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Stream: theory: category theory

Topic: categorification of the real numbers

Joseph Denman (Apr 26 2021 at 15:18):

I couldn't find a channel like this anywhere else. Seemed like a good place for it.

Joseph Denman (Apr 26 2021 at 15:22):

Is there a "standard" categorification of the real numbers in the literature? Google results for this are surprisingly sparse. It seems like an obvious starting point to categorifying aspects of real analysis.

Joe Moeller (Apr 26 2021 at 15:28):

I think the real numbers are used in so many different ways in mathematics, there could be no standard way of capturing it all in one other gizmo. I'll offer one direction to go: Lawvere metric spaces.

John Baez (Apr 26 2021 at 15:33):

I've worked on this for years and haven't really been satisfied. I keep hoping there should be a way to blend groupoid cardinality and Euler characteristic and get a class of topological spaces that have a well-defined real-valued "cardinality":

• John Baez, The mysteries of counting: Euler characteristic versus homotopy cardinality.

There's a no-go theorem:

• John D. Berman, Euler characteristic and homotopy cardinality.

Abstract. Baez asks whether the Euler characteristic (defined for spaces with finite homology) can be reconciled with the homotopy cardinality (defined for spaces with finite homotopy). We consider the smallest infinity category $\mathsf{Top}^{rx}$ containing both these classes of spaces and closed under homotopy pushout squares. In our main result, we compute the K-theory $K_0(\mathsf{Top}^{rx})$, which is freely generated by equivalence classes of connected p-finite spaces, as p ranges over all primes. This provides a negative answer to Baez's question globally, but a positive answer when we restrict attention to a prime.

But maybe we should try something a bit different.

Mike Shulman (Apr 26 2021 at 15:37):

There are also real sets (as John knows!).

Joseph Denman (Apr 26 2021 at 15:38):

I think the real numbers are used in so many different ways in mathematics, there could be no standard way of capturing it all in one other gizmo

Hmm. Wikipedia says that the go-to axiomatic definition is as the unique Dedekind-complete ordered field R. 2-algebraic structures are common in CT. Categorification of a field seems obvious. One way to think about it is as a model of the Lawvere theory of of a field in $\mathbf{Cat}$. The missing pieces would be ordering and Dedekind completeness.

Joe Moeller (Apr 26 2021 at 15:42):

I don't think there's a Lawvere theory for fields. There is one for rings.

Joseph Denman (Apr 26 2021 at 15:43):

Thanks @John Baez @Mike Shulman Excellent response time!

John Baez (Apr 26 2021 at 15:48):

Sure! I once was talking to Ross Street a bunch of real sets, posing a bunch of questions, but he got too busy to figure out the answer, and I don't know if real sets have made any progress since then.

John Baez (Apr 26 2021 at 15:49):

There is indeed no Lawvere theory for fields, as you can see from the fact that there is no initial field: the category of models of any Lawvere theory always has limits and colimits, and thus an initial object. The category of fields lacks most limits and colimits.

Joseph Denman (Apr 26 2021 at 15:54):

But the codomain of the model functors is $\mathbf{Set}$?

John Baez (Apr 26 2021 at 15:57):

Yes, there's no Lawvere theory whose category of models in $\mathsf{Set}$ is the category of fields.

John Baez (Apr 26 2021 at 15:57):

This is what we mean by saying "there's no Lawvere theory of fields".

Fawzi Hreiki (Apr 26 2021 at 16:02):

Although there is still a classifying category

Cole Comfort (Apr 26 2021 at 16:02):

This has always bothered me. I think that the set of all prime fields forms the multi-initial set of objects in the category of fields. Is there some sort of weakening of a Lawvere theory that can account for this?

Fawzi Hreiki (Apr 26 2021 at 16:03):

Lextensivity (which is what is necessary to define a field) is still a fairly ‘algebraic’ notion of logic since it’s mostly quantifier free

Joseph Denman (Apr 26 2021 at 16:23):

There are many generalizations of Lawvere theory. Enriched theories are the most general I've seen. While the category of fields is not algebraic, it is accessible, meaning that the theory of fields can at least be captured in a limit-colimit theory. This sounds like a more suitable starting point.

Reid Barton (Apr 26 2021 at 16:37):

One could argue that the category Set already has most of the aspects of the real numbers, or at least the nonnegative real numbers.

John Baez (Apr 26 2021 at 16:37):

Sets are like natural numbers; groupoids are like positive real numbers; the challenge is getting negatives.

Cole Comfort (Apr 26 2021 at 16:39):

People were saying on here before: the int construction applied to the natural numbers is the integers (hence the name). If FinSet is traced monoidal then wouldn't the right generalization be int(FinSet)

John Baez (Apr 26 2021 at 16:41):

If you want a category that's like the integers, you mean?

Cole Comfort (Apr 26 2021 at 16:41):

yes

John Baez (Apr 26 2021 at 16:42):

Okay, so someone should see what happens when you apply the int construction to the category of finite sets, or maybe to the groupoid of finite sets. (Are they both traced monoidal with disjoint union as monoidal structure?)

Cole Comfort (Apr 26 2021 at 16:44):

For reference to the earlier thread:
https://categorytheory.zulipchat.com/#narrow/stream/229136-theory.3A-category.20theory/topic/The.20compact.20closed.20category.20of.20integers

Reid Barton (Apr 26 2021 at 16:49):

FinSet isn't traced because $\mathrm{Hom}(A \amalg X, B \amalg X)$ might be nonempty when $\mathrm{Hom}(A, B)$ is empty.

Reid Barton (Apr 26 2021 at 16:50):

Maybe take pointed sets, or just isos as maps.

Cole Comfort (Apr 26 2021 at 16:50):

Yeah what I said before (and deleted) is wrong, when you trace out a multiplication (considering it as a generator as the canonical presentation for FinSet), then it results in something of type 1->0

John Baez (Apr 26 2021 at 16:51):

So is the groupoid of finite sets traced monoidal? If so, I suspect you get something pretty boring when you apply the int construction.

Joe Moeller (Apr 26 2021 at 16:53):

It is! You can just draw the traced structure too. If you draw a permutation you just draw an arc from an output back to the corresponding input.

Cole Comfort (Apr 26 2021 at 16:53):

I think it is, because the groupoid of finite sets is equivalent to the prop of permutations. Oh, you beat me to it

Joe Moeller (Apr 26 2021 at 16:53):

Then pull the strings straight so it looks like a permutation again.

Cole Comfort (Apr 26 2021 at 16:54):

So it seems like int applied to permuations would give you the free self dual compact closed category on one object?

Nathanael Arkor (Apr 26 2021 at 16:55):

Cole Comfort said:

This has always bothered me. I think that the set of all prime fields forms the multi-initial set of objects in the category of fields. Is there some sort of weakening of a Lawvere theory that can account for this?

Yes, there is: a multialgebraic theory. See the work of Diers, for instance.

Mike Shulman (Apr 26 2021 at 16:56):

Joseph Denman said:

The missing pieces would be ordering and Dedekind completeness.

The natural categorification of an order is, well, a category. Which raises the question: what sort of theory is there whose models in $\rm Poset$ are ordered fields?

Dedekind completion sort-of categorifies to a sort of Isbell completion.

Reid Barton (Apr 26 2021 at 16:58):

You could start with ordered rings. Already the fact that multiplication is not exactly order-preserving suggests you might need some extra ingredient.

John Baez (Apr 26 2021 at 16:59):

Cole Comfort said:

So it seems like int applied to permuations would give you the free self dual compact closed category on one object?

Okay, so that's not extremely boring, but it's also familiar: it's the category of 1d oriented cobordisms.

Cole Comfort (Apr 26 2021 at 17:00):

John Baez said:

Cole Comfort said:

So it seems like int applied to permuations would give you the free self dual compact closed category on one object?

Okay, so that's not extremely boring, but it's also familiar: it's the category of 1d cobordisms.

It still seems completely unrelated to the real numbers, although I don't really understand how groupoids correspond to the real numbers in the first place.

Mike Shulman (Apr 26 2021 at 17:01):

Reid Barton said:

You could start with ordered rings. Already the fact that multiplication is not exactly order-preserving suggests you might need some extra ingredient.

Yeah, that's a tricky one. Actually it reminds me of the problem of constructing multiplication on the surreal numbers in the absence of LEM...

John Baez (Apr 26 2021 at 17:39):

Cole Comfort said:

John Baez said:

Cole Comfort said:

So it seems like int applied to permuations would give you the free self dual compact closed category on one object?

Okay, so that's not extremely boring, but it's also familiar: it's the category of 1d cobordisms.

It still seems completely unrelated to the real numbers, although I don't really understand how groupoids correspond to the real numbers in the first place.

The groupoid cardinality of a finite groupoid can be any nonnegative rational number, and when we add or multiply groupoids their cardinalities add; also if we take the weak quotient of a groupoid $X$ by some action of a finite group $G$, its groupoid cardinality divides in the way you'd hope:

$|X//G| = |X|/|G|$

When we switch from finite groupoids to "tame" groupoids (defined at the link), the cardinality can be any nonnegative real number, and all the stuff I said still holds. For example, the cardinality of the groupoid of finite sets is $e$.

John Baez (Apr 26 2021 at 17:42):

There is a lot more you can do here, and a bunch is explained here:

• Jeffrey Morton, Categorified algebras and quantum mechanics.

Morton came up with a reasonably nice way of categorifying the complex numbers, suitable for quantum mechanics.

John Baez (Apr 26 2021 at 17:43):

He was building on this:

• John Baez and James Dolan, From finite sets to Feynman diagrams.

which is where we came up with groupoid cardinality and noticed that the cardinality of the groupoid of finite sets is $e$.

Fawzi Hreiki (Apr 26 2021 at 17:51):

I think Schanuel has a beautiful short paper on categorifying the integers.

Fawzi Hreiki (Apr 26 2021 at 17:52):

https://link.springer.com/chapter/10.1007/BFb0084232

John Baez (Apr 26 2021 at 18:08):

Yes, it's great. By the way, all these references and more are listed here.

John Baez (Apr 26 2021 at 18:09):

Another nice one is

• Daniel Loeb, Sets with a negative number of elements, Adv. Math. 91 (1992), 64-74.

John Baez (Apr 26 2021 at 18:10):

So there are lots of ideas out there, but they haven't quite coalesced.

Fawzi Hreiki (Apr 26 2021 at 18:12):

There’s also Lawvere’s ‘Taking Categories Seriously’ where he categorifies the Dedekind cut construction of the reals as a preliminary for metric space theory

John Baez (Apr 26 2021 at 18:18):

So what category does he get when he categorifies the reals? (If it's just the reals again, it's not much of a "categorification".)

Fawzi Hreiki (Apr 26 2021 at 18:35):

Well yes you’re right. It’s more a categorical take on the process

Amar Hadzihasanovic (Apr 26 2021 at 18:41):

John Baez said:

So is the groupoid of finite sets traced monoidal? If so, I suspect you get something pretty boring when you apply the int construction.

All of these are traced monoidal with disjoint union:

• finite sets and relations;
• finite sets and partial functions;
• finite sets and partial injections;
• finite sets and bijections.

See Section 5 of this paper.

Cole Comfort (Apr 26 2021 at 18:48):

Amar Hadzihasanovic said:

John Baez said:

So is the groupoid of finite sets traced monoidal? If so, I suspect you get something pretty boring when you apply the int construction.

All of these are traced monoidal with disjoint union:

• finite sets and relations;
• finite sets and partial functions;
• finite sets and partial injections;
• finite sets and bijections.

See Section 5 of this paper.

Spans of finite sets are matrices over the natural numbers. I wonder what happens when you apply the int construction. You don't get matrices over the integers, because this is not compact closed.

Cole Comfort (Apr 26 2021 at 19:40):

The cocommutative comonoid induced by a commutative monoid by bending around one of the input wires from the multiplication and transposing the unit forms a commutative Frobenius algebra with the original monoid, so it seems to me that this could perhaps give you something like interacting Hopf algebras over the rational numbers. That is to say, the prop generated by two bicommutative Hopf algebras interacting to form two Frobenius algebras where both compact closed structures agree.

Jade Master (Apr 26 2021 at 20:50):

Just proceeding by blind analogy, if finite groupoids categorify the positive rationals then maybe we need to transfinite (co?)limits of finite groupoids to get the categorification of the reals?

Jade Master (Apr 26 2021 at 20:52):

Maybe this is similar to Lawvere's construction of the reals that @Fawzi Hreiki was talking about?

Jade Master (Apr 26 2021 at 20:53):

This would be by analogy to the definition of the reals as equiv classes of cauchy sequences of rationals.

John Baez (Apr 26 2021 at 23:56):

Jade Master said:

Just proceeding by blind analogy, if finite groupoids categorify the positive rationals then maybe we need to transfinite (co?)limits of finite groupoids to get the categorification of the reals?

Certain colimits of finite groupoids, called "tame" groupoids, let you categorify the nonnegative reals. That's what James Dolan and I did in From Finite Sets to Feynman Diagrams.

Dmitri Pavlov (Apr 28 2021 at 22:08):

Nonnegative real numbers can be categorified by finite-dimensional modules over a type II factor. Arbitrary real numbers can be categorified by finite-dimensional super modules over a type II factor. Alternatively, one can also take the homotopy group completion of finite-dimensional modules.

This is completely analogous to how natural numbers can be categorified by finite-dimensional vector spaces and integer numbers can be categorified by finite-dimensional super vector spaces, or the homotopy group completion of vector spaces.

John Baez (Apr 28 2021 at 22:11):

Dmitri Pavlov said:

Nonnegative real numbers can be categorified by finite-dimensional modules over a type II factor. Arbitrary real numbers can be categorified by finite-dimensional super modules over a type II factor.

Nice! Do you know examples of people doing that: taking some interesting math with real numbers, and categorifying it this way? I can imagine some blend of Vaughan Jones and Mikhail Khovanov trying to do this.

Dmitri Pavlov (Apr 28 2021 at 22:31):

I am not aware of any genuine results in this direction that could be labeled as “categorification” in a strict sense. In a broad sense, Vaughan Jones's work on planar algebras heavily relies on the fact that dimensions of modules over type II factors are real numbers.

dusko (May 03 2021 at 10:29):

there is a construction of a category where the real numbers are the objects, and polarized simulations are the morphisms --- which gives a category where the addition of real numbers is the biproduct, and the multiplication is a tensor. the UPSHOT is that if you look at a real matrix as a discrete profunctor, then the two linear operators induced by the matrix are the two kan extensions. its in sec 6 here:
https://arxiv.org/abs/2007.10057

but i mentioned this here before, and no one seemed interested in a category of real numbers. now there is a 100 posts, and i am sorry but i only read a bit at the beginning and a bit at the end. maybe it is not what everyone is looking for. but if was fun to notice that the addition is the biproducts anyway :)

Jade Master (May 03 2021 at 13:41):

@dusko this is super interesting to me. What are polarized simulations?

dusko (May 07 2021 at 08:57):

ooh, sorry jade that i didn't login earlier.

i first explain what is polarized, then what is simulation.

you know how the INT-construction works, of compact from traced category? its morphisms are polarized functions. they are functions between sets with positive and negative elements, and they go forward between the positive elements and backward along the negative element. if you separate a polarized function into two, you get the chu construction.

simulations are functions extended in time. the domain streams the inputs and the function streams the outputs into the codomain. towers of functions.

with the hindsight, it is unsurprising that that should be the morphisms between the real numbers. he INT construction constructs the integers. they are like polarized numers. real numbers are streams of numbers. so just like the INT- functions are the morphisms between the integers, the streams of such INT-functions are the morphisms between the reals. si?

dusko (May 07 2021 at 09:12):

@Jade Master i am impressed with your name. we should petition your university to sometimes modify the title of Philosophiae Doctor into Of The Universe.

Jade Master (May 07 2021 at 15:59):

@dusko thank you don't wear it out :wink: I am aware of the INT construction and some sort of simulation. Reading your paper a bit it seems like the result of taking polarized simulations is you get the the standard poset of real numbers as the posetal collapse of asynchronous hypergames? I think I'm missing something though because + is definitely not a biproduct with that ordering.

dusko (May 07 2021 at 16:50):

@Jade Master well, if you are missing something it means that i did not present it clearly enough. it may also be that i had missed something, but that should be less likely, since i had first proved this in sometime before 1999 (when i presented it in edinburgh), and re-proved it 4 different ways, 3 of which for different coalgebra (Pow times Pow, not streams). but it is always possible that i made the same strange attractor mistake 4 times. the equivalence with the stream presentation here is not only in my paper about the reals with vaughan pratt, but is also the topic of gonshor's book...

i should have included more details in this paper, but i was 60 pages into the paper that had suggested length of 25 pages, and the reals are not its central topic... but i didn't find a simpler and roughly as compelling way to show that the model includes noncomputable stuff.

the crucial point which makes asynchronous hyperstrategies into biproduct is (75), which has is the matrix into which the biproducts of relations split, just "extended in time". it is the "disjoint union extended in time". the crucial place which shows that this is the addition of the reals is the equivalence of (79) with conway's "negative" preordering (80). hmm, maybe (79) should not be called "posetal collapse" of (75), because it does not "flatten out" a homset in one sweep, but coinductively, ie layer by layer of the ladder of relations extended in time...

i am sorry, i should not be explaining this here, but should have said it in the paper. it was a lot of work, and i was tired of it...

dusko (May 07 2021 at 16:57):

(i cannot afford to keep coming to networks too often, and then survive all meetings, but if you don't want to wait and email me, i will respond)

Jade Master (May 07 2021 at 17:39):

Okay thank you Dusko, the problem may be me. I will try again later when I've had some caffeine :)

Joseph Denman (Aug 16 2021 at 19:21):

I've had an idea for a while now about how this might be done, but I can't seem to make the time for it. It feels selfish not to share it in case anyone else does have the time to pursue it.

Basically, $\Bbb R$ is a field that satisfies some additional ordering axioms (total ordering and Dedekind completeness): https://en.wikipedia.org/wiki/Real_number

My original idea was the create a Lawvere theory for fields and then just model it in $\textbf{Cat}$. But there isn't a Lawvere theory for fields. There is however a finite limit theory (sketch) for fields. This is equivalent to saying that $\textbf{Field}$ is an accessible category. Of course the sketch for fields doesn't capture the additional axioms that the reals need to satisfy. We need a way to include them.

The main idea is to construct a $\textbf{Cat}$-enriched sketch for fields in a category $M$, then model that sketch in $\textbf{Cat}$. Let $R$ be the object of $M$ modeling the set of reals. Let $1$ be the terminal object such that $1 \xrightarrow{a} R$ can be considered a real number (an element). Then total ordering is captured:

IMG_0259.jpg

We simply impose the ordering condition on the category $M(1, R)$ as 2-morphisms and require that models of the sketch preserve them.

There are a lot of technical conditions that need to be met in order for this to work, which is why I haven't gotten around to it :)

Referenced for enriched sketches: http://www.tac.mta.ca/tac/volumes/1998/n3/n3.pdf

Thoughts appreciated.

Fawzi Hreiki (Aug 16 2021 at 19:56):

Just a point: the sketch for the theory of fields is not a finite limit sketch.

Fawzi Hreiki (Aug 16 2021 at 20:15):

E.g. the category of fields doesn’t have products

Nathanael Arkor (Aug 16 2021 at 20:26):

Fields are sketchable by a (finite product, finite coproduct) sketch.

Joseph Denman (Aug 17 2021 at 00:03):

Fawzi Hreiki said:

Just a point: the sketch for the theory of fields is not a finite limit sketch.

Products are finite limits :)

Keith Elliott Peterson (Aug 17 2021 at 00:38):

Instead of working with sets, could so-called "sets with negative cardinality" be defined using a geometric theory?

Zhen Lin Low (Aug 17 2021 at 01:02):

Joseph Denman said:

Fawzi Hreiki said:

Just a point: the sketch for the theory of fields is not a finite limit sketch.

Products are finite limits :)

The category of models of a finite limit sketch is not merely accessible but also locally finitely presentable. The category of fields is not.

Fawzi Hreiki (Aug 17 2021 at 13:14):

Joseph Denman said:

Fawzi Hreiki said:

Just a point: the sketch for the theory of fields is not a finite limit sketch.

Products are finite limits :)

Just to clarify, a 'finite limit theory/sketch' means a theory/sketch which is defined using at most finite limits, not a theory/sketch whose definition includes finite limits.

Fawzi Hreiki (Aug 17 2021 at 13:16):

The theory of fields is not axiomatisable/sketchable using just finite limits since, for example, every category of models of a finite limit theory has a terminal object while the category of fields does not.

John Baez (Aug 17 2021 at 20:39):

Maybe you should present your claimed sketch for fields, @Joseph Denman. There is no finite limits theory for fields; this is a well-known fact and various people here have sketched (!) proofs.

Joseph Denman (Aug 17 2021 at 22:19):

John Baez said:

Maybe you should present your claimed sketch for fields, Joseph Denman. There is no finite limits theory for fields; this is a well-known fact and various people here have sketched (!) proofs.

I see the claim once on the nCat page for fields: https://ncatlab.org/nlab/show/field

I see it again in volume II of Sketches of an Elephant (sketch puns abound!): IMG_1368.HEIC

Joseph Denman (Aug 17 2021 at 22:20):

But I haven't checked either for correctness, and they appear different.

Nathanael Arkor (Aug 17 2021 at 22:30):

The nLab page specifically says that you need a limit-colimit sketch, as opposed to just a limit sketch.

Nathanael Arkor (Aug 17 2021 at 22:32):

You can paste images into Zulip (many people won't be able to open a .heic file).

Joseph Denman (Aug 17 2021 at 22:35):

Nathanael Arkor said:

The nLab page specifically says that you need a limit-colimit sketch, as opposed to just a limit sketch.

Clearly my language is a little loose.

Nathanael Arkor said:

You can paste images into Zulip (many people won't be able to open a .heic file).

Thanks. Here's a PDF [Uploading IMG_1368.HEIC.pdf…]()

Nathanael Arkor (Aug 17 2021 at 22:36):

A "finite limit sketch" means something specific: namely that the cones are finite, and the set of cocones is empty. The Elephant says that fields form a "disjunctive sketch", which contains cocones.

David Michael Roberts (Aug 18 2021 at 02:34):

Dmitri Pavlov said:

Nonnegative real numbers can be categorified by finite-dimensional modules over a type II factor. Arbitrary real numbers can be categorified by finite-dimensional super modules over a type II factor. Alternatively, one can also take the homotopy group completion of finite-dimensional modules.

This is completely analogous to how natural numbers can be categorified by finite-dimensional vector spaces and integer numbers can be categorified by finite-dimensional super vector spaces, or the homotopy group completion of vector spaces.

Nice. I guess then a "categorified vector space" would then be a category with an action of the monoidal category of such type II factor modules?

David Michael Roberts (Aug 18 2021 at 02:42):

And can you remind me what "finite-dimensional" means in this context?

Dmitri Pavlov (Aug 18 2021 at 22:16):

@David Michael Roberts For modules over a type II_1 factor M, the dimension is given by a unique function from finitely-generated left M-modules to real numbers that sends directs sums of modules to sums of real numbers and sends M as a module over itself to 1. For modules over a type II_∞ factor M, the dimension is given by a unique up to rescaling function from finitely supported left M-modules to real numbers that sends directs sums of modules to sums of real numbers. More conretely, any finitely supported left module over a type II factor M is a direct sum of modules of the form Mp, where p is a projection in M. The dimension of Mp is equal to tr(p), where tr: M→R is some fixed trace on M.

Dmitri Pavlov (Aug 18 2021 at 22:25):

@David Michael Roberts “I guess then a "categorified vector space" would then be a category with an action of the monoidal category of such type II factor modules?”: Yes. A canonical example: take the W* -category of right A-modules, for some von Neumann algebra A. If M is a von Neumann algebra (e.g., a type II factor), then the category of left M-modules with direct sum acts on the category of right A-modules via the appropriate notion of a tensor product (W* -tensor product of W* -modules, or, equivalently, Connes fusion for representations on Hilbert spaces).

Dmitri Pavlov (Aug 18 2021 at 22:29):

Zulip's markdown syntax is brain-damaged: apparently, they manually removed escaping via backslash, so it is simply impossible to type an asterisk without a space after it: https://github.com/zulip/zulip/blob/master/docs/subsystems/markdown.md

David Michael Roberts (Aug 18 2021 at 23:16):

@Dmitri Pavlov I don't see that last statement. Given random von Neumann algebras A and M, what exactly does one do with an A-module and an M-module to get back an A-module? (that is, what is the object component of the action functor AMod x MMod -> AMod?)

David Michael Roberts (Aug 18 2021 at 23:42):

Hmm, that MMod might need to be on the left. I guess one takes the tensor product of the modules and ignores the M-action?

Dmitri Pavlov (Aug 18 2021 at 23:54):

@David Michael Roberts If you have a left M-module and a right A-module, you can tensor them together to get an M-A-bimodule. Furthermore, (X⊕X')⊗Y = X⊗Y ⊕ X'⊗Y, as required for a categorified module.

David Michael Roberts (Aug 19 2021 at 00:11):

@Dmitri Pavlov You know, that's what I was expecting. I read your message twice to make sure, and I was certain I read "right M-modules", hence my confusion! Thanks for fixing my misunderstanding.