The n-Category Café

Depending on what one is looking at, there are different notions of “passage of time” in different QFTs: in ordinary non-relativistic quantum mechanics of a point particle, time flows along the real numbers. In a topological QFT time would not really flow at all.

In the context of the $n$-particle, we model this by considering a 1-category $$\mathrm{Interval}$$ whose morphisms represent time steps. So for the non-relativistic particle we’d take $$\mathrm{Interval} = \Sigma \mathbb{R} \,,$$ the real numbers under addition, regarded as a 1-object category.

For a topological QFT we might take $\mathrm{Interval}$ to be the walking (co)-monad: the category with a single object and an endomorphism on it which is such that its composite with itself may be contracted back to just the morphism itself, and the other way around, such that everything is nicely associative.

$n=1$: Worldlines

For a 1-particle of shape $$\mathrm{par}$$ the worldlines it traces out should hence be the morphisms in the category $$\mathrm{worldvol} := \mathrm{par} \times \mathrm{Interval} \,.$$ So if the particle is just a point $\mathrm{par} = \{\bullet\}$, its “worldvolumes” are just Riemannian line segments, hence morphisms $$\bullet \stackrel{t}{\to} \bullet$$ in $\mathrm{Interval}$. But, for instance, we might take the 1-particle for instance to really be a collection of two particles, setting $\mathrm{par} = \{\bullet_1 \;\;\bullet_2\}$. Then its worldvolumes would consist either of time intervals for one or for the other of the other particle. I mention this just to demonstrate the idea.

$n=2$: Worldsheets

So far so trivial. The situation becomes somewhat more interesting as we pass to the 2-particle, modeled by a $(2-1 = 1)$-category of the form $\{\bullet_1 \to \bullet_2\}$. Assume $\mathrm{Interval} = \Sigma \mathbb{R}$, for definiteness.

Then we might be tempted to simply copy the construction for the 1-particle from above and say that $$\mathrm{worldvol} := \{\bullet_1 \to \bullet_2\} \times \mathrm{Interval}$$ is the worldvolume category of that string.

But does that make sense? Not if we take the ordinary cartesian tensor product of categories here: then $\{\bullet_1 \to \bullet_2\} \times \mathrm{Interval}$ would just contain a bunch of 1-morphisms and would in no way model the strip swept out by the string as it propagates along.

But then, we shouldn’t do that in the first place! It’s a violation of the dao. The 2-particle lives in the context of $2\mathrm{Cat}$ – even though $\mathrm{par}$ and $\mathrm{Interval}$ might both be degenerate in that no nontrivial 2-morphisms are present. But actually, as soon as we become more serious about all this, these will be 2-categories proper (for instance as we pass to the super $n$-particle). Certainly the walking monad is a 2-category, for instance.

As soon as we follow the dao again by properly working in 2$\mathrm{Cat}$, abstract nonsense alone turns out to produce the proper model for the worldsheet of the string for us:

as you all know, the “right” tensor product on $2\mathrm{Cat}$ to use is not the naive cartesian product $\times$, which is the ordinary cartesian product on the sets of morphisms. That’s because this would fail to be adjoint to the internal $\mathrm{Hom}$ of $2\mathrm{Cat}$ (the 2-category of 2-functors between two given 2-categories).

Rather, one defines a tensor product $$\otimes : 2\mathrm{Cat} \times 2\mathrm{Cat} \to 2\mathrm{Cat}$$ such that it is the adjoint functor to the internal Hom, i.e. such that for all 2-categories $A$, $B$ and $C$ we have an equivalence $$\mathrm{Hom}(A \otimes B, C) \simeq \mathrm{Hom}(A , \mathrm{Hom}(B, C))$$ of 2-categories.

If you don’t know how $\otimes$ is defined (I didn’t before we once talked about it here) it is actually easy and instructive to work it out in the simple cases which we are interested in here, where $$A = \mathrm{Interval} = \Sigma \mathbb{R}$$ and $$B = \mathrm{par} = \{\bullet_1 \to \bullet_2\}$$ and $C$ is something ($C = 2\mathrm{Vect}$ in our context, but that’s not relevant at all for what I am talking about here).

An object in $$\mathrm{Hom}(\mathrm{par},C)$$ is just a morphism $$V_1 \stackrel{f}{\to} V_2$$ in $C$ (which we would think of as a 2-space of state $V_i$ over each endpoint of the string, together with a 2-linear map between them associated to the bulk of the string). Now, since we are in $2\mathrm{Cat}$, a morphism in $\mathrm{Hom}(\mathrm{par},C)$ is, being a pseudonatural transformation of 2-functors on 1-categories, a square $$\array{ V_1 & \stackrel{f}{\to} & V_2 \\ \downarrow &\Downarrow^{U(t)}& \downarrow^t \\ V'_1 & \stackrel{f'}{\to} & V'_2 }$$ in $C$!

We already see intuitively that if the top and bottom horizontal lines of this square are supposed to be the spaces of 2-states of a string, then the square connecting them ought to be the propagation map associated to a pice of worldsheet along which our string travelled.

And indeed, that’s what demanding an equivalence $$\mathrm{Hom}(\mathrm{Interval} \otimes \mathrm{par}, C) \simeq \mathrm{Hom}(\mathrm{Interval} , \mathrm{Hom}(\mathrm{par}, C))$$ implies: the 2-category which we are supposed to address as $$\mathrm{Interval} \otimes \mathrm{par}$$ has to be such that a 2-functors from here to $C$ are given by squares in $C$ as above. But that means that $\mathrm{Interval} \otimes \mathrm{par}$ has to be the abstract such square of length $t$! $$\mathrm{Interval} \otimes \mathrm{par} = \left\lbrace \array{ \bullet_1 & \to & \bullet_2 \\ \downarrow &\Downarrow^\sim& \downarrow^t \\ \bullet_1 & \to & \bullet_2 } \;\; | t\in \mathbb{R} \right\rbrace \,.$$

(Physicists will have noticed that all I am talking here are the free propagators, where the $n$-particle just travels along its way, without interacting with other $n$-particles.)

Indeed, that’s how the Gray tensor product works: it says that the product $C \otimes D$ of two 2-categories is such that what used to be a commuting square $$\array{ (c_1,d_1) & \stackrel{g}{\to} & (c_1,d_2) \\ \downarrow^f && \downarrow^f \\ (c_2, d_1) & \stackrel{g}{\to} & (c_2,d_2) }$$ in $C \times D$ for any two morphisms $c_1 \stackrel{f}{\to} c_2$ and $d_1 \stackrel{g}{\to} d_2$ in $C$ and $D$ respectively, is replaced in $C \otimes D$ by a mere 2-isomorphism $$\array{ (c_1,d_1) & \stackrel{g}{\to} & (c_1,d_2) \\ \downarrow^f &\Downarrow^\sim_{\mu(f,g)}& \downarrow^f \\ (c_2, d_1) & \stackrel{g}{\to} & (c_2,d_2) } \,.$$

Here we see that these 2-morphisms added by using the Gray tensor product instead of the ordinary cartesian product provide us with the worldsheet of the string!

$n=3$: Membrane Worldvolumes

I don’t recall having seen the analogue of the Gray tensor product discussed for 3-categories (can anyone point me to some literature?) but from going throught the same kind of simple argument as before, for the simple parameter space categories which we are dealing with, it is clear how the pattern continues.

For instance for $\mathrm{par} = \{ \array{ & \nearrow \searrow \\ \bullet_1 &\Downarrow& \bullet_2 \\ & \searrow \nearrow } \}$ the disk-shaped membrane, I think one finds, with $\otimes$ denoting the tensor product in $3\mathrm{Cat}$ which is adjoint to the internal Hom there, that $$\mathrm{worldvol} := \{ \array{ & \nearrow \searrow \\ \bullet &\Downarrow& \bullet \\ & \searrow \nearrow } \} \otimes \mathrm{Interval}$$ is the 3-category whose 3-morphisms are filled cylinders over this disk.

And so on.

Finally: locally and globally extended QFT

With that said, the statement that local and global extension of QFT are mutually adjoint concepts is now most immediate:

The propagation $n$-functor of our $n$-particle $$q(\mathrm{tra}) : \mathrm{worldvol} \to n\mathrm{Vect}$$ is a locally extended QFT: it assigns data to points up to $n$-dimensional pieces of worldvolume.

And it is an object in $$q(\mathrm{tra}) \in \mathrm{Hom}(\mathrm{worldvol}, n\mathrm{Vect}) \,.$$ But since we have, by our definition now, $\mathrm{worldvol} = \mathrm{Interval} \otimes \mathrm{par}$ we have in fact $$q(\mathrm{tra}) \in \mathrm{Hom}(\mathrm{Interval} \otimes \mathrm{par}, n\mathrm{Vect}) \,.$$ This means that under the equivalence $$\mathrm{Hom}(\mathrm{Interval} \otimes \mathrm{par}, n\mathrm{Vect}) \simeq \mathrm{Hom}(\mathrm{Interval} , \mathrm{Hom}(\mathrm{par},n\mathrm{Vect}))$$ our locally refined extended QFT $q(\mathrm{tra})$ maps to an $n$-functor $$\tilde q(\mathrm{tra}) : \mathrm{Interval} \to n\mathrm{Vect}^{\mathrm{par}} \,.$$

But consider what this is like:

Assume, just for simplicity, again that $\mathrm{Interval} = \Sigma \mathbb{R}$. Then, first, $\tilde q(\mathrm{tra})$ looks very much like the naive Segal-like functor describing an $n$-dimensional QFT in the Schrödinger picture: it assignes some kind of space of states to the single point, and a propagator $U(t)$ to the time interval of length $t$.

But we need to remember that $\tilde q(\mathrm{tra})$ is really to be regarded as an $n$-functor: this implies that there may be morphisms between such assignments of propagators $U(t) \to U'(t)$.

I’d have trouble drawing all the pictures here which make this quite manifest, but if it isn’t quite clear already grab an envelope in your vicinity and picture this situation: it is indeed the one of globally extended QFT: there is a manifest assignment of data only to $(n-1)$-dimensional copies of our $n$-particle (a refined notion of space of state) and to the $n$-dimensional worldvolume swept out by it – but then there are higher morphisms relating the top-level assignments to each other.

This becomes more pronounced as we use more sophisticated models for $\mathrm{Interval}$.

On the other hand, by the general abstract nonsense at work here, the globally refined extended QFT $\tilde q(\mathrm{tra})$ encodes precisely the same information as the locally refined extended QFT $q(\mathrm{tra})$ which we started with. When you write it out, you’ll see that the globally extended QFT, which seems to be missing information on the level of lower dimensional cobordisms, actually has its own refined version of “spaces of states” which it assigns to the entire $(n-1)$-dimensional $n$-particle. This keeps all the information about what is going on in lower-dimensions.

Well, it’s all a game with abstract nonsense of course. But it is actually useful. I expect to talk about what this is useful for another time. Enough for now.