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Stream: learning: questions

Topic: DOTS, TQFTs and n-theories

David Corfield (Sep 24 2025 at 12:55):

Perhaps people can help me out here, if anything resonates with them. I'm looking to make sense of the central definition of a doctrine of system theories from @Sophie Libkind and @David Jaz Myers's Towards a double operadic theory of systems paper (hereafter DOTS),

image.png

I feel drawn to doing so via (1) the concept of a topological quantum field theory (TQFT), and (2) @Mike Shulman's What is an n-theory?.

I'll start with (1), just pointing out for now regarding (2) that Mike is placing the notion of a doctrine = 2-theory in the larger framework of $n$-theories. In David Jaz's earlier Categorical Systems Theory, he includes the remark:

image.png

which is at least in the same neck of the woods.

David Corfield (Sep 24 2025 at 13:05):

So why TQFTs? Recall that a [[TQFT]] is typically taken as a functor from some [[category of cobordisms]], $Z: Cob_n \to V,$ for some monoidal $V$ such as $k\text{-}Vect$. If you want that functor to be lax then no problem, people are also now considering lax TQFTs in mainstream mathematics (see, e.g., here).

That TQFTs might be thought to be of relevance to DOTS may be seen from its final words, where the authors speak of

... modules of Moore machines, POMDPs, and so on over double categories of directed wiring diagrams.

When you consider that wiring diagrams may be cast as cobordisms, as in

• David I. Spivak, Patrick Schultz, Dylan Rupel, String diagrams for traced and compact categories are oriented 1-cobordisms,

then it seems we have some form of modules being assigned to cobordisms.

Now maybe @John Baez is right to caution me that this may amount to little more than a kind of functorial semantics, but let's carry on a little.

David Corfield (Sep 24 2025 at 13:37):

Back to DOTS, first there's the issue of the interface, interaction, interface map double category. Here are a couple of examples from chap. 4. First in terms of undirected wiring diagrams:

image.png

Then a directed wiring interface/interaction:

image.png.

Then to such diagrams are assigned systems that have those interfaces, so in the top interaction of the first diagram we have:

image.png

The interaction provides a mapping from the module of systems with the first interface to that of the second. This is sitting over what they call the walking loose arrow

image.png

the right part annotated as

image.png.

Note that in this presentation we no longer show the identity morphisms and square for the left
object $\bullet$, because in a systems theory they do not label any non-identity morphisms and squares.

Now, I was reminded of the extension to TQFTs which allows for boundaries/singularities. Consider this section from Urs Schreiber's dcct, p. 522:

image.png

In such a situation with TQFTs, as Jacob Lurie in his 2009 paper remarks, this extra singular morphism is sent to a state of the module assigned to the codomain:

image.png

Having extended $\mathbf{Bord}_n$ by this extra morphism to $\mathbf{Bord}'_n$,

image.png

So it looks like we're dealing with some kind of double category of cobordisms/interfaces with a boundary. And each interface gets assigned a module of systems, while the singular boundary arrow picks out a particular system in that module.

David Corfield (Sep 24 2025 at 14:01):

A bit faster on (2) What is an n-theory?. A doctrine = a 2-theory at the very least involves some kind of 2-category. Just looking at the definition of DOTS (p. 49), we have a 2-category, $\mathcal{D}_{inter}$, which will be sent to the 2-category of double categories (and then systems to right loose modules).

It's looking like we're up a level from the passage where Mike is telling us about a syntactic 1-theory giving rise to a walking model of a semantic 1-theory which then takes values in a suitable category such as $Set$:

image.png

Would it make sense to see $\mathcal{D}_{Sys} \to \mathcal{D}_{Inter}$ as of the form of a semantic 2-theory, looking to be interpreted in double categories and loose right modules?

John Baez (Sep 24 2025 at 16:25):

David Corfield said:

So why TQFTs? Recall that a [[TQFT]] is typically taken as a functor from some [[category of cobordisms]], $Z: Cob_n \to V,$ for some monoidal $V$ such as $k\text{-}Vect$. If you want that functor to be lax then no problem, people are also now considering lax TQFTs in mainstream mathematics (see, e.g., here).

That TQFTs might be thought to be of relevance to DOTS may be seen from its final words, where the authors speak of

... modules of Moore machines, POMDPs, and so on over double categories of directed wiring diagrams.

When you consider that wiring diagrams may be cast as cobordisms, as in

• David I. Spivak, Patrick Schultz, Dylan Rupel, String diagrams for traced and compact categories are oriented 1-cobordisms,

then it seems we have some form of modules being assigned to cobordisms.

What I'm trying to argue is that directed wiring diagrams are quite unlike cobordisms and also quite unlike string diagrams for compact categories... while undirected wiring diagrams are very much related to these. Here you seem to be blurring the distinction, which makes me unhappy.

I could be wrong, but I think there are two quite different worlds, one including

• directed wiring diagrams, lenses, and the doctrine of input-output machines

and another including

• undirected wiring diagrams, cospans, and the doctrine of port-plugging systems.

TQFT live in something closer to the second world, although not really.

If you are, however, mainly interested in generalities about the DOTS formalism, then this distinction between worlds may not matter to you, since DOTS is supposed to be a big tent that holds many doctrines.

David Corfield (Sep 24 2025 at 17:14):

John Baez said:

What I'm trying to argue is that directed wiring diagrams are quite unlike cobordisms

I must be missing something. Do you (a) disagree with Spivak et al. that these below are cobordisms, or (b) have some other idea of directed wiring diagrams?

image.png

As for (b), aren't they talking about these things

image.png

Mike Shulman (Sep 24 2025 at 22:09):

Neither here nor there, but I enjoy how this paper has proclamations like "Explication 3.24" and "Attitude 4.3".

Mike Shulman (Sep 24 2025 at 22:13):

I think in my terminology I would say that each of their "modules of systems" is a "double theory" in a suitable doctrine, determined (partly) by slicing over the walking loose arrow. (Slicing is generally a good way to produce doctrines that separate out objects into different syntactic categories.) I'm not sure exactly what is "doctrinal" about their "doctrines of systems theories".

John Baez (Sep 24 2025 at 22:48):

David Corfield said:

John Baez said:

What I'm trying to argue is that directed wiring diagrams are quite unlike cobordisms

I must be missing something. Do you (a) disagree with Spivak et al. that these below are cobordisms, or (b) have some other idea of directed wiring diagrams?

Sorry, Zulip makes it very hard to show again a picture someone else uploaded.

Your picture shows an oriented 1-dimensional cobordism, and in that passage Spivak et al also call it a 'wiring diagram' - but he has called various things wiring diagrams in various papers. What Spivak and Rupel called and what we now all call a 'directed wiring diagram' is something more general that crucially allows for the splitting (but not joining) of directed wires. Here is an example of a directed wiring diagram:

directed_wiring_diagram.jpg

In a 1d cobordism a wire can't split like this!

On the other hand, when dealing with input-output machines it's essential that we allow wires to split like this, since we're working in a paradigm where signals can be duplicated. That's when Spivak's directed wiring diagrams become important.

John Baez (Sep 24 2025 at 23:02):

We need to distinguish two worlds, one that's all about

• directed wiring diagrams, lenses, and the doctrine of input-output machines

and that's all about

• undirected wiring diagrams, cospans, and the doctrine of port-plugging systems.

The operad of directed wiring diagrams is another way of talking about a certain symmetric monoidal category of lenses, while the operad of undirected wiring diagrams is another way of talking about the symmetric monoidal category of cospans of finite sets.

Libkind and Myers discuss both of these in their paper Toward a double operadic theory of systems, but they carefully distinguish between them.

Now that I think about it more carefully, I realize that TQFTs actually live in yet another world, or systems doctrine in Libkind and Myers' terminology. I could say more about this.

John Baez (Sep 24 2025 at 23:07):

Mike Shulman said:

I think in my terminology I would say that each of their "modules of systems" is a "double theory" in a suitable doctrine, determined (partly) by slicing over the walking loose arrow. (Slicing is generally a good way to produce doctrines that separate out objects into different syntactic categories.) I'm not sure exactly what is "doctrinal" about their "doctrines of systems theories".

Each of their "doctrines" has many "theories", and each "theory" has many "models".

Mike Shulman (Sep 24 2025 at 23:18):

The obvious thing that I would naturally call a doctrine in their setup is the 2-category of modules of systems. Are their "doctrines" basically just sub-2-categories of this? (Or, I guess, more general 2-categories over it?)

Kevin Carlson (Sep 25 2025 at 00:19):

Yeah, that $\ell\mathcal{M}\mathrm{od}_r$ guy that a doctrine of systems theories is over is exactly the 2-category of modules of systems, so I think that lines you up with David and Sophie.

David Corfield (Sep 25 2025 at 07:23):

John Baez said:

In a 1d cobordism a wire can't split like this!

So what stops us talking about 2d cobordisms in this regard? Or something like this kind of $2Thick$ cobordism from TQFTs and quantum computing:

image.png

(Incidently, I came across this paper wondering why more people don't think of cobordisms in terms of double categories. Pretty natural, no?)

(Didn't $2Thick$ originate with your old student, Aaron Lauda, in Frobenius algebras and planar open string topological field theories? That's what they confirm, having mention some of @Mike Shulman's work at the start of Sec. 2.)

Anyway, irrespective of whether to phrase what is common to these doctrines of systems theories as something TQFT-like, the larger issue for me is whether there is anything substantial to say about what is common. What would a 3-theory of these doctrines look like?

I see you're thoroughly invested in keeping these directed and undirected wiring diagram approaches separate, and as you guess I'm more interested in what you call the "big tent" of DOTS. At the very least a little probing of how to characterize what's allowed in the big tent may be worthwhile. As they admit:

Remark 5.3 (On the minimal notion of doctrine). Our definition of doctrine is rather minimal. We choose to go with this minimal definition so that it can act as an organizing principle, rather than as a mathematical object of study in its own right (which might require us to discover further structure carried by doctrines of systems theories in particular, above their 2-functoriality). That is to say, we will construct examples of doctrines as defined in Definition 5.1, but we will not investigate any higher category of doctrines.

Nevertheless, even with such a minimal definition of doctrine, the notion will help us organize the vast array of systems theories in use by applied category theorists. We may make use of morphisms of doctrines given by precomposition to express that one doctrine is a special case of another. In the upcoming Section 8, we will perform a construction at the doctrine level to restrict a doctrine to wiring diagrams or free processes.

If we had a tighter grip on this notion, maybe the tent wouldn't look so big, and perhaps we could even construct new doctrines.

David Corfield (Sep 25 2025 at 07:36):

That article co-authored by Khovanov, Automata and one-dimensional TQFTs with defects, I keep mentioning to you, which understands a non-deterministic finite state automaton as a kind of Boolean 1d-TQFT with defects, goes on to deal with splitting:

Motivated by these relations, one can introduce rigid symmetric monoidal category FCob with sign sequences as objects. It contains Cob as a subcategory and has additional trivalent vertex and inner endpoint generators as in Figure 4.2.1. Relations in Figures 4.2.2, 4.2.3, and 4.2.4 hold, as well as the standard relations from the symmetric structure on trivalent and univalent graphs, such as sliding a disjoint line over a vertex. One may or may not impose Figure 4.2.5 relation.

They call the morphisms "one-foams with boundary".

image.png

"Cob" is in "FCob" for a reason, :slight_smile: .

John Baez (Sep 25 2025 at 08:28):

I see you're thoroughly invested in keeping these directed and undirected wiring diagram approaches separate...

Yes, sort of like how I'm "thoroughly invested" in distinguishing between even and odd numbers. Please don't psychologize this: I'm just pointing out a logical distinction.

In fact I'm busy helping write software that will need to unify these two doctrines, because so far it uses the variable-sharing doctrine when composing models (e.g. identifying the population of infected people in one model with those in another), but we also want it to use the input-output doctrine (e.g. using the infection rate computed by one model as a parameter in another model).

perhaps we could even construct new doctrines....

Indeed, we do.

Sophie Libkind has already unified the input-output doctrine and variable sharing doctrine (which she called the resource sharing doctrine) in her paper

• Sophie Libkind, An algebra of resource sharing machines.

By the way, this paper has a good explanation of the distinction between the two doctrines, starting in the abstract:

To some, open dynamical systems are input-output machines which interact by feeding the input of one system with the output of another. To others, open dynamical systems are input-output agnostic and interact through a shared pool of resources. In this paper, we define an algebra of open dynamical systems which unifies these two perspectives.

Her unification feels a bit ad hoc to me, but surely it's a step in the right direction. I found the concept of 'undirected wiring diagram' complicated and ad hoc until Fong and Spivak discovered that the operad of undirected wiring diagrams is really just the symmetric monoidal category of cospans of finite sets:

• Brendan Fong and David Spivak, Hypergraph categories.

And I found the concept of 'directed wiring diagram' complicated and ad hoc until this paper showed the operad of directed wiring diagrams is really just a particular symmetric monoidal category of lenses:

• Sophie Libkind, Andrew Baas, Evan Patterson, James Fairbanks, Operadic modeling of dynamical systems: mathematics and computation.

I imagine something similar will happen with Sophie's resource sharing machines: either they'll be formalized as a doctrine with a spiffy definition, or something a bit more general will be.

David Corfield (Sep 25 2025 at 10:01):

John Baez said:

Yes, sort of like how I'm "thoroughly invested" in distinguishing between even and odd numbers. Please don't psychologize this: I'm just pointing out a logical distinction.

Sorry, I didn't mean to offend. But logical distinctions may be transcended in a sense. To use your example, 'even' and 'odd' participate in one of Lawvere's 'unity of opposites' via adjoint modalities. But I'm new to all this, maybe there's a good sense already of what kind of structure the collection of systems doctrines possesses and little hope of anything as lovely as adjoint modalities.

John Baez (Sep 26 2025 at 10:07):

Thanks.

I don't know anything about the collection of systems doctrines - you'd have to talk to David Jaz about that.

I have my nose closer to the pavement. I'm more interested in how various frameworks you're talking about allow for various kinds of information flow, obeying various rules. You're making me realize that we can get fine-grained control over which variables can get duplicated, deleted, 'fed back', 'affected by what they affect', and all that, in a way that lets us put variable-sharing, input-output, TQFTs, and other frameworks into a more flexible but still not too floppy framework.

Btw, TQFTs are different than variable-sharing/port-plugging because in a TQFT if you split a wire and then re-merge the two pieces, that's usually not the identity - while in variable sharing this is demanded. This was one of the things I was grumbling about earlier. But this is the sort of thing we should be allowed to make up our minds about on a case-by-case basis, if we want.

David Corfield (Sep 26 2025 at 10:19):

So far from the DOTS paper we see the collection (3-category?) of doctrines breaking into two, as you suggest, with restrictions on what they call the doctrine of initial process including port-plugging and variable-sharing, and then, on the directed side, the doctrine of generalized Moore machines and restrictions.

I've come across 'circular port graphs' in

• Sophie Libkind, Andrew Baas, Evan Patterson, James Fairbanks, Operadic Modeling of Dynamical Systems: Mathematics and Computation,

(a paper which plays a confusing trick of referring to an earlier version (v2) of itself as [11]).

The operad of these is a suboperad of the operad of directed wiring diagrams. Is there a kind of machine/system that corresponds to them?

David Corfield (Sep 26 2025 at 10:26):

One reason I'm interested in the big picture here is that DOTS sets out a way of doing a kind of formal metaphysics, Sec 1.3 in particular. What else were systems going to be but things sitting over $\mathbb{L}oose$, the walking loose arrow?

John Baez (Sep 26 2025 at 10:28):

The operad of these is a suboperad of the operad of directed wiring diagrams. Is there a kind of machine/system that corresponds to them?

I don't know.

So far from the DOTS paper we see the collection (3-category?) of doctrines breaking into two, as you suggest, with restrictions on what they call the doctrine of initial process including port-plugging and variable-sharing, and then, on the directed side, the doctrine of generalized Moore machines and restrictions.

I wouldn't dare say it "breaks in two", as if there were nothing else out there. My wisecrack about even and odd numbers was not meant to say that we have an exhaustive classification. Those two (or three) doctrines are the examples Sophie and David happen to talk about. They're extremely important examples, but it should be pretty easy to come up with lots of others if one wants.

David Jaz Myers (Sep 26 2025 at 13:46):

Hi all,

I'm chuffed to see this discussion about our paper here.

Mike Shulman said:

The obvious thing that I would naturally call a doctrine in their setup is the 2-category of modules of systems. Are their "doctrines" basically just sub-2-categories of this? (Or, I guess, more general 2-categories over it?)

This is correct: a doctrine as we define it is a cartesian pseudo-functor into the 2-category of loose right modules of double categories. We take the attitude that symmetric monoidal loose right modules of systems (i.e. pseudo-monoid objects in this cartesian 2-category) are modules of systems over double categories of interfaces and interactions. So a "doctrine" is just a 2-functorial way to produce modules of systems from simpler data.

We really only mean this as a stepping-stone definition to help organize the variety of structures used in categorical systems theory. I don't think we've quite "carved nature at it's joints" with this one yet.

Nevertheless, here's how I would connect this to the more usual notion of doctrine. A doctrine is usually something like a variety of 2-algebraic structures which enables so-structured categories to act as (the invariant content of) a theory. For example, the doctrine of essentially algebraic theory is the 2-algebraic theory of finitely complete categories. To hew closer to this sense of the term, we should just consider the 2-category of symmetric monoidal loose right modules as the doctrine of modules of systems; but this is not quite how I think about it.

A doctrine describes what it means to be a theory. But it also describes what it means to be a model of a theory, since in all the usual cases we have a functorial semantics: a model is just a morphism of theories (into, perhaps, a fixed "large" theory). But if we didn't have a functorial semantics, we should also include what it means to be a model of a theory into our notion of doctrine. This means that instead of taking just some 2-category $\mathcal{K}$ of 2-algebraic structures to be a doctrine; we should also include the construction of "categories of models" in the form of a 2-functor $\mathsf{Mod} : \mathcal{K} \to \mathcal{S}$.

In the usual case, we would have $\mathcal{S} = \mathcal{C}\mathsf{at}$ and $\mathsf{Mod} = \mathcal{K}(-, \mathsf{Set})$, so this extra data is implicit. Sometimes we even find that the categories of modules all uniformly carry some structure, and that this may even suffice to reproduce the (invariant content of the) theories themselves; this is the case for essentially algebraic theories, whose categories of models are all locally finitely presentable; or, coherent theories, whose categories of models all have some sort of ultrastructure; etc.

We don't have any of this extra richness, but this is the sense in which we mean that a 2-functor into symmetric monoidal loose right modules is a "doctrine"; the domain 2-category is the 2-category of theories, while the 2-functor itself sends every theory to its symmetric monoidal loose right module of models. In as much as our starting theory is a "systems theory", we end up with a "module of systems". Usually it is the case that the domain of one of our doctrines is a category-with-structure in the usual sense of a doctrine; we're just supplying them with a different notion of "model".

David Jaz Myers (Sep 26 2025 at 13:55):

John Baez said:

Those two (or three) doctrines are the examples Sophie and David happen to talk about. They're extremely important examples, but it should be pretty easy to come up with lots of others if one wants

This is totally right, we just had time for these three. But we hope that we've laid some of the ground to make it easy to spin up new ones as they become more interesting.

For example, we can define a doctrine of "flavored Petri nets" in the sense of Fabrizio Genovese and David Spivak by sending each symmetric monoidal double category $\mathbb{C}$ "of color spaces and guarded transitions" to the symmetric monoidal loose right module

$\mathbb{Q}\mathsf{t}(\mathcal{S}\mathsf{ymMon}(\mathcal{D}\mathsf{bl}))^{\mathsf{co}}(\mathbb{C}, \mathsf{Proc}(-))$

where $\mathsf{Proc}(P)$ is Joachim Kock's symmetric monoidal (double) category of *processes* associated to a Petri net $P$. I describe this in two recent talks; but the point is that once you can form hom-bimodules and restrict them, you can come up with new modules of systems quite readily.

David Jaz Myers (Sep 26 2025 at 14:11):

One thing that feels different to me about TQFTs and the modules of systems we describe in the paper --- and forgive me for speaking vaguely here --- the way that temporality is handled. Roughly speaking, in one the time variation is of the system, while in the other it is in the system. We could think of a TQFT as giving rise to a symmetric monoidal loose right module by restricting the hom of the target (double) category by the functor from the bordism (double) category; but in this case the "time-evolution" of the system is through the bordisms. So this is an evolution of the system by composition / interaction.

In contrast, in for example Moore machines, a behavior of the system is a map into it from an appropriately chosen "clock" system. So that time variation is in the system. Composition of systems is on a different axis to time-variation within a system.

David Corfield (Sep 26 2025 at 15:03):

David Jaz Myers said:

In contrast, in for example Moore machines, a behavior of the system is a map into it from an appropriately chosen "clock" system. So that time variation is in the system. Composition of systems is on a different axis to time-variation within a system.

That sounds like it tallies with Khovanov et al in Automata and one-dimensional TQFTs with defects, right? The time variation there is along labelled (internal) 1-d cobordisms, taking a Boolean vector of initial states to a vector of final states.

But by the time people have taken TQFTs to, say, arithmetic, as here, do we care about giving a temporal interpretation to them?

John Baez (Sep 26 2025 at 15:07):

We could make TQFTs more like the other systems David's formalism describes by working with extended TQFTs. Then instead of a cobordism describing spacetime, it could describe space - with a boundary, making the fields on that space into an "open system".

The big difference from the port-plugging doctrine is that now instead of hypergraph categories (as you expect from the port-plugging doctrine), you get compact closed categories.

David Jaz Myers (Sep 26 2025 at 15:53):

David Corfield said:

That sounds like it tallies with Khovanov et al in Automata and one-dimensional TQFTs with defects, right? The time variation there is along labelled (internal) 1-d cobordisms, taking a Boolean vector of initial states to a vector of final states.

Yep, that's what I mean. It also vies with section 1.2.10 of Urs's DCCT (arXiv version).

David Corfield (Sep 27 2025 at 08:09):

David Jaz Myers said:

A doctrine describes what it means to be a theory. But it also describes what it means to be a model of a theory, since in all the usual cases we have a functorial semantics: a model is just a morphism of theories (into, perhaps, a fixed "large" theory). But if we didn't have a functorial semantics, we should also include what it means to be a model of a theory into our notion of doctrine. This means that instead of taking just some 2-category $\mathcal{K}$ of 2-algebraic structures to be a doctrine; we should also include the construction of "categories of models" in the form of a 2-functor $\mathsf{Mod} : \mathcal{K} \to \mathcal{S}$

How do we choose the $\mathcal{S}$?

If we take this thought down to the 1-level, say we're considering monoids, and we have the Lawvere theory of monoids, then, as you say, there's a standard fixed theory to map into for models, here $Set$. But of course there are other codomains to provide models, and we have the [[microcosm principle]] to guide us . So here something like

for a Lawvere theory $L$, an internal $L$-algebra in a categorical $L$-algebra, $A$, (an $L$-algebra in Cat) is a lax $L$-morphism from the terminal such algebra to $A$.

Monoids make sense internal to monoidal categories. Presumably there's a similar account with 2-categorical $L$-algebras for some algebraic 2-theory.

David Corfield (Sep 27 2025 at 08:16):

While I'm on the subject, for TQFTs, how do we know where to find models of, say, $1\text{-}Cob$? Not a Lawvere theory, but in some doctrine (like symmetric monoidal categories), there's the semantic 1-theory $1\text{-}Cob$. Does the microcosm principle point me to interpret it in $1\text{-}Cob$-algebras in $Cat$?

Cole Comfort (Sep 27 2025 at 15:13):

David Corfield said:

While I'm on the subject, for TQFTs, how do we know where to find models of, say, $1\text{-}Cob$? Not a Lawvere theory, but in some doctrine (like symmetric monoidal categories), there's the semantic 1-theory $1\text{-}Cob$. Does the microcosm principle point me to interpret it in $1\text{-}Cob$-algebras in $Cat$?

I think that the only 1-cob algebra in Cat is the terminal category.
But I think that 1-cob algebras in Prof should be categories which are isomorphic to their opposite category, in some coherent way, up to size considerations.

David Corfield (Sep 28 2025 at 07:52):

Thanks. That fits neatly with the generalization of functorial semantics to $Rel$ in

• Filippo Bonchi, Dusko Pavlovic, Pawel Sobocinski, Functorial Semantics for Relational Theories, (arXiv:1711.08699)

Whereas the universe for models of Lawvere theories is the category of sets and functions, or more generally cartesian categories, Frobenius theories take their models in the category of sets and relations, or more generally in cartesian bicategories.

David Jaz Myers (Sep 28 2025 at 11:40):

David Corfield said:

How do we choose the $\mathcal{S}$?

It's usually a 2-category of some flavor of structured categories itself. I would say that you start with $\mathcal{C}\mathsf{at}$ and then record the important extra structure of the category of models that you need. In the DOTS case, this means recording a coloring in (i.e. coloring a system by its interface) and action by (i.e. composing systems through interaction) a symmetric monoidal double category that you must also produce from the theory.

John Baez (Sep 28 2025 at 11:57):

Cole Comfort said:

David Corfield said:

While I'm on the subject, for TQFTs, how do we know where to find models of, say, $1\text{-}Cob$? Not a Lawvere theory, but in some doctrine (like symmetric monoidal categories), there's the semantic 1-theory $1\text{-}Cob$. Does the microcosm principle point me to interpret it in $1\text{-}Cob$-algebras in $Cat$?

I think that the only 1-Cob algebra in Cat is the terminal category.

That sounds believable. The basic reason $1\mathsf{Cob}$ algebras in $\mathbf{Cat}$ are bad is this:

$1\mathsf{Cob}$ is a compact closed symmetric monoidal category, so any symmetric monoidal functor

$F: 1\mathsf{Cob} \to X$

will send all objects of $1\mathsf{Cob}$ to objects that have duals.

But if we take $X$ to be a cartesian monoidal category, I believe the only object in $X$ with a dual is the terminal object. Is that right?

We could take $X$ to be a cartesian monoidal 2-category like $\mathbf{Cat}$, but I don't think that changes this story much.

This is all about classical versus quantum logic. A quantum-like theory will have models in a quantum-like environment. So we should look for models of $1\mathsf{Cob}$ not in $(\mathbf{Cat},\times)$ but something like the category $(\mathsf{Vect}, \otimes)$ or the 2-category $(2\mathsf{Vect}, \otimes)$.

Cole Comfort (Sep 28 2025 at 13:27):

@John Baez

In some sense profunctors are categorified modules, which is why I suggested them over Cat.

If we looked at TQFT from oriented 1-cobordisms into Prof then this should pick out a category, its opposite category, as well as the evaluation and coevaluation profunctors linking them together and forcing the category to be small.

On the other hand if we looked at a TQFT into Vect this would pick out a vector space, its dual vector space, and require both to be adjoints in the sense of compact closure, forcing the vector space and its dual to be finite.

So I would say that TQFT type things are naturally interpreted in categories of finite dimensional modules.

Mike Shulman (Sep 28 2025 at 14:55):

In what sense does a duality in Prof "force a category to be small"?

Cole Comfort (Sep 28 2025 at 15:02):

Oops, I mean locally small, because the evaluation and coevaluation are witnessed by the yoenda and coyoneda embedding. Although I guess profunctors don't make sense between non-locally small categories?

John Baez (Sep 28 2025 at 15:04):

Does anyone know if my guess is right: an object with a dual in a cartesian category must be terminal?

Mike Shulman (Sep 28 2025 at 15:09):

You can define (small-set-valued) profunctors between any kind of category, but only locally small categories have identity profunctors, and only profunctors between small categories can be composed. So large categories and profunctors form a virtual double category with restrictions, locally small categories and profunctors form a virtual equipment, and small categories and profunctors form an equipment.

So in particular, since the zigzag laws of a duality already require composing profunctors, I would say you already have to have small categories before you can even ask what a duality means.

Mike Shulman (Sep 28 2025 at 15:13):

John Baez said:

Does anyone know if my guess is right: an object with a dual in a cartesian category must be terminal?

Yes, I believe that's right.

Cole Comfort (Sep 28 2025 at 15:14):

John Baez said:

Does anyone know if my guess is right: an object with a dual in a cartesian category must be terminal?

If I recall correctly, cartesian, compact closed categories are isomorphic to the terminal category. But what is kind of cool is that cartesian *-autonomous categories are Boolean algebras... so if you weaken what you mean by "duals" then things get a little bit less degenerate.

Cole Comfort (Sep 28 2025 at 15:28):

@John Baez Although I have no idea if there is any reasonable notion of *-autonomous "cobordisms" with which to defined some sort of generalized TQFT out of. The closest kind of thing that I could find were the surfaces in this paper:
https://arxiv.org/abs/1601.05372

Me an my colleagues were thinking of a similar question for generalizing TQFTS to rig categoires, but we couldn't find a satisfying answer. We were wondering how this paper could be framed in a similar language to TQFTs:
https://arxiv.org/abs/2307.03075

John Baez (Sep 28 2025 at 15:39):

Once I wanted to write a paper with @Paul-André Melliès on how *-autonomous categories, giving symmetric Frobenius pseudomonoids in Prof, thereby yield something like Prof-valued 2d TQFTs. I had a bunch of ideas about how various cobordisms give various operations in linear logic, and what this actually means. But it never came to pass.

Cole Comfort (Sep 28 2025 at 15:49):

@John Baez That would be cool to know. Do you have any idea what kind of geometric object is captured by the *-autonomous category generated by a Frobenius algebra, as this is the obvious generalization of 2-Cob to the linearly distributive setting. Maybe some polarized version of cobordisms? One reason I wanted to understand this, is because you can take a *-autonomous functor from the walking Frobenius algebra into finiteness modules, and then you get some sort of infinitary version of a TQFT.

For rig categories, we were motivated by understanding if control flow in quantum mechanics can be framed in similar terms to TQFTs.

Mike Shulman (Sep 28 2025 at 17:19):

"The $\ast$-autonomous category generated by a Frobenius algebra" is actually ambiguous, since there are two (actually three) notions of "Frobenius algebra" in a $\ast$-autonomous category. On one hand there is the usual notion of Frobenius algebra relative to the underlying monoidal structure. But there's also a notion of Frobenius algebra in the underlying [[linearly distributive category]], where the algebra and coalgebra structure use the two different monoidal structures. And then of course you could have an ordinary Frobenius algebra relative to the dual monoidal structure.

Arguably, the linearly distributive version is in some sense the most natural. For instance, the terminal [[polycategory]] is the same as the polycategory freely generated by a Frobenius algebra in the linearly-distributive (= polycategorical) sense.

Cole Comfort (Sep 28 2025 at 20:49):

@Mike Shulman I was thinking of the notion where you have a tensor commutative monoid and par commutative comonoid distributing over each other; ie, a linear point. This is the linearly distributive version which you are describing. And I agree this is the most natural definition of a Frobenius algebra because in a sense, it subsumes the monoidal notion. I guess you could go further and argue that the polycategorical notion is even more natural...

We understand that the equations of monoidal (commutative) Frobenius algebras are captured by diffeomorphism classes of 2-dimensional cobordisms. But is there some analogous geometric object which captures the notion of a linearly distributive (commutative) Frobenius algebra? It seems like a completely natural question but I have no idea what the answer is.

Mike Shulman (Sep 28 2025 at 21:45):

I guess you don't count the terminal polycategory as "geometric"?

Cole Comfort (Sep 28 2025 at 22:05):

Maybe that just is the answer, and I am trying to fit a polycategory peg into a manifold shaped hold

John Baez (Sep 29 2025 at 07:45):

Due to my ignorance I have no idea what the terminal polycategory "looks like", or what "the polycategory freely generated by Frobenius algebra in the linearly-distributive sense" looks like, so I can't see whether it looks at all like a category of cobordisms. If I saw a bunch of generators and relations, I could try to match them up with cobordism-like things.

Does this terminal gadget map in some way to the free symmetric monoidal category on a Frobenius algebra? So its morphisms could be seen as 2d cobordisms with extra properties, structure or stuff?

Mike Shulman (Sep 29 2025 at 07:58):

The terminal polycategory has exactly one object, say $\star$, and exactly one morphism $(\overbrace{\star,\dots,\star}^m) \to (\overbrace{\star,\dots,\star}^n)$ for each possible natural number lengths of the input and output.

Mike Shulman (Sep 29 2025 at 07:59):

I think you could think of that morphism as a connected genus-zero 2-cobordism from m circles to n circles.

Mike Shulman (Sep 29 2025 at 08:01):

I guess that's what it would map to in 2Cob, where the map comes from regarding a symmetric monoidal category as a linearly distributive category with the two tensors the same, and then taking its underlying polycategory, which process takes "symmetric monoidal" Frobenius algebras to "polycategorical" ones.

David Corfield (Sep 29 2025 at 08:13):

Mike Shulman said:

exactly one morphism $(\overbrace{\star,\dots,\star}^m) \to (\overbrace{\star,\dots,\star}^n)$ for each possible natural number lengths of the input and output.

So the kind of thing showing up in Joachim Kock's paper on whole-grain Petri nets

image.png

John Baez (Sep 29 2025 at 08:18):

Mike Shulman said:

I think you could think of that morphism as a connected genus-zero 2-cobordism from m circles to n circles. I guess that's what it would map to in 2Cob...

Hmm. You're making it sound like if you follow the unique morphism $1 \to 2$ by $2 \to 1$ in the terminal polycategory you get the identity $1 \to 1$. But in 2Cob, composing the genus-zero connected cobordisms $1 \to 2$ and $2 \to 1$ you don't get the identity: you get a cobordism $1 \to 1$ with genus 1.

So this idea doesn't seem to give a functor from the terminal polycategory to 2Cob.

Cole Comfort (Sep 29 2025 at 08:39):

John Baez said:

Mike Shulman said:

I think you could think of that morphism as a connected genus-zero 2-cobordism from m circles to n circles. I guess that's what it would map to in 2Cob...

Hmm. You're making it sound like if you follow the unique morphism $1 \to 2$ by $2 \to 1$ in the terminal polycategory you get the identity $1 \to 1$. But in 2Cob, composing the genus-zero connected cobordisms $1 \to 2$ and $2 \to 1$ you don't get the identity: you get a cobordism $1 \to 1$ with genus 1.

In a polycategory you cant compose a map of type [1]->[1,1] with a map of type [1,1]->[1] to get one of type [1,1], you can only cut in two ways to get maps of type [1,1]->[1,1]. In the terminal polycategory, these two cuts are equated, which is Frobenius law. In other words, a map in the terminal polycategory is completely determined by its domain and codomain. Interpreting this in 2Cob, as Mike points out, one obtains connected, genus 0 cobordisms.

Contrast this to the terminal multicategory, which went sent to 2Cob produces exactly trees with many inputs and a single output; in the terminal polycategory, things are sent to trees with multiple inputs and multiple outputs.

For the walking linearly distributive commutative Frobenius algebra, then things get a bit more interesting, because you can tensor or par things together, so the image of the functor into 2Cob would contain the disjoint union of genus 0 cobordisms. The geometric interpretation here is more elusive, because you now have an extra dimension with which to tensor and par things together. So in the linearly distributive category, you can't form "cycles" which would increase the genus in the image, because the "pants" and the "copants" split in two different dimensions.

John Baez (Sep 29 2025 at 08:51):

Okay. I don't really understand this stuff, so I'll just note that genus zero cobordisms, and disjoint unions of those, are precisely the ones that avoid the "divergences" (infinities) that show up when we try to copy the TQFT formalism using infinite-dimensional Hilbert spaces. The same thing shows up in quantum field theory: Feynman diagrams with "loops" (nonzero genus) are the ones that give the infinities. Thus, physicists often restrict to "tree level" QFT when they want to avoid these infinities.

So, practically, what I'm saying is that if you copy the 2d TQFT formalism using the terminal polycategory, it might provide a nice framework for certain "genus-zero TQFTs" that people like to study in 2 dimensions, such as 2d BF theory with a compact Lie group as gauge group. These are like 2d TQFTs, but they give undefined results for cobordisms with genus > 0.

I don't think you'll get much market for this, at least not right away, because the overhead is too high and there's no obvious payoff yet. But maybe it could lead to other ideas.

The case where you have both tensor and par - it's less obvious what that could be good for in physics.

Mike Shulman (Sep 29 2025 at 09:04):

That's interesting. If you want to allow disjoint unions of cobordisms without bringing in the tensor or par operations, you could use a "mix polycategory".

I'm not sure if this is a definition that exists in the literature, but it should be a straightforward semantic version of linear logic with the [[mix rule]], and a polycategorical version of the (iso)mix linearly distributive categories that do exist in the literature. A mix polycategory would also be the complementary halfway point ("conceptual pushout complement") to properads between polycategories and props.

Explicitly, in a polycategory you can only compose along one object. In a properad you can compose along any positive number of objects. In a prop you can compose along any number of objects, including zero, giving a side-by-side disjoint union operation. A mix polycategory would allow compositions along a subunary number of objects, hence a polycategory with a disjoint union operation.

The free mix polycategory generated by a Frobenius algebra would not be the terminal mix polycategory, but it ought to consist of genus-zero not-necessarily-connected cobordisms.

John Baez (Sep 29 2025 at 09:16):

Nice. This business of being allowed to compose cobordisms only along a subunary number of components (0 or 1, for us hidebound classical reasoners) is important not only for 2d genus-zero TQFT but a lot of higher-dimensional theories like 3d quantum gravity. The infinities only show up when you glue together cobordisms along 2 or more components, because that can create a noncontractible loop, which is related to taking the trace of the identity operator on an infinite-dimensional vector space.

Perhaps even more importantly, the same issue shows up when gluing together Feynman diagrams in ordinary QFT.

I think people usually formalize this issue using operads... not sure.

Cole Comfort (Sep 29 2025 at 13:58):

@John Baez Do you have any good references for this? Trying to intepret the terminal polycategory into Hilb will cause problems, because it seems that you would still be picking out a commutative Frobenius algebra. I would be very interested if TQFT people work with different semantic categories than (2-)Hilbert spaces which are dioperads, polycategories, *-autonomous categories.

John Baez (Sep 29 2025 at 14:31):

Do you have any good references for this?

I'm afraid most of the references that leap to mind are deep in the weeds of specific examples. One thing that might help a little is Steve Sawin's axiomatic description of TQFTs in Theorem 5 here. He breaks the operation of composing cobordisms into two cases, "sewing" and "mending", explained most clearly by Figure 4. In the ill-behaved theories I'm talking about, the infinities show up when you do "mending", while "sewing" is still okay.

Trying to intepret the terminal polycategory into Hilb will cause problems, because it seems that you would still be picking out a commutative Frobenius algebra.

Let me explain an example of the kind of ill-behaved theories that physicists run into.

You have an infinite-dimensional Hilbert space $H$ with an orthonormal basis $e_i$, and you define a comultiplication $\Delta$ by

$\Delta(e_i) = e_i \otimes e_i$

and a multiplication by

$\mu(e_i \otimes e_j) = \delta_{i j} e_i$

where $\delta_{i j}$ is the Kronecker delta. In the finite-dimensional case this makes your Hilbert space into a commutative Frobenius algebra, but in the infinite-dimensional case that's impossible, since there are no infinite-dimensional Frobenius algebras.

What goes wrong? I believe it's that the counit should have

$\epsilon(e_i) = 1$

for all $i$, but no bounded linear map between Hilbert spaces can do that. This formula only gives a densely defined map $\epsilon : H \to \mathbb{C}$.

I think it's best to understand an example of the ill-behaved theories I'm talking about before going too far in trying to formalize them.

David Corfield (Sep 30 2025 at 14:43):

Back to machines as systems, presumably transducers acting on finite state automata will find a place in the DOTS set-up. Here in their extension, Formal Languages and TQFTs with Defects, of Khovanov et al.'s work to include transducers, Boateng and Marcolli have categorical transducers acting on categorical finite state automata as follows:

image.png

Presumably directly relevant, I see @fosco was speaking about 2-transducers, the other day, concerning his Two-dimensional transducers. And no doubt we could also learn from Paul-André Melliès and Noam Zeilberger's The categorical contours of the Chomsky-Schützenberger representation theorem. Always an impossible amount of relevant things to read.

fosco (Sep 30 2025 at 16:15):

I know that those papers exist but I had no idea they contained this, very cool

fosco (Sep 30 2025 at 16:15):

Always an impossible amount of relevant things to read.

precisely.

fosco (Sep 30 2025 at 16:32):

Let's see if there is some vague connection between the two notions. For me a "2-transducer" is (among other possible definitions) a profunctor $A^*\times Q \to Q\times B^*$ where $A,B,Q$ are categories, and $A^*,B^*$ are the free monoidal categories over A,B

Their definition has the delicious smell of profunctorial spice, because a bijective on object functor out of ${\cal Q}_T$ is a [[promonad]] on ${\cal Q}_T$, and a "ULF" functor into $\cal C$ is a "Conduché" functor, i.e. a pseudofunctor $\cal C \to {\bf Prof}$. But at the moment I see no connection between 2-transducers and these gadgets.

fosco (Sep 30 2025 at 16:56):

I was also thinking about Khovanov-Im's
"Topological theories and automata" https://arxiv.org/abs/2202.13398
"From finite state automata to tangle cobordisms: a TQFT journey from one to four dimensions" https://arxiv.org/abs/2309.00708
and
"Topological Kleene Field Theories: A new model of computation" https://arxiv.org/abs/2503.16100

that probably have aready been mentioned?

fosco (Sep 30 2025 at 16:56):

That's why I was using the plural: these papers are very alluring and I would like to understand their story.

David Corfield (Sep 30 2025 at 18:57):

Thanks.

fosco said:

I was also thinking about Khovanov-Im's

Yes, I mentioned the first of these. There's an intriguing suggestion at the end of the Boateng and Marcolli paper:

Further directions: higher dimensional automata and TQFTs with defects. There is a notion of higher dimensional automata (see [8]), based on James Roger’s “higher dimensional trees” [19], and allowing for a rich categorical formulation and for algebra and coalgebra structures. It seems then natural to investigate the question of possible relations between these higher dimensional automata and TQFTs with defects, especially in the 2-dimensional case where ordinary TQFTs admit an algebraic description in terms of Frobenius algebras, and versions with defects have been analyzed, for instance, in [2].

That would connect to a vibrant field of mathematics/mathematical physics. Plenty of opportunity for your:

Paré’s work embodies an aesthetic perspective that resonates strongly with the author, which is the idea that a mathematician should never shy away from asking seemingly naïve questions (or even ‘two silly ones’, as in [Par21b]), since the answers may reveal unexpected analogies.

fosco (Sep 30 2025 at 19:12):

awesome! I also happen to be an ex student of one of the biggest experts on TQFTs on the market ;-) my dear advisor Domenico Fiorenza worked a lot on the topic

David Corfield (Oct 01 2025 at 08:00):

fosco said:

my dear advisor Domenico Fiorenza worked a lot on the topic

A name I know well from his collaboration with Sati and Schreiber:

image.png