The n-Category Café

Thanks for this indoctrination!

So instead of admitting that I am talking about a linear representation of some group $G$, I could equivalently say that I am talking about a model in $\mathrm{Vect}$ of the theory $\Sigma(G)$ with respect to the doctrine $\mathrm{Cat}$ of no structure.

As an exercise, try to make the general concept of ‘doctrine’ more precise.

Oh dear. I will fail miserably.

Since $\mathrm{Cat}$ may be addressed as the doctrine of no structure, a first guess would be that a doctrine is a 2-category of categories with certain structure, structure preserving functors between them and natural transformations between these.

Instead of trying to make that more precise, I ask a counter question:

whatever the right answer is, it seems clear that “doctrine” is just the “n=1”-version of a general concept.

What is a 0-doctrine? By my above guess, it would just be any category of sets with structure and structure preserving functions between them.

Maybe I should remind our physically inclined $n$-Café readers that my questions cited by John above were part of an effort to understand what it means to say that

The syntax of quantum mechanics is quantum $\lambda$-calculus. #

This is a pretty cool statement!

In my attempts to decode this statement I first caught a mutant strain # of the true concept; and the $n$-category doctor diagnosed conceptual sickness.

(I don’t know if it’s related, but over the week-end I also must have cought a more ordinary virus and became sick in the more traditional sense. Poor me.)

Given what John said here maybe I can phrase things like this:

A particular quantum theory is this:

1) a collection of Hilbert spaces $\{H_i\}$, which are the spaces of $i$-states.

For instance for the ordinary quantum mechanics of a single nonrelativistic particle there is just a single such Hilbert space, which is the space of states of precisely that particle.

As another example, in the quantum theory of a linearly extended thing like a string, we may have in general at least two different Hilbert spaces, $H_c$ of closed and $H_o$ of open string states.

Usually we want to admit several copies of the “single” quantum object we are describing, and hence we also consider all the tensor powers of the Hilbert spaces $\{H_i\}$. For instance $H_c\otimes H_c$ is the space of states of two closed strings.

It is a little easier to illustrate the following concepts in the string example than in the particle example, so I’ll stick to the string. If you don’t like strings you are invited to replace “string” by “1-dimensional spacetime” without loss of anything substantial here.

2) A quantum object in a given state may usually evolve into some other state. There are in general different “channels” along which to involve. Each such channel is characterized by the space of states of objects coming into the channel and the space of states of objects coming out of the channel on the other side.

So we are tempted to regard a “channel” (it’s really a common term in particle physics) as a morphism

For instance, a cylinder (usually called a cylinder diagram in this context) can be regarded as a channel

which describes a closed string coming in, just propagatinng without interaction and coming out again.

Slightly more interestingly, a sphere with three disks cut out (usually called the pair-of-pants diagram) may be regarded as a channel

In order to emphasize the relation of this business to the theory of (quantum) computation, we simply go ahead and rename everything encountered so far.

Instead of spaces of states we’ll talk about types. The elementary spaces of states (like $H_c$ and $H_o$ above) we call generating types.

Instead of channels we now say operations.

Composition of different channels usually obeys certain rules. For instance in the theory of “topological strings” we want a rule that the two ways to compose the pair-of-pants channel with itself lead to the same result. Such conditions we now call equations between functions.

Finally, we want to be able to consider different theories of topological strings, say, each of which involves the same sort of channels. In a lucky coincidence of terminology, different such theories are often called different models of the given quantum theory (like the “A-model” or the “B-model” of the topological string).

This “model”-terminology now becomes precise as follows.

We realize that all such theories can be regarded as functors

from 2-dimensional (open/closed) cobordisms to Hilbert spaces.

Whereas at the beginning I was thinking of quantum theories in all generality, right now I am making some sort of restriction to what are usually called “topological theories”.

As far as I understand, the crucuial point of this restriction for our current purpose is that the category $n\mathrm{Cob}$ is cartesian closed. For instance, the internal $\mathrm{hom}$-object of morphisms from the circle to the circle is just

where the overbar denotes orientation reversal.

This now finally is getting close to answering the question what it means to say that quantum $\lambda$-calculus is the syntax of quantum mechanics.

Namely, $n\mathrm{Cob}$ is a CCC (a cartesian closed category), and, using the above language of types, functions and equations, a functor (preserving the relevant structure) from any $CCC$ $C$ to $\mathrm{Hilb}$ we would call a model of the $\lambda$-theory $C$.

For the special case that $C = n\mathrm{Cob}$ we know that we may address such a functor equivalently as a model of topological “$n$-particles (= $(n-1)$-branes). For other CCCs we may still imagine addressing the functor as a quantum theory, though it might be an exotic sort of quantum theory that no physicist has ever dreamed of, I guess.

That’s currently roughly my understanding of what John and Mike are trying to tell me.

One thing I don’t understand yet is this:

is it reasonable to restrict the notion of “models in quantum mechanics” to functors whose domain is cartesian closed?

Doesn’t that exclude any “non-topological” theory?

For instance in ordinary non-relativistic QM of a single point particle we actually need 1-dimensional Riemannian cobordisms as domain.

Or the non-relativistic string. It needs 2-dimensional conformal cobordisms.

But these categories of cobordisms with extra structure are not cartesian closed, are they?

Is it therefore maybe better to say that “Quantum $\lambda$-calculus is the syntax of topological quantum mechanics”?

This is interesting. I had the idea that a doctrine is a 2-monad on Cat, but that is not consistent with your referring to various 2-categories as doctrines.

So perhaps a doctrine is the 2-category of (pseudo-)algebras for some 2-monad on Cat. But that isn’t right, because you mentioned the doctrine of CCCs.

Could it be that a doctrine is the 2-category of (pseudo-)algebras for some 2-monad on Cat_g, the 2-category of categories, functors and natural isomorphisms? That can’t be right either, because in some of your examples the 1-cells preserve all the relevant structure, but in others they preserve only some of it (e.g. the maps of CCCs preserve finite products up to isomoprhism, but not the exponentials).

So I’m guessing that the answer is something much simpler, that ignores distinctions of this sort. Perhaps a doctrine is a 2-category with finite products, together with a finite-product-preserving functor to Cat. That seems to fit your examples, but I’m sure it’s not the right answer. I look forward to learning what the right answer is. (Or else I’ll cheat and look in the Big Yellow Book tomorrow, as you suggested.)

I think I’ll keep posting the “main line” of my Socratic dialog with Urs as comments that will appear on the very bottom of this page, instead of commenting on a comment on a comment….

John wrote approximately:

Can you say anything about what it means to say $D$ is a doctrine, or 1-doctrine - that is, a 2-category whose objects are “categories with extra structure”?

Urs wrote:

Yes, yes, I can. I have heard somebody (was it you? ;-) ) talk about the yoga of stuff, structure and property once in a while.

Yes - most recently, in this thread I wrote:

We probably shouldn’t talk about this much now; it’s sort of distracting…

I was trying to tell you not to bring the yoga of stuff, structure and property into this game! It’s relevant, but only in a subtle way. It’s not the royal road to the official “right answer” - that is, Lawvere’s definition of “doctrine”.

However, even if we don’t march down the royal road, we can still get there pretty quick by sneaking down back alleys. I will try not to let you get too lost.

So - you wisely decategorified my question and tackled this warmup question first:

What does it mean to say $C$ is a 0-doctrine - that is, a category whose objects are “sets with extra structure”?

You answered approximately:

It is any category $C$ with a functor $$U : C \to \mathrm{Set}$$ such that $U$ is faithful.

Great! This is the correct answer if you follow the yoga of stuff, structure and properties. Here $U$ is a functor that sends any object of $C$ to its underlying set - I’m using $U$ here because that’s the traditional letter: $U$ for “underlying”. You used $F$, but $F$ stands for something else that’s about to be very important.

You are right that $U$ forgets at most structure iff $U$ is faithful - in other words, the objects of $C$ don’t have any extra “stuff”, they’re just a set with extra structure and properties.

And indeed, your answer is part of the official right answer, so I really should not complain about your use of the yoga!

However, the official right answer is much better, because it’s extremely elegant yet allows us to prove lots more theorems. It relies on noticing something that lots of examples have in common.

Here are a few classic examples:

• Groups: $$U : \mathrm{Grp} \to \mathrm{Set}$$
• Rings: $$U : \mathrm{Ring} \to \mathrm{Set}$$
• Complex vector spaces: $$U : \mathrm{Vect}_{\mathbb{C}} \to \mathrm{Set}$$

There’s something very nice and simple you can do in all these cases that you can’t do in these cases:

• Let $C$ be the category of 5-element sets and functions between them, and let $$U : C \to \mathrm{Set}$$ be the obvious functor.
• Let $\mathrm{Set}_0$ be the group of sets and bijections betwen them, and let $$U : \mathrm{Set}_0 \to \mathrm{Set}$$ be the obvious functor.

In both these cases $U$ is faithful - but there’s something that doesn’t work for these that works for the previous ones. And, I hope you’ll agree that intuitively, these cases feel different than the previous ones!

So, I’ve given you a clue, but I’m not sure you’ll get it. If you don’t get it maybe someone else can chime in - or else I can say two magic words that will instantly make it all clear.

Urs writes:

Hm, now I am not sure what precisely I am supposed to say.

Having another look at John’s lecture on universal algebra, I see that, as Robin indicates, too, we are really talking about $n$-monads.

So from an adjunction we get a monad, and we can always assume the context to be such that every monad comes from an adjunction (p. 21 of the above lectures).

So maybe I should just say that a 0-doctrine is a monad on $\mathrm{Set}$.

Does that imply that the adjunction

(1)$$\mathrm{Set} \stackrel{L}{\to} D \stackrel{R}{\to} \mathrm{Set}$$

corresponding to that monad has faithful $R$? Or is that an extra condition? (I have never thought about this, but maybe faithfulness is in fact a necessary condition for there being an adjoint?)

No, but… you’ve got the right idea, Urs!

A 0-doctrine is just a monad on $\mathrm{Set}$! This implies that the forgetful functor from the category of “algebras” of this monad to $\mathrm{Set}$ is faithful - but it’s much more powerful.

Let me sketch how this beautiful story goes - for details, people can read my universal algebra notes or else get serious and look at Toposes, Triples and Theories. (A “triple” is the same as a monad.)

As explained in “week89” of This Week’s Finds, a monad on a category $C$ is just a monoid $T$ in the monoidal category of functors from $C$ to itself. This monoidal category acts on $C$, so we can say what it means for $T$ to act on an object in $C$, and such an action is called an algebra of $T$.

Don’t worry if this seems mysterious. The first time someone told me this, here’s what I looked like:

Let me unpack what I just said. A monad is a functor

$$T : C \to C$$

with a multiplication

$$m: T^2 \Rightarrow T$$

and unit

$$i: 1 \Rightarrow T$$

satisfying the usual laws of a monoid: associativity and the left/right unit laws. A great example is the functor

$$T: \mathrm{Set} \to \mathrm{Set}$$

sending any set $S$ to the underlying set of the free group on $S$. Here we are building $T$ as the “free” functor (left adjoint)

$$F : \mathrm{Set} \to \mathrm{Grp}$$

followed by the “underlying” functor (right adjoint)

$$U : \mathrm{Grp} \to \mathrm{Set}$$

The properties of adjoint functors make $T$ into a monad. Indeed, any pair of adjoint functors gives a monad, and there’s an utterly simple picture proof of this using string diagrams, given in “week92”.

Continuing to unpack what I said, an algebra of a monad is an object $c \in C$ together with a morphism

$$a: T c \to c$$

satisfying the usual laws for an “action” of a monoid: the associative law and the left unit law.

It’s a great exercise to check that the algebras of the monad $$T = U F$$ built from

$$F : \mathrm{Set} \to \mathrm{Grp}$$

and

$$U : \mathrm{Grp} \to \mathrm{Set}$$

are precisely groups. There’s no possible way to truly understand what I’m saying until one does this exercise.

For any monad $T: C \to C$ we can define a category of algebras, often called $C^T$ for some silly reason. This is just like the usual category of actions of a monoid. But it’s a cool idea: in the example above it’s the category of groups!

There’s an obvious forgetful functor from algebras of our monad to the category we started with, say

$$U': C^T \to C$$

This is always faithful, since being an algebra of a monad is just extra structure on an object of $C$! The proof is not too hard - just axiom-juggling.

See, Urs? We get the faithfulness for free, from a very elegant set of ideas - we don’t need to stick it in by hand!

Furthermore, the functor $U'$ always has a left adjoint

$$F': C \to C^T$$

sending each object $c \in C$ to the “free” algebra $T c$.

So, from a monad we get a pair of adjoint functors

$$F' : C \to C^T$$

and

$$U' : C^T \to C$$

and $U'$ is always faithful.

You’ll note we went around a kind of loop. If you give me a pair of adjoint functors

$$F: C \to D$$

$$U: D \to C$$

I can cook up a monad

$$T = U F: C \to C$$

Then I can cook up the category of algebras $C^T$ of this monad, and then I get an adjunction

$$F' : C \to C^T$$

and

$$U' : C^T \to C$$

Do I get back the adjoint functors I started with? Not always! But I do in the example of groups and sets, and all the examples we’re really interested in now. So, that’s the case we want to focus on. We call such adjoint functors monadic.

We can also run around the loop starting with a monad! Start with a monad $C$, build its category of algebras $C^T$, get a pair of adjoint functors between $C$ and $C^T$, and then use this to define a monad!

In this case we always do get back where we started. So this is the nicest way to play the game. Say we take $C = \mathrm{Set}$ to be specific:

We define a 0-doctrine to be a monad

$$T: \mathrm{Set} \to \mathrm{Set}$$

This allows us to define a certain “category of sets with extra structure”, namely namely the category of algebras of $T$, or $C^T$ for short. Then we get a pair of adjoint functors

$$F' : C \to C^T$$

and

$$U' : C^T \to C$$

and $U'$ is always faithful.

Actually, nobody ever talks about 0-doctrines. But this is the idea that Lawvere categorified to define doctrines, or what we would call “1-doctrines”. In brief:

A doctrine is a monad on $\mathrm{Cat}$.

We can play the whole game as before, defining algebras of these and so on.

There’s a lot more to say, but it’s late - and this is probably far more than anyone can digest who doesn’t sort of know this stuff already!

Older patrons of the n-Category Café may recall this 70’s hit song - we used to play it all the time here:

Doctor Doctrine, give me the news!
I’ve got a hard case of logic blues!

Anyway, Mike Stay seems to have a case of logic blues here, so Dr. Doctrine had better try to cure it:

John wrote:

A monad on a category $C$ is just a monoid $T$ in the monoidal category of functors from $C$ to itself.

Mike wrote:

I’m having some trouble reconciling what you said in week89 with what you said in this thread, so I’m going to try to unpack this, too.

Okay… but you seem intent on “unpacking” each phrase by treating it as a special case of something more general, creating a truly frightening slag-heap of abstraction:

$\mathrm{hom}(C,C)$ is the image of a product-preserving functor $\mathrm{Th}(\mathrm{Mon}) \to \mathrm{Cat}$ and a monad is the image of a product-preserving functor $\mathrm{Th}(\mathrm{Mon}) \to \mathrm{hom}(C,C)$.

Zounds! I’ve created a monster! It’s not clear that taking every concept and expressing it in terms of algebraic theories actually helps here. I’m going to have to break this down into tiny shards to understand it.

Anyway, the first half is correct:

$\mathrm{hom}(C,C)$ is the image of a product-preserving functor $\mathrm{Th}(\mathrm{Mon}) \to \mathrm{hom}(C,C)$.

Translating into slightly less scary language, this says

$\mathrm{hom}(C,C)$ is a model in Cat of the algebraic theory of monoids.

Here you are using the fact that Cat has finite products to describe monoids in Cat using the algebraic theory of monoids. But you’re really just saying this:

$\mathrm{hom}(C,C)$ is a monoid in Cat.

A monoid in Cat is usually called a strict monoidal category: it has a tensor product

$$\otimes : C \times C \to C$$

and unit

$$1 \in C$$

for which the monoid laws (associativity, right/left unit laws) hold “strictly”, as equations. So, you’re saying:

$\mathrm{hom}(C,C)$ is a strict monoidal category.

And indeed, this is true if we take composition

$$\circ : \mathrm{hom}(C,C) \times \mathrm{hom}(C,C) \to \mathrm{hom}(C,C)$$

as the tensor product, and the identity functor

$$1 \in \mathrm{hom}(C,C)$$

as the unit object. So, you’re really just saying this:

Composition in $\mathrm{hom}(C,C)$ is associative, and $1: C \to C$ serves as a left and right unit for composition.

Whew! Now it doesn’t sound so scary.

The second part, alas, is wrong:

A monad is the image of a product-preserving functor $\mathrm{Th}(\mathrm{Mon}) \to \mathrm{hom}(C,C)$.

And, you explain why: $\mathrm{hom}(C,C)$ doesn’t have finite products! It’s a monoidal category - as we’ve just seen - but it’s tensor product is not the “product” in the categorical sense.

It’s not a big problem. You’re just using the wrong doctrine to describe monoids in $\mathrm{hom}(C,C)$: you’re trying to use categories with finite products (= algebraic theories), but you should be using monoidal categories (= PROPs). There’s an algebraic theory for monoids, but there’s also a PROP for monoids, and we should use that. So, you should have said:

A monad is the image of a monoidal functor $\mathrm{PROP}(\mathrm{Mon}) \to \mathrm{hom}(C,C)$.

but this is just a fancy way of saying

A monad is a monoid in $\mathrm{hom}(C,C)$.

A monoid in $\mathrm{hom}(C,C)$ would be, as usual, an object $T$ together with a multiplication

$$m : T T \Rightarrow T$$

(note the tensor product in $\mathrm{hom}(C,C)$ is composition, and morphisms are natural transformations) and a unit

$$i: 1 \Rightarrow T$$

(note the unit object in $\mathrm{hom}(C,C)$ is $1: C \to C$), satisfying associativity and the left/right unit laws. So, you’re saying

A monad is a functor

$$T : C \to C$$

together with natural transformations

$$m : T^2 \Rightarrow T$$

$$i: 1 \Rightarrow T$$

satisfying the monoid laws.

I could unpack further and show what the monoid laws look like here - but they’re hard to draw, and you can find them in Tom Leinster’s notes or any decent book on category theory.

For the present purposes, “decent books on category theory” include Categories for the Working Mathematician and Toposes, Triples and Theories. The latter calls monads “triples”, and it’s tough going for the beginner, but it’s free, and it’s a great place to read if you’re interested in both monads and algebraic theories as ways of describing algebraic gadgets.

There are lots of nice relations between monads and algebraic theories, which we haven’t gotten to yet…

(By the way, Mike: I know most of what I said in this post you already figured out yourself in yours. I just thought it would be good to go through everything in detail, so everyone could get a nice workout!)