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Stream: theory: category theory

Topic: Operads for compound structures

Patrick Nicodemus (May 13 2026 at 16:55):

Hi all,
I am aware that Tom Leinster's notion of "generalized operad" can be used to characterize categories as algebras of a generalized operad in Graph.

I am interested in whether "generalized multicategories" or a related notion are documented for characterizing the following standard concepts as algebras of a generalized operad / multicategory.

• a functor
• a pair of functors together with a natural transformation between them
• an adjunction
• a Grothendieck fibration
• a presheaf

I am currently working through these characterizations on my own and I have characterized functors as generalized multicategory algebras to my own satisfaction. I would appreciate pointers to existing literature which model these "compound" structures involving multiple sorts as models of a certain "algebraic theory" which shares some nice relationship with operads - it need not be Leinster's formulation.

El Mehdi Cherradi (May 16 2026 at 09:35):

Hello,
https://arxiv.org/abs/0907.2460 is relevant here in case you have not looked at it.

Patrick Nicodemus (May 20 2026 at 17:13):

Hi all.

As I said above, it is well known that categories are algebras for a generalized multicategory in $(\mathbf{Graph}, \mathbf{FreeCat})$, which Leinster calls the "suspension" of the operad for monoids. (It took me a long time to realize that what Leinster calls the "suspension" is simply the horizontal categorification of the theory.)

I have come to the conclusion that

• functors, thought of as triples $(C, D, F : C\to D)$, are also models of a certain generalized multicategory over $(\mathbf{Graph}, \mathbf{FreeCat})$
• natural transformations $(C,D, F, G, \tau : F \Rightarrow G)$ are also models of a certain generalized multicategory over $(\mathbf{Graph}, \mathbf{FreeCat})$

But I was unable to cast adjunctions in this light. I don't claim it is impossible, but if we recall that generalized multicategories over $(\mathbf{Graph}, \mathbf{FreeCat})$ are exactly the virtual double categories, and that algebras for these g.m.c's are essentially functors on the V.D.C. valued in the V.D.C. of sets and spans, then it becomes a bit more clear why it is hard to express adjunctions as a VDC - the map $Hom(x, Gy) \to Hom(Fx,y)$ involves restriction of the Hom functor, which means that we would really like to express this theory as a virtual equipment.

This is nice but it seems to involve a departure from the existing theory of operads and generalized multicategories. If our goal is to find common frameworks for talking about categorical coherence in different situations, we know that "contractible operads" are a useful tool. The paper linked above by Cruttwell and Shulman contains on page 6 a table of many different kinds of generalized multicategories in their framework, but virtual equipments are not among them. Is the conclusion here that the theory of an adjunction is not well expressed an "operadic" theory because it involves restriction? Is there any well known class of algebraic structures which include certain generalized multicategories and also certain virtual equipments?

Patrick Nicodemus (May 20 2026 at 21:58):

I think I have worked something out that unifies these notions.

Patrick Nicodemus (May 28 2026 at 13:02):

I have drafted a short paper which observes that, just as virtual double categories are a special case of generalized multicategories, virtual double categories with all restrictions are a special case of some kind of "generalized multicategory with restrictions", which I define in the paper. I also define the notion of an algebra for a generalized multicategory with restriction, which generalizes functors from a virtual equipment into Set preserving Cartesian 2-cells.

I would be interested in comments on the mathematical content - I understand it is not especially polished, but please say something if any proof is hard to follow or unclear.

operads_w_restriction.pdf

@John Baez Would you be interested in skimming this and suggesting a next direction? I don't think the diagrams are especially intuitive or clarifying, but being able to extend the theory of operads with "reindexing" strikes me as plausibly powerful.

John Baez (May 28 2026 at 16:11):

Hi! Why did you ask me to suggest a next direction? If I knew that, I might be better able to think of one. There are other people here who are much more deeply involved in multicategories and/or virtual double categories.

Nathanael Arkor (May 28 2026 at 16:18):

On the other hand, there is a virtual equipment O_Adj such that morphisms of equipments O_Adj → Set do indeed correspond to adjunctions.

What do you mean by this? What virtual equipments are you thinking of with O_Adj and Set?

Nathanael Arkor (May 28 2026 at 16:21):

(I will look at the rest later, but cc @James Deikun, who has thought about similar ideas in the past.)

Patrick Nicodemus (May 28 2026 at 17:17):

Thanks Nathanael, I will expand on that.

Patrick Nicodemus (May 28 2026 at 18:56):

I think the claim I made is mistaken that adjunctions are not algebras for a generalized multicategory. I will fix this mistake, I need to fix some things here.

James Deikun (May 30 2026 at 04:16):

I've tried this approach of "cups" for generalized restrictions before; it is attractive because it ties in to the same concept of "representability" that is expressed by restrictions in virtual equipments. Nonetheless I came to believe that the actual correct concept for T-categories in general is a generalization of absolute (co)limits, along the same lines as Bryce Clarke's talk "Companions and conjoints are (co)limits". I even came to believe this in a very general way before I knew about the work in question, although I hadn't gotten to the point of trying to recreate any of the details involved that make it real math rather than just a possible approach to doing some math.

Something seemingly beyond the scope of this work, and thus likely to remain in need of doing, is figuring out the appropriate notion of what a (co)limit is in a $T$-category in general. This may, however, require the notion of a "normal $T$-category", which is a quite thorny problem to characterize, presently being worked on by several groups of researchers in parallel.

Patrick Nicodemus (May 30 2026 at 18:31):

John Baez said:

Hi! Why did you ask me to suggest a next direction? If I knew why you asked me, I might be better able to think of one. There are other people here who are much more deeply involved in multicategories and/or virtual double categories.

It's not so specific to you, it could have been directed at anyone who is interested in operads or multicategories. (You have at least one paper on operads.)
I could have been more specific with my question. The basic point of the paper is that some algebraic structures involving "dependent types" or fibering - for example, categories, which involve arrows indexed by objects - have operations, either primitive or derived, whose domain / codomain types involve reindexing - for example, the transpose operation of an adjunction is a map $\operatorname{Hom}_C(x,Gy) \to \operatorname{Hom}_D(Fx,y)$, and you cannot really express this map internally within any kind of "algebraic theory of adjunctions" unless that algebraic theory allows you to reindex the type $\operatorname{Hom}_C(x,y)$ along $(id_C : C \to C, G : D\to C)$. Thus, it is possible to define a virtual double category which is the "algebraic theory of an adjunction", but this virtual double category does not contain the transpose map anywhere. On the other hand, if one freely adjoins Cartesian restrictions to that virtual double category, in this richer setting one is able to talk about the transpose map and some equations that relate it to other cells.

The paper takes this as motivation to define a notion of "restriction" in a generalized multicategory. When I ask for a next direction, I think more concretely I am looking for examples of algebraic theories which seem to inherently involve reindexing and would thus be good candidates to formalize using the model. So, any kind of commonsense suggestion of dependently typed algebraic theories where one needs reindexing to describe the basic operations would be appreciated.

Patrick Nicodemus (May 30 2026 at 19:38):

operads_w_restriction.pdf
@Nathanael Arkor I have added a section clarifying the remarks about adjunctions, but I have not done further work on the main ideas.

John Baez (May 31 2026 at 08:14):

Thanks, I'll think about that. The "theory of cartesian closed categories" bothered me for years, but I'm not sure this is relevant here.

Nathanael Arkor (Jun 01 2026 at 07:40):

It would be nice to give some more examples of what "having restrictions" looks like for other kinds of generalised multicategories.

Patrick Nicodemus (Jun 01 2026 at 11:16):

Indeed. I suppose this is easily done by going to Leinster and working through some examples

John Onstead (Jun 02 2026 at 00:20):

Probably won't be much help here, but I think it's somewhat relevant and I always like sharing things I'm working on. Recently I was learning more about "triangular categories". Ordinary categories are the algebras for the path monad on $\mathrm{Psh}(G_2)$ where $G_2 = s,t: E \to V$ is the walking graph. A "triangular category" is an algebra for the path monad on $\mathrm{Psh}(G_3)$, where $G_3 = a,b,c: T \to V$, replacing edges (E) with triangles (T). Unlike a category with homsets of morphisms $\mathrm{Hom}(X, Y)$, a triangular category has homsets of triangles $\mathrm{Hom}(X, Y, Z)$. What is important about triangular categories is that their triangular profunctors act as trinary relation and are represented by 3d matrices, unlike usual profunctors that are based on binary relations and are represented by 2d matrices.

Just as a profunctor between two categories assigns a set of "heteromorphisms" between objects in one and objects in the other, a triangular profunctor between three triangular categories assigns a set of "hetero-triangles" between objects across the three triangular categories. This means that triangular categories and triangular profunctors themselves form a triangular category, and in fact, they form a "virtual triangular double category", which are the generalized multicategories for the path monad on $\mathrm{Psh}(G_3)$.

John Onstead (Jun 02 2026 at 00:20):

In a virtual double category with restrictions, a proarrow is meant to act like a binary relation, which is encapsulated by how restrictions take in a proarrow and two tight arrows and return a "scalar". For instance, given a profunctor $P: C \to D$ and two functors $X,Y: * \to C,D$, the restriction basically is the command to look up the entry $(X, Y)$ in the 2d matrix for $P$, which is the heteromorphism set $P(X, Y)$. In a virtual triangular double category with restrictions, a triangular proarrow is meant to act like a trinary relation. In the virtual triangular double category of triangular categories, a restriction here takes in a triangular profunctor $P: C \to D \to E$ and three triangular functors $X,Y,Z: * \to C,D,E$ and returns the entry $(X, Y, Z)$ in the 3d matrix for $P$, the set of hetero-triangles $P(X, Y, Z)$.

Of course, one can generalize to higher simplicial categories. These are the algebras for the path monad on $\mathrm{Psh}(G_n)$ (with $G_n$ the category with two objects and n morphisms from one to the other) and form virtual n-simplicial double categories, which are the generalized multicategories for those path monads. The appropriate notion of restriction for these virtual n-simplicial double categories would involve the input of one n-simplicial proarrow and n different tight arrows, and return what can be thought of as like an entry in an n-dimensional matrix.

James Deikun (Jun 02 2026 at 00:22):

How is the path monad on $G_3$ described?

Patrick Nicodemus (Jun 02 2026 at 00:26):

I was just wondering that. I'm conjecturing that a "word" is built by looking at a triangular number of objects https://en.wikipedia.org/wiki/Triangular_number and filling it in with triangles between those objects in the obvious way.

James Deikun (Jun 02 2026 at 00:32):

That description makes a lot of sense, but it seems unsatisfactory when talking about triangular profunctors since it wouldn't allow for a meaningful action of triangular categories on their profunctors. There are no triangles of that shape you can make that include hetero-triangles because the two adjacent triangles to one side would have to meet and the geometry just doesn't work.

Patrick Nicodemus (Jun 02 2026 at 00:34):

Yes also there are no edges in this model which makes things weird. A triangle doesn't have a boundary composed of 1-cells.

James Deikun (Jun 02 2026 at 00:36):

(Also: it can't be the whole story, because it isn't a monad by itself, you have to include things like a 3-triangle array with a smaller 3-triangle array set into one corner, or shapes like approximations of the Sierpinski triangle, in order to get the monad laws.)

John Onstead (Jun 02 2026 at 21:22):

Ah, you're right. I admit didn't think too much of that aspect on how things would work. I'm still very much a beginner at all this!
Maybe the point I was trying to get at was that if you have some notion of "generalized multicategory with restrictions" where the equivalent of "loose arrows" are meant to be interpreted as higher dimensional matrices/higher arity relations, then the analogous notion of "restriction" would need to change from requiring two "tight arrow" inputs to possibly many such inputs. Maybe this would still be relevant for generalized multicategories for any path-like or familial style monad on $\mathrm{Psh}(G_3)$ supposing one exists (even if it doesn't work the way I was thinking...) since the algebras would at the very least all have underlying "triangular graphs".