.  In the next few days I plan to write up many of the talks, but it was quite a lot.  For the moment, I’d like to do a little writeup on the talk I gave.  I wasn’t originally expecting to speak, but the organizers wanted the grad students and postdocs who weren’t talking in the scheduled sessions to give little talks.  So I gave a short version of this one which I gave in Ottawa but as a blackboard talk, so I have no slides for it.

Now, the workshop had about ten people from Oxford’s Comlab visiting, including Samson Abramsky and Bob Coecke, Marni Sheppard, Jamie Vicary, and about half a dozen others.  Many folks in this group work in the context of dagger compact categories, which is a nice abstract setting that captures a lot of the features of the category $Hilb$ which are relevant to quantum mechanics.  Jamie Vicary had, earlier that day, given a talk about n-dimensional TQFT’s and n-categories – specifically, n-Hilbert spaces.  I’ll write up their talks in a later,  but it was a nice context in which to give the talk.

The point of this talk is to describe, briefly, $Span(Gpd)$ – as a category and as a 2-category; to explain why it’s a good conceptual setting for quantum theory; and to show how it bridges the gap between Hilbert spaces and 2-Hilbert spaces.

History and Symmetry

In the course of an afternoon discussion session, we were talking about the various approaches people are taking in fundamentals of quantum theory, and in trying to find a “quantum theory of gravity” (whatever that ends up meaning).  I raised a question about robust ideas: basically, it seems to me that if an idea shows up across many different domains, that’s probably a sign it belongs in a good theory.  I was hoping people knew of a number of these notions, because there are really only two I’ve seen in this light, and really there probably should be more.

The two physical  notions that motivate everything here are (1) symmetry, and (2) emphasis on histories.  Both ideas are applied to states: states have symmetries; histories link starting states to ending states.  Combining them suggests histories should have symmetries of their own, which ought to get along with the symmetries of the states they begin and end with.

Both concepts are rather fundamental. Hermann Weyl wrote a whole book, “Symmetry”, about the first, and wrote: As far as I can see, all a-priori statements in physics are based on symmetry. From diffeomorphism invariance in general relativity, to gauge symmetry in quantum field theory, to symmetric tensor products involved in Fock space, through classical examples like Noether’s theorem. Noether’s theorem is also about histories: it applies when a symmetry holds along an entire history of a system: in fact, Langrangian mechanics generally is all about histories, and how they’re selected to be “real” in a classical system (by having a critical value of the action functional). The Lagrangian point of view appears in quantum theory (and this was what Richard Feynman did in his thesis) as the famous “sum over histories”, or path integral. General relativity embraces histories as real – they’re spacetimes, which is what GR is all about. So these concepts seem to hold up rather well across different contexts.

I began by drawing this table:

 $Sets$ $Span(Sets) \rightarrow Rel$ $Grpd$ $Span(Grpd)$

The names are all those of categories. Moving left to right moves from a category describing collections of states, to one describing states-and-histories. It so happens that it also takes a cartesian category (or 2-category) to a symmetric monoidal one. Moving from top to bottom goes from a setting with no symmetry to one with symmetry. In both cases, the key concept is naturally expressed with a category, and shows up in morphisms. Now, since groupoids are already categories, both of the bottom entries properly ought to be 2-categories, but when we choose to, we can ignore that fact.

Why Spans?

I’ve written a bunch on spans here before, but to recap, a span in a category $C$ is a diagram like: $X \stackrel{s}{\leftarrow} H \stackrel{t}{\rightarrow} Y$. Say we’re in $Sets$, so all these objects are sets: we interpret $X$ and $Y$ as sets of states. Each one describes some system by collecting all its possible (“pure”) states. (To be better, we could start with a different base category – symplectic manifolds, say – and see if the rest of the analysis goes through). For now, we just realize that $H$ is a set of histories leading the system $X$ to the system $Y$ (notice there’s no assumption the system is the same). The maps $s,t$ are source and target maps: they specify the unique state where a history $h \in H$ starts and where it ends.

If $C$ has pullbacks (or at least any we may need), we can use them to compose spans:

$X \stackrel{s_1}{\leftarrow} H_1 \stackrel{t_1}{\rightarrow} Y \stackrel{s_2}{\leftarrow} H_2 \stackrel{t_2}{\rightarrow} Z \stackrel{\circ}{\Longrightarrow} X \stackrel{S}{\leftarrow} H_1 \times_Y H_2 \stackrel{T}{\rightarrow} Z$

The pullback $H_1 \times_Y H_2$ – a fibred product if we’re in $Sets$ – picks out pairs of histories in $H_1 \times H_2$ which match at $Y$. This should be exactly the possible histories taking $X$ to $Z$.

I’ve included an arrow to the category $Rel$: this is the category whose objects are sets, and whose morphisms are relations. A number of people at CQC mentioned $Rel$ as an example of a monoidal category which supports toy models having some but not all features of quantum mechanics. It happens to be a quotient of $Span(Sets)$. A relation is an equivalence class of spans, where we only notice whether the set of histories connecting $x \in X$ to $y \in Y$ is empty or not. $Span(Sets)$ is more like quantum mechanics, because its composition is just like matrix multiplication: counting the number of histories from $x$ to $y$ turns the span into a $|X| \times |Y|$ matrix – so we can think of $X$ and $Y$ as being like vector spaces.

In fact, there’s a map $L : Span(Sets) \rightarrow Hilb$ taking an object $X$ to $\mathbb{C}^X$ and a span to the matrix I just mentioned, which faithfully represents $Span(Sets)$. A more conceptual way to say this is: a function $f : X \rightarrow \mathbb{C}$ can be transported across the span. It lifts to $H$ as $f \circ s : H \rightarrow \mathbb{C}$. Getting down the other leg, we add all the contributions of each history ending at a given $y$: $t_*(s \circ f) = \sum_{t(h)=y} f \circ s (h)$.

This “sum over histories” is what matrix multiplication actually is.

Why Groupoids?

The point of groupoids is that they represent sets with a notion of (local) symmetry. A groupoid is a category with invertible morphisms. Each such isomorphism tells us that two states are in some sense “the same”. The beginning example is the “action groupoid” that comes from a group $G$ acting on a set $X$, which we call $X /\!\!/ G$ (or the “weak quotient” of $X$ by $G$).

This suggests how groupoids come into the physical picture – the intuition is that $X$ is the set (or, in later variations, space) of states, and $G$ is a group of symmetries.  For example, $G$ could be a group of coordinate transformations: states which can be transformed into each other by a rotation, say, are formally but not physically different.  The Extended TQFT example comes from the case where $X$ is a set of connections, and $G$ the group of gauge transformations.  Of course, not all physically interesting cases come from a single group action: for the harmonic oscillator, the states (“pure states”) are just energy levels – nonnegative integers.  On each state $n$, there is an action of the permutation group $S_n$ – a “local” symmetry.

One nice thing about groupoids is that one often really only wants to think about them up to equivalence – as a result, it becomes a matter of convention whether formally different but physically indistinguishable states are really considered different.  There’s a side effect, though: $Gpd$ is a 2-category.  In particular, this has two consequences for $Span(Gpd)$: it ought to have 2-morphisms, so we stop thinking about spans up to isomorphism.  Instead, we allow spans of span maps as 2-morphisms.  Also, when composing spans (which are no longer taken up to isomorphism) we have to use a weak pullback, not an ordinary one.  I didn’t have time to say much about the 2-morphism level in the CQC talk, but the slides above do.

In any case, moving into $Span(Gpd)$ means that the arrows in the spans are now functors – in particular, a symmetry of a history$h$  now has to map to a symmetry of the start and end states, $s(h)$ and $t(h)$.  In particular, the functors give homomorphisms of the symmetry groups of each object.

Physics in Hilb and 2Hilb

So the point of the above is really to motivate the claim that there’s a clear physical meaning to groupoids (states and symmetries), and spans of them (putting histories on an even footing with states).  There’s less obvious physical meaning to the usual setting of quantum theory, the category $Hilb$ – but it’s a slightly nicer category than $Span(Gpd)$.  For one thing, there is a concept of a “dual” of a span – it’s the same span, with the roles of $s$ and $t$ interchanged.  However (as Jamie Vicary pointed out to me), it’s not an “adjoint” in $Span(Gpd)$ in the technical sense.  In particular, $Span(Gpd)$ is a symmetric monoidal category, like $Hilb$, but it’s not “dagger compact”, the kind of category all the folks from Oxford like so much.

Now, groupoidification lets us generalize the map $L : Span(Sets) \rightarrow Hilb$ to groupoids making as few changes as possible.  We still use Hilbert space $\mathbb{C}^X$, but now $X$ is the set of isomorphism classes of objects in the groupoid.  The “sum over histories” – in other words, the linear map associated to a span – is found in almost the same way, but histories now have “weights” found using groupoid cardinality (see any of the papers on groupoidification, or my slides above, for the details).  This reproduces a lot of known physics (see my paper on the harmonic oscillator; TQFT’s can also be defined this way).

While this is “as much like” linearization of $Span(Set)$ as possible in some sense, it’s not exactly analogous.  It also is rather violent to the structure of the groupoids: at the level of objects it treats $X /\!\!/ G$ as $X/G$. At the morphism level, it ignores everything about the structure of symmetries in the system except how many of them there are.   Since a groupoid is a category, the more direct analogy for $\mathbb{C}^X$ – the set of functions (fancier versions use, say, $L^2$ functions only) from $X$ to $\mathbb{C}$ is $Hilb^G$ – the category of functors from a groupoid into $Hilb$.  That is, representations of $X$.

One of the attractions here is that, because of a generalization of Tanaka-Krein duality, this category will actually be enough to reconstruct the groupoid if it’s reasonably nice.  The representation of $Span(Gpd)$ in $2Hilb$, unlike in $Hilb$ is actually faithful for objects, at least for compact or finite groupoids.

Then you can “pull and push” a representation$F$ across a span to get $t_*(F \circ s)$ – using $t_*$, the adjoint functor to pulling back.  This is the 1-morphism level of the 2-functor I call $\Lambda$, generalizing the functor $L$ in the world of sets.  The result is still a “direct sum over histories” – but because we’re dealing with pushing representations through homomorphisms, this adjoint is a bit more complicated than in the 0-category world of $\mathbb{C}$.  (See my slides or paper for the details).  But it remains true that the weights and so forth used in ordinary groupoidification show up here at the level of 2-morphisms.  So the representation in $2Hilb$ is not a faithful representation of the (intuitively meaningful) category $Span(Gpd)$ either.  But it does capture a fair bit more than Hilbert spaces.

One point of my talk was to try to motivate the use of 2-Hilbert spaces in physics from an a-priori point of view.  One thing I think is nice, for this purpose, is to see how our physical intuitions motivate $Span(Gpd)$ – a nice point itself – and then observe that there is this “higher level” span around:

$Hilb \stackrel{|\cdot |}{\leftarrow} Span(Gpd) \stackrel{\Lambda}{\rightarrow} 2Hilb$

Further Thoughts

Where can one take this?  There seem to be theories whose states and symmetries naturally want to form n-groupoids: in “higher gauge theory“, a sort of  gauge theory for categorical groups, one would have connections as states, gauge transformations as symmetries, and some kind of  “symmetry of symmetries”, rather as 2-categories have functors, natural transformations between them, and modifications of these.  Perhaps these could be organized into n-dimensional spans-of-spans-of-spans… of n-groupoids.  Then representations of an n-groupoid – namely, n-functors into $(n-1)-Hilb$ – could be subjected to the kind of “pull-push” process we’ve just looked at.

Finally, part of the point here was to see how some fundamental physical notions – symmetry and histories – appear across physics, and lead to $Span(Gpd)$.  Presumably these two aren’t enough.  The next principle that looks appealing – because it appears across domains – is some form of an action principle.

But that would be a different talk altogether.

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