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Stream: theory: applied category theory

Topic: Category Coherence as Admissibility Constraints

John Dai (Sep 23 2026 at 14:04):

Hello, everybody, I am new to the Zulip server, so forgive me if I may have created a new thread that could coincide with an existing thread. There are too much stuff here and reading them all will take me years. I thought it might be easier if somebody can point me to an existing thread if the topic already exists. Thank you very much.

John Dai (Sep 23 2026 at 14:28):

I created this thread because I have repeatedly encountered a structural pattern in several very different mathematical settings, and I would like to understand whether it already has a standard categorical formulation.

My motivation comes from comparing two independently developed theories—one concerning the relational realization of physical structure and another concerning the organization of living systems. Although their concrete mathematics is quite different, both appear to require more than the commutativity of individual diagrams.

In both cases, the admissible objects, morphisms, and transformations are constrained by a larger governing structure. Individual coherent diagrams are necessary, but they are not sufficient. A candidate construction may satisfy local coherence while still being inadmissible because it fails to remain compatible with the surrounding family of structures.

This led me to the tentative idea that coherence may sometimes play the role of an admissibility constraint rather than merely a consistency condition. I am interested in whether this intuition corresponds to an established categorical notion—perhaps involving double categories, fibrations, indexed categories, coherence spaces, or something else entirely—or whether it is simply a misleading way of looking at familiar constructions.

At this stage I am not advocating new terminology or proposing a theorem. My goal is to understand whether category theory already provides the right language for this phenomenon, and, if so, where I should be looking.

Any pointers to existing concepts, examples, or literature would be greatly appreciated.

Matteo Capucci (he/him) (Sep 23 2026 at 15:15):

What do you mean by an 'admissibility constraint'?

John Dai (Sep 23 2026 at 15:16):

Can coherence itself function as an admissibility constraint, rather than merely as a consistency condition?

The motivation is a recurring pattern where locally coherent diagrams are still not considered admissible unless they remain compatible with a larger governing family of structures. I am interested in whether this already has a standard categorical formulation or whether I am simply rediscovering an existing idea under different language.

Rather than duplicate the discussion across multiple forums, I would be grateful for comments on the original thread here:

https://categorytheory.zulipchat.com/#narrow/channel/229156-theory.3A-applied-category-theory/topic/Category.20Coherence.20as.20Admissibility.20Constraints/with/626329058

Pointers to relevant concepts, examples, or literature would be especially appreciated.

John Dai (Sep 23 2026 at 15:24):

I am trying to understand whether monoidal double categories provide the right language for a distinction that has arisen in my work.

At the local level, I have squares whose objects and boundary morphisms satisfy the requirements needed for each individual square to be admissible.

But I do not necessarily want to impose the full coherence structure on every object or morphism independently from the outset. Instead, when several individually admissible squares are assembled, I want additional coherence conditions to constrain the objects, morphisms, and squares jointly.

So schematically I have in mind:

admissible individual squares
→ assemblies of squares
→ higher coherence selecting admissible assemblies

Is this naturally expressible using a monoidal double category, or are the coherence conditions in a monoidal double category of a different character because they are already built into the ambient structure?

I am particularly interested in whether there is a standard framework in which local square-level admissibility is distinguished from higher-order admissibility of a whole configuration.

John Baez (Sep 23 2026 at 15:49):

What do you mean by "admissibility"?

John Baez (Sep 23 2026 at 15:51):

Heh, I hadn't known that @Matteo Capucci (he/him) asked the exact same question!

It goes to show: if you talk in ways that sound technical but don't have any meaning to mathematicians, they will want to know exactly what you mean.

Matteo Capucci (he/him) (Sep 23 2026 at 15:58):

@John Dai please don't double post. People who want to reply to your thread will do so here

John Dai (Sep 23 2026 at 16:43):

Understand. I posted a shorter version of my question here after I realized it might be relevant to this discussion.

John Dai (Sep 23 2026 at 16:50):

Matteo Capucci (he/him) said:

What do you mean by an 'admissibility constraint'?

By “admissibility” I mean admissibility relative to extra coherence conditions imposed on a functor.

Suppose we have two categories A and B and a functor

F: A -> B.

As an ordinary functor, F sends every object and morphism of A to B. But now suppose we require F to be symmetric monoidal. Then we also require coherent structure maps like

F(X) tensor F(Y) ~= F(X tensor Y)

and

1_B ~= F(1_A),

with the usual compatibility conditions for associativity, units, and symmetry.

So the issue is not whether an object or morphism has an image under F. The issue is whether a proposed collection of structures in A can be carried to B in a way compatible with all of those extra coherence conditions.

That is what I mean by admissible: a structure is admissible if its image can participate in the full symmetric-monoidal structure of F.

So, schematically:

ordinary functorial image
vs.
image compatible with the required coherence.

Equivalently, some data may map perfectly well under the underlying functor, but fail to extend to a symmetric-monoidal realization because the tensor products, symmetries, or compositions cannot all be made compatible at once.

John Dai (Sep 23 2026 at 17:15):

My point is simply that, if B is viewed as the space of realizations of A under a functor F, then requiring F merely to preserve the relevant squares admits a relatively broad class of realizations in B. Requiring F to be symmetric monoidal imposes additional compatibility conditions across those images, thereby selecting a smaller class of admissible realizations.

John Dai (Sep 23 2026 at 17:30):

John Baez said:

Heh, I hadn't known that Matteo Capucci (he/him) asked the exact same question!

It goes to show: if you talk in ways that sound technical but don't have any meaning to mathematicians, they will want to know exactly what you mean.

Please do press me for clarification. It is a delicate balance between defining everything as if nobody knows anything and presuming everybody already knows everything. I will try to be sensitive and conscientious. I promise.

John Baez (Sep 23 2026 at 17:37):

We know about (symmetric) monoidal categories, functors, etc. So: what's an "admissible square"? You wrote

At the local level, I have squares whose objects and boundary morphisms satisfy the requirements needed for each individual square to be admissible.

but I don't think you've yet defined an admissible square.

John Dai (Sep 23 2026 at 17:54):

John Baez said:

We know about (symmetric) monoidal categories, functors, etc. So: what's an "admissible square"? You wrote

At the local level, I have squares whose objects and boundary morphisms satisfy the requirements needed for each individual square to be admissible.

but I don't think you've yet defined an admissible square.

Who are “we”? A card-carrying mathematician should know what “admissibility” means by now after my clarification :) I do not wish to over-burden true mathematicians …

John Dai (Sep 23 2026 at 18:02):

But it is a fair point that “admissibility” is not a standard categorical term. But that is why the question is in an “applied” thread. By an “admissible square” I mean a commutative square whose objects and morphisms satisfy whatever local structural constraints are imposed in the problem at hand.

What I am trying to isolate is the distinction between local admissibility of individual squares and stronger compatibility conditions that arise only when a family of such squares is considered together.

So the question is: starting from a class of locally valid squares, what categorical structure lets us impose additional coherence conditions on the family as a whole, thereby selecting a smaller class of jointly realizable configurations?

John Dai (Sep 23 2026 at 18:04):

I should clarify that I am not asking whether this two-stage construction is possible; I have a constructive example in which locally admissible squares are assembled and then subjected to additional higher-order coherence conditions that restrict the jointly realizable configurations.

My question is whether category theorists have already developed a systematic framework for doing this in general: starting from locally admissible cells, then imposing higher coherence as an additional selection principle on their assemblies.

John Dai (Sep 24 2026 at 07:18):

John Baez said:

We know about (symmetric) monoidal categories, functors, etc. So: what's an "admissible square"? You wrote

At the local level, I have squares whose objects and boundary morphisms satisfy the requirements needed for each individual square to be admissible.

but I don't think you've yet defined an admissible square.

John: you are right I could be more clear about what I meant by “admissibility”. I was using some terms that were natural and intuitive for the specifics problems I have been working on, but can be meaningless in a more general setting, because “admissibility” could mean different things in different settings. I think you were really pressing me for a specific example.

John Dai (Sep 24 2026 at 07:31):

For an example, if we naively model a natural language as a mapping between generic syntactic rules without strong typing to semantic expressions then the semantic image may contain funny expressions like “an ant can eat a universe”. This expression has to be inadmissible. But how? One way to do this is to think about collecting all possible expressions involving ants and universes that can legitimately descend from the syntactic category and show that they can not be logically consistent as a whole. This is one admissibility problem. Its resolution may not require appealing to higher order coherence conditions.

Joe Moeller (Sep 24 2026 at 20:40):

I doubt that the nonsense-ness of the sentence "an ant can eat a universe" is something that can be expressed syntactically. It is a sentence I could imagine reading in e.g. a Terry Pratchett book. It's not on the level of nonsense of "Colorless green ideas sleep furiously." Am I misunderstanding your point?

David Michael Roberts (Sep 24 2026 at 22:52):

By an “admissible square” I mean a commutative square whose objects and morphisms satisfy whatever local structural constraints are imposed in the problem at hand.

this is so general so as to not mean much to me. Please give a pair of contrasting examples coming purely from mathematics using terminology that a category theorist can be assumed to be familiar with.

David Michael Roberts (Sep 24 2026 at 22:53):

locally valid squares

does valid = admissible?

John Dai (Sep 25 2026 at 05:41):

Joe Moeller said:

I doubt that the nonsense-ness of the sentence "an ant can eat a universe" is something that can be expressed syntactically. It is a sentence I could imagine reading in e.g. a Terry Pratchett book. It's not on the level of nonsense of "Colorless green ideas sleep furiously." Am I misunderstanding your point?

Syntactically, that nonsense sentence is just Eat: NP x NP -> sentence, so it is legit. But at the semantic level, it becomes eat: Entity x Entity -> proposition with each Entity assigned a specific meaning. Now the crucial point is that a specific meaning entails logical obligations. So the sentence “an ant can eat the universe” is inadmissible not because it is illegitimate by itself, but because it can not be compatible with all other individually/locally legitimate semantic expressions involving an ant and a universe.

John Dai (Sep 25 2026 at 05:54):

David Michael Roberts said:

By an “admissible square” I mean a commutative square whose objects and morphisms satisfy whatever local structural constraints are imposed in the problem at hand.

this is so general so as to not mean much to me. Please give a pair of contrasting examples coming purely from mathematics using terminology that a category theorist can be assumed to be familiar with.

My bad. I think you, John and Matteo are all correct in questioning whether I have defined the “inadmissible square” clearly enough. The existence of a square itself should imply commutativity so the notion of an admissible square is redundant. what I should have said is this: consider a collection of squares each representing a point-wise object/morphism square, then such a collection may not be admissible once the objects on one side of the functorial mapping are assigned specific semantic meaning (human language, physics, etc).

John Dai (Sep 25 2026 at 06:11):

David Michael Roberts said:

By an “admissible square” I mean a commutative square whose objects and morphisms satisfy whatever local structural constraints are imposed in the problem at hand.

this is so general so as to not mean much to me. Please give a pair of contrasting examples coming purely from mathematics using terminology that a category theorist can be assumed to be familiar with.

Basically the way I put it earlier I was conflating the notion of categorical admissibility / validity with the notion of semantic admissibility / validity. My clarification relating to the non-sense sentence “an ant can eat the universe” is a good illustrative example. If “ant” and “universe” remain abstract labels without any semantic meaning with logical consequences, then we can form a legitimate square mapping a syntactic expression to this semantic expression. Now, even after ant and universe have been assigned specific meanings, there is a question of whether we should assert that the syntactic to semantic square is still admissible / valid on its own but not compatible with other admissible / valid squares or whether we should assert that given the assigned semantic meanings this square is simply not admissible / valid any more. Joe Moeller appears to be leaning toward the latter, and I was too. But I now believe that I am leaning toward the former: the squares are locally admissible / valid. What makes the sentence a non-sentence is because it is not compatible with some other squares under a consistent assignment of semantic meaning to all objects and morphisms.

David Michael Roberts (Sep 25 2026 at 06:28):

These are not mathematical examples, so I don't know what you mean. Perhaps you are not intending there to be examples from mathematics, in which case I have no feedback to give

David Michael Roberts (Sep 25 2026 at 06:29):

the notion of categorical admissibility / validity

you haven't specified what these words mean, except perhaps implicitly as "commuting square".

John Dai (Sep 25 2026 at 06:42):

David Michael Roberts said:

These are not mathematical examples, so I don't know what you mean. Perhaps you are not intending there to be examples from mathematics, in which case I have no feedback to give

David, I need some help here. I am not a mathematician. I am a theoretical physicist with descent grounding in complex analysis, differential geometry, group symmetry and topology, and I have a keen interest in category theory as a way to constrain physical models. If I am not expressing my ideas rigorously in category language, it may be the very reason why I am here. The problem I am interested in is a physics problem but I think it has a clean mapping to the language problem at the conceptual level. The idea is that you have a syntactic category and a functorial mapping to a semantic category. The question is: what makes the semantic category a logically consistent language that describes reality?

John Dai (Sep 25 2026 at 06:46):

John Dai said:

David Michael Roberts said:

These are not mathematical examples, so I don't know what you mean. Perhaps you are not intending there to be examples from mathematics, in which case I have no feedback to give

David, I need some help here. I am not a mathematician. I am a theoretical physicist with descent grounding in complex analysis, differential geometry, group symmetry and topology, and I have a keen interest in category theory as a way to constrain physical models. If I am not expressing my ideas rigorously in category language, it may be the very reason why I am here. The problem I am interested in is a physics problem but I think it has a clean mapping to the language problem at the conceptual level. The idea is that you have a syntactic category and a functorial mapping to a semantic category. The question is: what makes the semantic category a logically consistent language that describes reality?

I am not bringing in the physics example yet because it involves too much machinery. The language example is nice because it could be a prototype that can be mapped not only to a physics problem but also to some other interesting problems.

David Michael Roberts (Sep 25 2026 at 07:18):

You are assuming people here don't know physics? I have a physics degree.

I asked if you could supply a mathematical example and the short answer would have been "sorry, no, that's not what I'm trying to do"

David Michael Roberts (Sep 25 2026 at 07:18):

Just give us the technical details and we can wind you back.

David Michael Roberts (Sep 25 2026 at 07:19):

Talking about plain language semantics of nonsensical sentences when you are intending to do physics is less helpful than eg giving equations with carefully defined meaning

John Dai (Sep 25 2026 at 07:29):

David Michael Roberts said:

You are assuming people here don't know physics? I have a physics degree.

I asked if you could supply a mathematical example and the short answer would have been "sorry, no, that's not what I'm trying to do"

No I am not assuming that people don’t know physics here. What I said was that that the physics example involves bringing in a lot of machinery that may obscure the core conceptual problem I am trying to resolve. I am simply trying to reduce the problem to its core. And the language problem captures that core very well. It is likely that the problem I am struggling with really just boils down to a typing problem. Once both the syntactic category and semantic category are properly typed, nonsense sentences can be ruled out already. In physics terms, the issue is whether strong typing can select a sub-representation naturally from a full representation of an upstream group symmetry.

David Michael Roberts (Sep 25 2026 at 07:37):

I'm sorry, but actual rigorous physics would be clearer than the material so far. I hope someone else can give you a good answer.

John Dai (Sep 25 2026 at 07:48):

Thank you David. Much appreciated for your feedback.

John Dai (Sep 25 2026 at 08:26):

Let me try to define the language problem mathematically:

For language, the mathematical problem is not merely to define an interpretation functor. It is to determine which syntactically valid expressions remain semantically admissible once interpreted meanings impose type and consistency constraints.

A concise formulation is:

Given a syntactic category C_syn, a semantic category C_sem, an interpretation functor

F: C_syn -> C_sem,

and a category of consistent semantic typings

p: E_cons -> C_sem,

determine the subcategories D of C_syn for which the restricted interpretation

F|D: D -> C_sem

admits a lift

F_hat_D: D -> E_cons

with

p o F_hat_D = F|D.

Then study the admissibility poset

A = { D subseteq C_syn : F|D admits such a lift }.

The core questions are:

• when does a global lift exist on all of C_syn?
• if not, do maximal admissible subcategories exist?
• is there a unique largest admissible subcategory?
• how can it be constructed from local semantic typing constraints?
• what conditions guarantee closure of admissibility under composition?

In plain language:

The problem is to determine the largest fragment of syntactically valid language whose semantic interpretation can be assigned mutually compatible meanings everywhere at once.

John Dai (Sep 25 2026 at 09:22):

I have isolated this problem also because it may have some interesting implications for LLM training and/or interpretations

John Dai (Sep 25 2026 at 10:25):

I was hoping that mathematicians would say: hey John, you are not stating the problem in a form that we can understand. But based on what you are saying, perhaps this is the mathematical problem you are trying to solve? As a DYI plumber, I know that once I have defined the problem, I have already solved 85-90% of that problem. So repeatedly asking me, a non-mathematician, to state a problem in a mathematically prestine fashion feels like shutting me down for being a non-mathematician — as if I don’t belong here. That is not the best attitude toward a fellow truth seeker. :blush:

Joe Moeller (Sep 25 2026 at 19:58):

@John Dai As you could probably imagine, we've had a few instances in this community of new people showing up and then simply copy pasting nonsense from some LLM. To earlier instances of this, we largely responded with genuine attempts to meaningfully engage with the person. We actually ended up spending a lot of time chasing things down that turned out to be pure mist. After this happening more times, we've sorta landed on a strategy of identifying the core concepts that the poster seems to be working around, and then asking for a rigorous definition. This seems to be a good strategy for weeding out people that have no idea what's going on. To be super clear, the community doesn't have a single unified opinion on the use of LLMs, but we do agree that nobody is interested in sifting through tomes of slop hoping to find a kernel of mild interest. To be ultra-mega clear, I'm not accusing you of this at all. I'm not evaluating your post in any way. I just felt like sharing where our attitude might be coming from. We do ultimately want to be welcoming to people of all backgrounds interested in category theory. We can always do better on being more welcoming, and we should also remain vigilant on keeping garbage out.

John Dai (Sep 26 2026 at 02:24):

Hi Joe,

Thank you for explaining your attitude. So I was correct in detecting
certain kind of attitude. I guess that proves that I am not an LLM. :) I
will let you guys play with your own games in your own protected domain.
You won’t hear from me again.

John

Rémy Tuyéras (Sep 26 2026 at 12:13):

Hi @John Dai,

I believe I understand what you are after, so I will try to help formalize your idea.

In particular, I think the following intuition you expressed can be made mathematically precise:

This led me to the tentative idea that coherence may sometimes play the role of an admissibility constraint rather than merely a consistency condition.

and similarly:

Can coherence itself function as an admissibility constraint, rather than merely as a consistency condition?

First, every monoid can be viewed as a category with a single object. Using this, I will denote the multiplicative monoid $\mathbb{Z}/2\mathbb{Z}=\{0,1\}$ by $\mathcal{C}_{\mathrm{sem}}$, referring to the semantic category that you mentioned.

Then I will denote by $\mathcal{W}$ the list of words in my language, say a tokenization of English, and by $\Sigma\mathcal{W}$ the free monoid generated by $\mathcal{W}$.

For example,

$\mathcal{W}=\{\texttt{"An"},\texttt{" ant"},\texttt{" can"},\texttt{" eat"},\texttt{" a"},\texttt{" universe"},\ldots\}.$

Because each of these words makes sense individually, we can assign a valid semantic value to each of these atomic syntactic components:

$F(\texttt{"An"})=1,\quad F(\texttt{" ant"})=1,\quad\dots$

However, making the mapping $F$ preserve the underlying monoid structures becomes more difficult as we move from these local pieces toward more global expressions.

We may certainly have

$F(\texttt{"An ant"})=1,\quad F(\texttt{" can eat"})=1,\quad F(\texttt{" a universe"})=1,\quad\dots$

and we can even consider intersecting segments such as

$F(\texttt{"An ant can eat"})=1,\quad F(\texttt{" can eat a universe"})=1,\quad\dots$

But the multiplication $1\cdot 1=1$ in $\mathcal{C}_{\mathrm{sem}}$ is not compatible with the complete local-to-global transition when we build the larger sentence, since we would want

$F(\texttt{"An ant can eat a universe"})=0.$

I think this is very close to what you mean when you say that locally admissible structures may fail to remain admissible once they are considered as part of a larger family.

This is where categories become useful, because they allow us to restrict the monoid multiplication to sparser compositions.

It is possible to define a category $\mathcal{C}$ whose set of objects is $\Sigma\mathcal{W}$ and whose arrows are triples in $\Sigma\mathcal{W}^{\times 3}$, where

$(m,a,b):a\to b$

is an arrow in $\mathcal{C}$.

Composition is defined by

$(n,b,c)\circ(m,a,b)=(m\cdot n,a,c).$

Examples of arrows in $\mathcal{C}$ are

$(\texttt{"An ant"},\texttt{"An"},\texttt{" can"}):\texttt{"An"}\to\texttt{" can"}$

and

$(\texttt{" can eat"},\texttt{" can"},\texttt{" a"}):\texttt{" can"}\to\texttt{" a"}.$

Their composition is therefore

$(\texttt{"An ant can eat"},\texttt{"An"},\texttt{" a"}):\texttt{"An"}\to\texttt{" a"}.$

Note: This category is inherently related to autoregressive language models because the category-theoretic target of an arrow $(m,a,b): a \to b$ corresponds to what we call a "target" during the training of an LLM for a given prompt $m$ and context $a$.

The categorical structure of $\mathcal{C}$ now gives us more local control over semantics because we can control which contexts and targets are allowed.

For instance, we may impose

$F(\texttt{" ant can"}\mathop{\longrightarrow}\limits^{\texttt{" ant can eat a"}}\texttt{" universe"})=0.$

This means that if we compose any arrow targeting $\texttt{" ant can"}$ with the arrow above, then the resulting sentence will be semantically invalid.

This seems to connect quite directly with another formulation you gave:

What I am trying to isolate is the distinction between local admissibility of individual squares and stronger compatibility conditions that arise only when a family of such squares is considered together.

In this toy construction, the corresponding distinction is between assigning semantic values to individual pieces and requiring those values to remain compatible under the compositions that assemble them into larger structures.

The main problem then becomes:

• Does $F$ even exist and, if so, is it a functor?

• If not, does there exist a subcategory $D\subseteq\mathcal{C}$ such that the restriction of $F$ to $D$ is a functor $D \to \mathcal{C}_{\mathrm{sem}}$?

This also seems very close to your later question about finding the largest fragment of a syntactically valid language whose semantic interpretations can remain mutually compatible everywhere at once.

Rémy Tuyéras (Sep 26 2026 at 12:28):

There are a few points where I am less certain, and I would be interested to know whether they match what you have in mind.

First, I am wondering whether some of the squares you mentioned could arise from the different ways of composing smaller linguistic pieces into the same larger expression.

For example, in the category $\mathcal{C}$ above, consider the arrows

$f=(\texttt{"An ant"},\texttt{"An"},\texttt{" can"}):\texttt{"An"}\to\texttt{" can"}$

$g=(\texttt{" can eat a"},\texttt{" can"},\texttt{" universe"}):\texttt{" can"}\to\texttt{" universe"}$

and

$h=(\texttt{"An ant can"},\texttt{"An"},\texttt{" eat"}):\texttt{"An"}\to\texttt{" eat"}$

$k=(\texttt{" eat a"},\texttt{" eat"},\texttt{" universe"}):\texttt{" eat"}\to\texttt{" universe"}.$

Then we have two different compositional paths:

$\texttt{"An"}\mathop{\longrightarrow}\limits^{\texttt{"An ant"}}\texttt{" can"}\mathop{\longrightarrow}\limits^{\texttt{" can eat a"}}\texttt{" universe"}$

and

$\texttt{"An"}\mathop{\longrightarrow}\limits^{\texttt{"An ant can"}}\texttt{" eat"}\mathop{\longrightarrow}\limits^{\texttt{" eat a"}}\texttt{" universe"}.$

Both compositions give

$g\circ f=k\circ h=(\texttt{"An ant can eat a"},\texttt{"An"},\texttt{" universe"}).$

So $f,g,h,k$ form a commutative square. In this example, the square expresses the fact that two different local decompositions assemble into the same larger syntactic construction.

I am wondering whether this is close to what you mean when you write:

At the local level, I have squares whose objects and boundary morphisms satisfy the requirements needed for each individual square to be admissible.

But I do not necessarily want to impose the full coherence structure on every object or morphism independently from the outset. Instead, when several individually admissible squares are assembled, I want additional coherence conditions to constrain the objects, morphisms, and squares jointly.

admissible individual squares
→ assemblies of squares
→ higher coherence selecting admissible assemblies

If so, then I would interpret one square such as the one above as expressing a local compatibility between two ways of composing the same expression.

The next step would be to consider many such squares at once. Different squares could overlap by sharing objects or arrows, and the same longer expression could appear through several different chains of compositions. The question would then no longer be only whether each individual square commutes, but whether all of these local commutative squares can be assembled into one jointly compatible structure.

That seems very close to your distinction between an individually admissible square and an admissible assembly of squares.

I am less certain about whether a monoidal double category is the right structure for expressing that second step, but I think this example may at least isolate the distinction you are trying to capture.

John Dai (Sep 27 2026 at 05:13):

Rémy Tuyéras said:

Hi John Dai,

I believe I understand what you are after, so I will try to help formalize your idea.

In particular, I think the following intuition you expressed can be made mathematically precise:

This led me to the tentative idea that coherence may sometimes play the role of an admissibility constraint rather than merely a consistency condition.

and similarly:

Can coherence itself function as an admissibility constraint, rather than merely as a consistency condition?

First, every monoid can be viewed as a category with a single object. Using this, I will denote the multiplicative monoid $\mathbb{Z}/2\mathbb{Z}=\{0,1\}$ by $\mathcal{C}_{\mathrm{sem}}$, referring to the semantic category that you mentioned.

Then I will denote by $\mathcal{W}$ the list of words in my language, say a tokenization of English, and by $\Sigma\mathcal{W}$ the free monoid generated by $\mathcal{W}$.

For example,

$\mathcal{W}=\{\texttt{"An"},\texttt{" ant"},\texttt{" can"},\texttt{" eat"},\texttt{" a"},\texttt{" universe"},\ldots\}.$

Because each of these words makes sense individually, we can assign a valid semantic value to each of these atomic syntactic components:

$F(\texttt{"An"})=1,\quad F(\texttt{" ant"})=1,\quad\dots$

However, making the mapping $F$ preserve the underlying monoid structures becomes more difficult as we move from these local pieces toward more global expressions.

We may certainly have

$F(\texttt{"An ant"})=1,\quad F(\texttt{" can eat"})=1,\quad F(\texttt{" a universe"})=1,\quad\dots$

and we can even consider intersecting segments such as

$F(\texttt{"An ant can eat"})=1,\quad F(\texttt{" can eat a universe"})=1,\quad\dots$

But the multiplication $1\cdot 1=1$ in $\mathcal{C}_{\mathrm{sem}}$ is not compatible with the complete local-to-global transition when we build the larger sentence, since we would want

$F(\texttt{"An ant can eat a universe"})=0.$

I think this is very close to what you mean when you say that locally admissible structures may fail to remain admissible once they are considered as part of a larger family.

This is where categories become useful, because they allow us to restrict the monoid multiplication to sparser compositions.

It is possible to define a category $\mathcal{C}$ whose set of objects is $\Sigma\mathcal{W}$ and whose arrows are triples in $\Sigma\mathcal{W}^{\times 3}$, where

$(m,a,b):a\to b$

is an arrow in $\mathcal{C}$.

Composition is defined by

$(n,b,c)\circ(m,a,b)=(m\cdot n,a,c).$

Examples of arrows in $\mathcal{C}$ are

$(\texttt{"An ant"},\texttt{"An"},\texttt{" can"}):\texttt{"An"}\to\texttt{" can"}$

and

$(\texttt{" can eat"},\texttt{" can"},\texttt{" a"}):\texttt{" can"}\to\texttt{" a"}.$

Their composition is therefore

$(\texttt{"An ant can eat"},\texttt{"An"},\texttt{" a"}):\texttt{"An"}\to\texttt{" a"}.$

Note: This category is inherently related to autoregressive language models because the category-theoretic target of an arrow $(m,a,b): a \to b$ corresponds to what we call a "target" during the training of an LLM for a given prompt $m$ and context $a$.

The categorical structure of $\mathcal{C}$ now gives us more local control over semantics because we can control which contexts and targets are allowed.

For instance, we may impose

$F(\texttt{" ant can"}\mathop{\longrightarrow}\limits^{\texttt{" ant can eat a"}}\texttt{" universe"})=0.$

This means that if we compose any arrow targeting $\texttt{" ant can"}$ with the arrow above, then the resulting sentence will be semantically invalid.

This seems to connect quite directly with another formulation you gave:

What I am trying to isolate is the distinction between local admissibility of individual squares and stronger compatibility conditions that arise only when a family of such squares is considered together.

In this toy construction, the corresponding distinction is between assigning semantic values to individual pieces and requiring those values to remain compatible under the compositions that assemble them into larger structures.

The main problem then becomes:

• Does $F$ even exist and, if so, is it a functor?

• If not, does there exist a subcategory $D\subseteq\mathcal{C}$ such that the restriction of $F$ to $D$ is a functor $D \to \mathcal{C}_{\mathrm{sem}}$?

This also seems very close to your later question about finding the largest fragment of a syntactically valid language whose semantic interpretations can remain mutually compatible everywhere at once.

Hello Remy, thank you for your response! Finally substance is penetrating through a palpable paranoia against LLM. A true expert does not need to fear LLMs — a true expert is always smarter than an LLM. He should make LLM enhance his productivity by 100 times.

Remy, I think you got my question almost perfectly. When we have F: C -> L, where C denote the category of “syntactic possibilities” and L the category of semantic realizations, we have basically two possible approaches: (1) fix F, and restrict C to to a sub-category D so that L is not only coherent locally (each square makes sense) but also globally (all squares make sense together). (2) fix C and F, and ask: what are the subsets of L (select stacks of squares) that make sense jointly (since “admissibility” seems to be such a toxic word here, I will avoid using that word again). Once we know how to answer (2), we can then ask what happens if we strengthen F — say adding symmetric braiding and/or hexagon etc (while always keeping C fixed)? I am most interested in this direction, and I believe this is also a direction you want from LLM training perspective. You don’t really want to restrict the source. You want to select the outcome.

John Dai (Sep 27 2026 at 05:14):

Rémy Tuyéras said:

There are a few points where I am less certain, and I would be interested to know whether they match what you have in mind.

First, I am wondering whether some of the squares you mentioned could arise from the different ways of composing smaller linguistic pieces into the same larger expression.

For example, in the category $\mathcal{C}$ above, consider the arrows

$f=(\texttt{"An ant"},\texttt{"An"},\texttt{" can"}):\texttt{"An"}\to\texttt{" can"}$

$g=(\texttt{" can eat a"},\texttt{" can"},\texttt{" universe"}):\texttt{" can"}\to\texttt{" universe"}$

and

$h=(\texttt{"An ant can"},\texttt{"An"},\texttt{" eat"}):\texttt{"An"}\to\texttt{" eat"}$

$k=(\texttt{" eat a"},\texttt{" eat"},\texttt{" universe"}):\texttt{" eat"}\to\texttt{" universe"}.$

Then we have two different compositional paths:

$\texttt{"An"}\mathop{\longrightarrow}\limits^{\texttt{"An ant"}}\texttt{" can"}\mathop{\longrightarrow}\limits^{\texttt{" can eat a"}}\texttt{" universe"}$

and

$\texttt{"An"}\mathop{\longrightarrow}\limits^{\texttt{"An ant can"}}\texttt{" eat"}\mathop{\longrightarrow}\limits^{\texttt{" eat a"}}\texttt{" universe"}.$

Both compositions give

$g\circ f=k\circ h=(\texttt{"An ant can eat a"},\texttt{"An"},\texttt{" universe"}).$

So $f,g,h,k$ form a commutative square. In this example, the square expresses the fact that two different local decompositions assemble into the same larger syntactic construction.

I am wondering whether this is close to what you mean when you write:

At the local level, I have squares whose objects and boundary morphisms satisfy the requirements needed for each individual square to be admissible.

But I do not necessarily want to impose the full coherence structure on every object or morphism independently from the outset. Instead, when several individually admissible squares are assembled, I want additional coherence conditions to constrain the objects, morphisms, and squares jointly.

admissible individual squares
→ assemblies of squares
→ higher coherence selecting admissible assemblies

If so, then I would interpret one square such as the one above as expressing a local compatibility between two ways of composing the same expression.

The next step would be to consider many such squares at once. Different squares could overlap by sharing objects or arrows, and the same longer expression could appear through several different chains of compositions. The question would then no longer be only whether each individual square commutes, but whether all of these local commutative squares can be assembled into one jointly compatible structure.

That seems very close to your distinction between an individually admissible square and an admissible assembly of squares.

I am less certain about whether a monoidal double category is the right structure for expressing that second step, but I think this example may at least isolate the distinction you are trying to capture.

Don’t worry about the double monoidal category linkage. Take that as a red herring :)

John Dai (Sep 27 2026 at 05:32):

@Rémy Tuyéras so aside from the “double monoidal” digression, which we should ignore, your second post is closer to what I have been thinking.

John Dai (Sep 27 2026 at 08:54):

@Joe Moeller now that Reme’s response brought me back here, let me add a more formal response to your last post here.

I appreciate the context, but I still don’t think that is a good default way to treat someone you don’t know.

If something I write is unclear, ask me to define it. If it is mathematically wrong, point out where. That is completely fair. But previous experiences with people posting LLM-generated nonsense don’t really justify treating a new person as suspicious from the outset, or explaining the tone toward them by reference to those unrelated cases.

And I don’t think the use of LLMs is the real issue anyway. People use all kinds of tools to think, write, code, search, and communicate more efficiently. The relevant question is whether the person understands what they are saying and can defend it when challenged.

So yes, by all means be rigorous. I welcome that. I just think rigor and basic courtesy should go together.

Matteo Capucci (he/him) (Sep 27 2026 at 09:04):

What @Joe Moeller said is true regardless of LLMs. I'll be more blunt than him. People coming here deserve to be listened and we grow and renew the community by welcoming new members. At the same time, attention, patience, good will, these are commons that are easily abused by those who, often in good faith, come and exact to be entertained in their musings. Sorry, but nobody here owes you attention. I know for a fact all the people that replied here pour their heart in mantaining the community (the very fact they bothered replying proves this). So if you are not getting the consideration you expect, try changing the way you engage and communicate.

Matteo Capucci (he/him) (Sep 27 2026 at 09:04):

(for the record, I am a mod of this Zulip but the above is an exclusively personal position)

John Dai (Sep 27 2026 at 11:04):

Matteo Capucci (he/him) said:

What Joe Moeller said is true regardless of LLMs. I'll be more blunt than him. People coming here deserve to be listened and we grow and renew the community by welcoming new members. At the same time, attention, patience, good will, these are commons that are easily abused by those who, often in good faith, come and exact to be entertained in their musings. Sorry, but nobody here owes you attention. I know for a fact all the people that replied here pour their heart in mantaining the community (the very fact they bothered replying proves this). So if you are not getting the consideration you expect, try changing the way you engage and communicate.

Ok, I think the mod has the last word on this :)

James Deikun (Sep 27 2026 at 11:19):

He just said he wasn't exercising his last word as a mod here.

Anyway, I think it is fundamentally fair to ask someone coming here to get an answer to a question to put in at least as much work communicating their question as the community has to put in to answer it. Basically, meet us halfway. The reason people were reacting hostilely is because your question was coming across as really unclear and unclear questions take a lot of effort to answer.

James Deikun (Sep 27 2026 at 11:23):

FWIW later down the thread I came to think that probably the thing you're looking for is a type refinement in the sense of Zeilberger: that is, a span of functors where one end is your original syntax, the other end is your semantics, and the middle is the original syntax decorated with a proof that the particular term respects the more stringent constraints of the semantics. But when I figured that out was the same time I read that you were done reading the thread, so I didn't bother saying anything.

John Dai (Sep 27 2026 at 12:37):

James Deikun said:

FWIW later down the thread I came to think that probably the thing you're looking for is a type refinement in the sense of Zeilberger: that is, a span of functors where one end is your original syntax, the other end is your semantics, and the middle is the original syntax decorated with a proof that the particular term respects the more stringent constraints of the semantics. But when I figured that out was the same time I read that you were done reading the thread, so I didn't bother saying anything.

I like your span idea more than the Zeilberger-style type refinement idea. The latter seems to start with something I am trying to derive. Your span idea is closer to what I was pointing at: strengthening the functorial coherence requirements should progressively reduce the admissible objects and morphisms in the target category.

I suppose what I am really looking for is a general formal result that captures this span picture.

John Dai (Sep 27 2026 at 13:04):

There is probably some semantic confusion that can be cleared up once and for all: if I identify my source as Zeilberger’s source, then the the span idea may be equivalent to a composition of the Zeilberger’s functor and another functor that takes his refined category to the ultimate target. Interpreted this way, the Zeilberger’s idea and the span idea are formally equivalent. But the Zeilberger approach risks putting answers into the questions in my setting: wiring admissibility directly into the refinement functor.

John Dai (Sep 27 2026 at 14:56):

Suppose every component carries a valid local certificate, but their interconnection creates new reciprocal constraints that are not visible componentwise. In the simple feedback system \dot x=-x+ky, \dot y=-y+kx, both components are locally stable, while the composite becomes unstable for k>1. Does the categorical assume-guarantee formalism derive the additional global compatibility condition on the wiring itself, or must that condition be supplied as part of the contracts?

John Dai (Sep 27 2026 at 15:01):

John Dai said:

Suppose every component carries a valid local certificate, but their interconnection creates new reciprocal constraints that are not visible componentwise. In the simple feedback system \dot x=-x+ky, \dot y=-y+kx, both components are locally stable, while the composite becomes unstable for k>1. Does the categorical assume-guarantee formalism derive the additional global compatibility condition on the wiring itself, or must that condition be supplied as part of the contracts?

Did @John Baez just delete his post about assume-guarantee reasoning? I was responding to it now I can not see the original post any more.

John Baez (Sep 27 2026 at 15:04):

I was not trying to discuss this in your thread! I have my own thread going on, and accidentally dropped this piece of text (not written by me) in here. Sorry.

John Dai (Sep 27 2026 at 15:04):

The fact that the composite system in my simple feedback example is not always stable is somewhat analogous to the “admissibility” question I raised.

John Dai (Sep 27 2026 at 15:06):

Understood @John Baez it happens to be somewhat relevant so I thought you were making a comment.

Rémy Tuyéras (Sep 27 2026 at 20:59):

@John Dai To expand on coherence and the construction of the functor $F$, we can take a similar approach to the previous one, but where we want to retain some non-associative compositional structure.

Let $\Mu\mathcal{W}$ be the unital magma freely generated by the set $\Sigma\mathcal{W}$.

Then, let us denote by $\mathcal{M}$ the unital magmoid, which is like a category but without the associativity requirement, whose set of objects is $\Sigma\mathcal{W}$ and whose set of arrows is given by $\Mu\mathcal{W}\times\Sigma\mathcal{W}^{\times 2}$.

If $e$ denotes the unit of $\Mu\mathcal{W}$, the identity at an object $a$ is $(e,a,a)$, and composition is defined by

$(n,b,c)\circ(m,a,b)=((m,n),a,c).$

The difference between $\mathcal{M}$ and $\mathcal{C}$ is that $\mathcal{M}$ remembers how we compose arrows, including the bracketing of successive compositions. In fact, a component $m$ in an arrow $(m,a,b): a\to b$ in $\mathcal{M}$ is given by a binary tree whose leaves are labelled by elements of $\Sigma\mathcal{W}$. We will denote the tuple of ordered leaves of $m$ by $\mathrm{Leaf}(m)$.

For example, if $m=((\texttt{"An ant can"},\texttt{" eat a"}),\texttt{" universe"})$, then we have

$\mathrm{Leaf}(m)=(\texttt{"An ant can"},\texttt{" eat a"},\texttt{" universe"}).$

Now, let us consider an arbitrary function $F:\Sigma\mathcal{W}\to \{0,1\}$. We can use this function to define an equivalence relation on the hom-set $\mathcal{M}(a,b)$ for every pair $(a,b)$ of objects as follows:

Two parallel arrows $(m,a,b):a\to b$ and $(n,a,b):a\to b$ are equivalent under a relation $\sim_F$ if and only if $F(m_1)\times\cdots\times F(m_k)=F(n_1)\times\cdots\times F(n_p),$ where $\mathrm{Leaf}(m)=(m_1,\ldots,m_k)$ and $\mathrm{Leaf}(n)=(n_1,\ldots,n_p)$. For the unit $e$, we take the corresponding empty product to be $1$.

Then, we can quotient each hom-set of $\mathcal{M}$ by $\sim_F$. Since $\sim_F$ is compatible with composition, meaning that $f\sim_F g$ and $h\sim_F i$ imply $f\circ h\sim_F g\circ i$ whenever these compositions are defined, this gives us a category (associativity follows from the associativity of the multiplication in $\{0,1\}$). We can denote the resulting category by $\mathcal{M}/F$.

Now, the difference between $\mathcal{M}/F$ and $\mathcal{C}$ is that, with $\mathcal{M}/F$, we do not forget how an arrow was composed if it contradicts "admissibility". Specifically, we cannot contract a subtree into the corresponding concatenated expression if doing so leads to an inconsistency with the way we defined admissibility through the assignment $F$.

For example, if

$F(\texttt{"ant can eat a"})=1,\qquad F(\texttt{" universe"})=1$,

but

$F(\texttt{"ant can eat a universe"})=0$,

then

$(\texttt{"ant can eat a"},\texttt{" universe"})\not\sim_F\texttt{"ant can eat a universe"}$.

Thus, the 2-level tree cannot be contracted to the single expression $\texttt{"ant can eat a universe"}$ in $\mathcal{M}/F$.

Finally, we can check that the assignment

$(m,a,b)\longmapsto F(m_1)\times\cdots\times F(m_k)$

defines a functor

$\mathcal{M}/F\to\mathcal{C}_{\mathrm{sem}}=\{0,1\}$.

John Dai (Sep 28 2026 at 08:11):

@Rémy Tuyéras i would like to see F upgraded to a meaning-producing realization and make admissibility a consequence of whether these meanings compose coherently. So F: M->C, and then A: C -> {0,1}. Attacking it via associativity is quite interesting (at semantic level). Also, I think M should encode syntactic rules somehow, rather than a flat list atomic expressions in W.

Rémy Tuyéras (Sep 28 2026 at 10:28):

@John Dai could you expand a little more on each idea?

Also, note that it would make sense to add the condition $m_1 \cdot \dots \cdot m_k = n_1 \cdot \dots \cdot n_p$ (in $\Sigma\mathcal{W}$) to the definition of $\sim_F$ in order to restrict equivalence classes to sentences that say the same thing

John Dai (Sep 28 2026 at 12:29):

Yes, I agree with adding
m_1 … m_k=n_1 … n_p.
That makes the question much cleaner: you are comparing different decompositions/compositional histories of the same expression, rather than identifying unrelated expressions merely because F gives them the same value.

John Dai (Sep 28 2026 at 12:30):

On my first suggestion, I am wondering whether {0,1} is being asked to do two jobs at once: represent meaning and judge admissibility.

John Dai (Sep 28 2026 at 12:32):

My second suggestion concerns M. At the moment, as I understand it, M remembers how expressions are bracketed, but does not distinguish syntactic types. I would like the source to encode which compositions are actually syntactically legitimate. For example, noun phrases, verbs, predicates, sentences, etc. should not all be interchangeable generators. Composition should be constrained by their types.

Rémy Tuyéras (Sep 29 2026 at 02:13):

On my first suggestion, I am wondering whether {0,1} is being asked to do two jobs at once: represent meaning and judge admissibility

For now, yes, but we could also temptatively take the reverse route: consider an arbitary equivalence relation $\sim$, for which we would ask to have a functor

$(\mathcal{M}/\!\sim )\to \mathcal{C}$.

Then we could try to infer the truth value associated with a morphism $(m,a,b):a \to b$ by defining a sheaf-like mapping:

$\mathcal{F}((m,a,b)) = \{(f_1,\dots,f_k) ~|~ (m,a,b) = f_k \circ \dots \circ f_1\textrm{ in }\mathcal{M}/\!\sim\}$

and computing some sort of homology class associated with it (whatever it means). The intuition here is that our earlier functor $F:\Sigma \mathcal{W} \mathcal \to \{0,1\}$ allowed us to classify coherent contractions between morphisms, so our hope would be that we could do the reverse: start with the coherence axioms (equalities between paths of arrows) and infer the classifiers (but going beyond only just $0$ and $1$).

Basically, we are looking into the possibility of eliminating imaginaries in the category's internal logic.

For example, noun phrases, verbs, predicates, sentences, etc. should not all be interchangeable generators. Composition should be constrained by their types.

To me, this is equivalent to taking a subcategory of $\mathcal{M}$. But, at this point, I would not see it as a huge blocker and would even suggest to keep the whole category $\mathcal{M}$ because it could contain more than modern English (1800s English, jargon, slang, baby English, etc.). Also, note that untrained Transformers would spit out outputs living in $\Sigma \mathcal{W}$, beyond modern English.

John Dai (Sep 29 2026 at 05:47):

@Rémy Tuyéras for LLM training, this reverse logic could be more practical … but then we don’t know if LLM has actually learned about the internal structure of M or the functor

John Dai (Sep 29 2026 at 06:02):

@Rémy Tuyéras your thinking is more LLM centric, which is fine. You are saying that an unrestricted M may contain different syntactic systems and by ruling out inadmissible semantic expressions the LLM learns how to impose these restrictions upstream — even though we don’t know how it does it exactly.

John Dai (Sep 29 2026 at 06:07):

@Rémy Tuyéras I think how to systematically impose admissibility restrictions in the semantic category is itself a very interesting problem. Maybe that is the attacking angle of your Leaf construction?

Rémy Tuyéras (Sep 30 2026 at 02:12):

@John Dai You are right on all three points.

One point to note, which you are already alluding to, is that the restriction to a category $D$ will affect the semantics that we can infer from the syntax.

For example, given the morphisms

$f=\texttt{"An"} \mathop{\longrightarrow}\limits^{\texttt{"An ant can eat a"}} \texttt{" universe"}$

$g=\texttt{"An"} \mathop{\longrightarrow}\limits^{(\texttt{"An ant"}) (\texttt{" can eat a"})} \texttt{" universe"}$

$i \circ h =\texttt{"An"} \mathop{\longrightarrow}\limits^{\texttt{"An ant"}} \texttt{" can eat"} \mathop{\longrightarrow}\limits^{\texttt{" can eat a"}} \texttt{" universe"}$

if we restrict $D$ to the morphisms $g,h,i$, then we require only a single semantic value. However, if we restrict $D$ to the morphisms $f,g,h,i$ with $f \neq g$, then we want two semantic values.