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Stream: theory: category theory

Topic: Looking for interesting examples of enriched sites

Ari Rosenfield (Jan 27 2025 at 19:53):

Enriched sheaves and enriched sites were defined in the 90s in this paper of Borceux's and Quinteiro's, but the paper doesn't contain any examples, so I'm trying to collect examples.

It's easy enough to take the definitions there and write down all the axioms in the setting of a particular enriching category $\mathcal{V}$, but it's often not clear to me how to interpret the result of doing so. The one example I feel like I understand is the case where $\mathcal{V}$ is the category of abelian groups and $\mathcal{C}$ is a ring (i.e., a 1-object Ab-category). In this case, a $\mathcal{V}$-enriched sheaf is a certain localization of $\mathcal{C}$ (its Gabriel localization), and an enriched site is a Giraud subcategory of $\text{Mod}_\mathcal{C}$. The other simple example I know of is the case of enrichment over $\{0,1\}$.

I'm wondering if anyone knows of other examples. In particular, I wonder if anyone has worked out how to interpret sheaves or sites in the case of enrichment over the quantale $[0,\infty]$, but really any examples would be good to see.

Fernando Yamauti (Jan 29 2025 at 16:49):

I have yet to read that paper properly, but I'm wondering if their definition is really the correct one whenever one considers non-cartesian monoidal structures. It seems their condition of left exactness doesn't require the left adjoint to be monoidal, right? But if tensor products are to substitute products, that should be a necessary condition. It's also not clear to me whether the induced monoidal structure on the cat of sheaves (if that exists) will induce a commutative monoid in $\mathbf{Pr}^L$.

Also, consider some $\omega$-accessible locally presentably monoidal category $\mathcal{V}$ and some topos $\mathcal{X}$. Is $\mathcal{X} \otimes \mathcal{V}$ supposed to be an example of enriched topos in their conception?

Another thing. It seems, from Gabriel-Popescu, any Grothendieck abelian category will be an enriched Borceux-Quinteiro topos. But if that's the case, we don't have a priori any monoidal structure.

Btw, sorry to barge in asking further questions.

Kevin Carlson (Jan 29 2025 at 17:40):

Who said tensor products are to substitute products, though? This certainly isn’t what happens in the case of sheaves of abelian groups or modules, and would be a vastly more complicated kind of thing to try to do.

Kevin Carlson (Jan 29 2025 at 17:41):

The (fibered) product serves two roles in set-enriched toposes, namely to compose morphisms in the enrichment base and to describe matching families for sheaves. Those functions are split between the monoidal products and the fiber product in Borceux-Quintero, not both made monoidal.

Kevin Carlson (Jan 29 2025 at 17:44):

For the question about $\mathcal X\otimes \mathcal V,$ I’m not sure I know what construction you mean. Objects are pairs of an $X$ and a $V$ and homs are the tensoring, using cocompleteness of $\mathcal V$? It doesn’t seem like a terribly natural construction…I would think we’d rather try to talk about $\mathcal V$-objects internal to $\mathcal X$, to reproduce the passage from a topos to its Grothendieck category of abelian group objects. But maybe that’s not what you’re thinking?

Fernando Yamauti (Jan 29 2025 at 18:08):

Kevin Carlson said:

For the question about $\mathcal X\otimes \mathcal V,$ I’m not sure I know what construction you mean.

That's the tensor product of locally presentable cats. So, yes, it's the same thing as internal models of an essentially algebraic theory inside the topos.

Perhaps Borceux-Quinteiro's generalisation was never supposed to model the usual objects appearing in noncommutative geometry: certain monoid objects in the cat of locally presentable $\mathcal{V}$-cats. That's what I'm trying to understand...

Kevin Carlson (Jan 29 2025 at 18:56):

Fernando Yamauti said:

Kevin Carlson said:

For the question about $\mathcal X\otimes \mathcal V,$ I’m not sure I know what construction you mean.

That's the tensor product of locally presentable cats. So, yes, it's the same thing as internal models of an essentially algebraic theory inside the topos.

Right, I see.

Kevin Carlson (Jan 29 2025 at 19:02):

Fernando Yamauti said:

Perhaps Borceux-Quinteiro's generalisation was never supposed to model the usual objects appearing in noncommutative geometry: certain monoid objects in the cat of locally presentable $\mathcal{V}$-cats. That's what I'm trying to understand...

This doesn’t strike me as an obvious notion of topos relative to $\mathcal{V}$, no. Separately, I think that’s a rather idiosyncratic use of the word “usual”! I haven’t looked into this in quite a while but I believe that many approaches to noncommutative geometry try to get by without the monoidal structure since it simply does not exist for, say, left modules over a noncommutative ring.

Kevin Carlson (Jan 29 2025 at 19:03):

In particular notice that your notion doesn’t specialize to anything close to an ordinary topos in the case $\mathcal V=\mathbf{Set}.$

Fernando Yamauti (Jan 29 2025 at 19:19):

Kevin Carlson said:

In particular notice that your notion doesn’t specialize to anything close to an ordinary topos in the case $\mathcal V=\mathbf{Set}.$

Hmm... I think it does, no? Every topos is a commutative monoid in $\mathbf{Pr}^L$. I'm thinking more along the general idea that topoi are certain $2$-rings (or rather, the underlying logoi in Joyal notation).

But, yes, maybe the monoidal structure shouldn't always be present if one wants to consider arbitrary noncommutative rings. Still the sort of noncommutativity I was thinking about would be things like representations of a group or, more generally, quasi-coherent sheaves over some stack. All of those are locally presentable tensor cats.

Kevin Carlson (Jan 29 2025 at 19:20):

There are many monoids in $\mathbf{Pr}^L$ that are not topoi, was my point.

Fernando Yamauti (Jan 29 2025 at 19:36):

Kevin Carlson said:

There are many monoids in $\mathbf{Pr}^L$ that are not topoi, was my point.

Yes. That's why I said certain monoids as opposed to all of them!

But I do agree regarding my idiosyncracy since I didn't specify what I meant by noncommutative geometry. Still I have never seen any object that comes from deforming some notion of commutative space that doesn't have a tensor structure in its cat of modules... but, perhaps, something like that might exist (?).

Kevin Carlson (Jan 29 2025 at 19:38):

I admit I missed the "certain", and certainly it all hinges on which "certain" you mean, but I think we've established that you and Borceux are thinking of two different things.

Kevin Carlson (Jan 29 2025 at 19:39):

I don't think everybody considers deformations of commutative spaces to be exhaustive of the concept of noncommutative space. There's no reason a priori why every noncommutative space should be close to a commutative one!

Fernando Yamauti (Jan 29 2025 at 19:48):

Kevin Carlson said:

I don't think everybody considers deformations of commutative spaces to be exhaustive of the concept of noncommutative space. There's no reason a priori why every noncommutative space should be close to a commutative one!

Generalizations usually come from abstracting properties the main examples satisfy. But if the main examples are such deformations...

Also, I think another possible justification might be that perhaps we can't recover a commutative space from its cat of modules when we don't have a tensor structure, but I'm not sure about that. For instance, can we recover a group from its cat of representations without the monoidal structure? I think we need to consider only the automorphisms of a fiber functor that preserve the monoidal structure.

Kevin Carlson (Jan 29 2025 at 19:56):

I mean, the single most famous example of a formulation of noncommutative geometry is Connes' via $C^*$-algebras, and certainly modules over noncommutative $C^*$-algebras lack a tensor product.

Ari Rosenfield (Jan 29 2025 at 19:57):

Btw, sorry to barge in asking further questions.

No worries at all, thanks for the questions. I'm a bit isolated in my grad program as nobody else at my school (including my advisor) really thinks about this stuff.

I need to do a little reading to understand all the concerns being raised, but can one of y'all clarify what's meant by $\mathbf{Pr}^L$? This notation and the idea of toposes as certain monoids within it aren't familiar to me yet.

Kevin Carlson (Jan 29 2025 at 19:58):

$\mathbf{Pr}^L$ is the category of locally presentable categories and cocontinuous functors between them. Locally presentable categories satisfy an optimal adjoint functor theorem, so these are equivalently the left adjoint functors.

Fernando Yamauti (Jan 29 2025 at 21:09):

Kevin Carlson said:

I mean, the single most famous example of a formulation of noncommutative geometry is Connes' via $C^*$-algebras, and certainly modules over noncommutative $C^*$-algebras lack a tensor product.

It seems all examples coming from quantisation have a tensor product (?). But perhaps the physicists around here can hop in to clarify that (maybe @John Baez ).

All of the examples in practice I've seen are convolution algebras of groupoids $G$. Certain Fredholm modules over that should correspond to $G$-bundles and I'm tempted to think those things always have a canonical monoidal structure. All those things are bialgebras(oids) though and that seems exactly what is giving us the monoidal structure.

Also correcting one thing I said. There are examples where one can recover a commutative space from its cat of modules without the tensor structure (Rosenberg's spectrum for instance), but whenever one wants to recovers also automorphisms (so higher stuff like $BG$ I'had mentioned before) it seems the monoidal structure is used. So it seems it's exactly the other way around: the monoidal structure is likely indispensable when one wants to recover a stack from its cat of modules. But that's of course only under the assumptions that stacks should embed fully faithfully into noncommutative spaces and noncommutative spaces should be given by certain module categories.

Fernando Yamauti (Jan 29 2025 at 21:16):

Ari Rosenfield said:

No worries at all, thanks for the questions. I'm a bit isolated in my grad program as nobody else at my school (including my advisor) really thinks about this stuff.

I need to do a little reading to understand all the concerns being raised, but can one of y'all clarify what's meant by $\mathbf{Pr}^L$? This notation and the idea of toposes as certain monoids within it aren't familiar to me yet.

This "topoi are affine schemes" and "locally pres. cats are abelian groups" analogy probably started with Joyal (?). It's described in http://mathieu.anel.free.fr/mat/doc/Anel-Joyal-Topo-logie.pdf for instance.

Taking more general monoids as possible noncommutative spaces is described in https://arxiv.org/abs/1105.3104 for instance.

Fernando Yamauti (Jan 29 2025 at 21:27):

I think it's not only you who feel isolated. There seems to be no unified consensus on what should be a "linear topos". Just a bunch of disparate examples appearing in K-theory (localising invariants), Tannakian theory and quantale theory that should all supposedly be part of common concept. And people working on those fields are usually interested in solving only particularly already well stablished problems in those fields

John Baez (Jan 30 2025 at 01:37):

Fernando Yamauti said:

Kevin Carlson said:

I mean, the single most famous example of a formulation of noncommutative geometry is Connes' via $C^*$-algebras, and certainly modules over noncommutative $C^*$-algebras lack a tensor product.

It seems all examples coming from quantisation have a tensor product (?). But perhaps the physicists around here can hop in to clarify that (maybe John Baez ).

All of the examples in practice I've seen are convolution algebras of groupoids $G$.

The most basic example of a C*-algebra is the algebra of $n \times n$ complex matrices, and we usually don't think of the category of modules of this as having a tensor product, since this algebra is used to describe observables of a system with $n$ states, and when you tensor two $n$-state systems you should not get another system with $n$ states: you should get one with $n^2$ states.

Basically you want your C* -algebra to be a bialgebra, making its modules into a monoidal category, when there's a physically relevant way to "combine systems" whose observables lie in this C*-algebra.

Fernando Yamauti (Jan 30 2025 at 15:38):

John Baez said:

The most basic example of a C*-algebra is the algebra of $n \times n$ complex matrices, and we usually don't think of the category of modules of this as having a tensor product, since this algebra is used to describe observables of a system with $n$ states, and when you tensor two $n$-state systems you should not get another system with $n$ states: you should get one with $n^2$ states.

Basically you want your C* -algebra to be a bialgebra, making its modules into a monoidal category, when there's a physically relevant way to "combine systems" whose observables lie in this C*-algebra.

Thanks for chiming in! Yes, I'm aware that in order to get a monoidal structure we need at least a bialgebra. My question was if there are examples of algebras coming from deformation quantisation of commutative ones that are not bialgebras. Is that example of $n \times n$ complex matrices such a case? I mean is there a commutative algebra with a Poisson structure that deforms to that? Perhaps it's so obvious and that's why you didn't mention anything...

Fernando Yamauti (Jan 30 2025 at 15:40):

@Ari Rosenfield If the discussion is getting too much out of the original scope, I think I (or some admin?) can split into another thread (no idea how to do that though).

Ari Rosenfield (Jan 30 2025 at 16:01):

@Fernando Yamauti It's fine with me to keep it here

John Baez (Jan 30 2025 at 21:58):

Fernando Yamauti said:

My question was if there are examples of algebras coming from deformation quantisation of commutative ones that are not bialgebras.

Yes, most of them are not bialgebras, at least not in any way I'd understand. We'd expect the deformation quantization to be a bialgebra, or hope for them to be, if the original commutative algebra being deformed was a bialgebra. That usually happens when its an algebra of functions on a topological monoid (e.g. a topological group). But if we apply deformation quantization to the algebra of functions on an arbitrary Poisson manifold, I don't see why the resulting algebra should get a bialgebra structure.

Maybe the simplest example is the algebra of functions on $S^2$, which is a symplectic manifold. When you apply deformation quantization to this you should get a 'fuzzy sphere'. I say "a" rather than "the" fuzzy sphere because there's been a huge amount of work on fuzzy spheres, and there are different approaches. I'm not finding anything particular elegant right now, looking at the 384 papers on the arXiv about fuzzy spheres. But I'd be surprised if the fuzzy sphere is a bialgebra because the algebra of functions on the sphere isn't a bialgebra (in any interesting way that I know).

Fernando Yamauti (Jan 30 2025 at 22:11):

John Baez said:

Fernando Yamauti said:

My question was if there are examples of algebras coming from deformation quantisation of commutative ones that are not bialgebras.

Yes, most of them are not bialgebras, at least not in any way I'd understand. We'd expect the deformation quantization to be a bialgebra, or hope for them to be, if the original commutative algebra being deformed was a bialgebra. That usually happens when its an algebra of functions on a topological monoid (e.g. a topological group). But if we apply deformation quantization to the algebra of functions on an arbitrary Poisson manifold, I don't see why the resulting algebra should get a bialgebra structure.

I'm confused. I always thought that for a Poisson manifold, quantisation could be computed as follows: first consider the underlying Poisson Lie algebroid, then, (possibly only formally) integrate to a symplectic (possible higher) groupoid, and, finally, take the convolution C* algebra. If so we get a Hopf algebroid.

John Baez (Jan 30 2025 at 22:15):

You are very sophisticated: for me, I just take the functions on the Poisson manifold and seek a new noncommutative but associative product called $\star$ with the property that

$f \star g = f g + \frac{1}{2} \hbar \{f,g\} + \text{terms of order } \hbar^2$

John Baez (Jan 30 2025 at 22:17):

That's what they talk about in the nLab article on deformation quantization.

John Baez (Jan 30 2025 at 22:21):

I was just saying that I don't expect the resulting algebra to be a bialgebra except in cases where the original Poisson manifold was a Poisson Lie group or more generally a 'Poisson Lie monoid'.

Fernando Yamauti (Jan 30 2025 at 22:36):

John Baez said:

You are very sophisticated: for me, I just take the functions on the Poisson manifold and seek a new noncommutative but associative product called $\star$ with the property that

$f \star g = f g + \frac{1}{2} \hbar \{f,g\} + \text{terms of order } \hbar^2$

Hmm...the nlab seems to suggest that all those things should coincide and everything should be an instance of pushforward quantization, for instance (see also Nuiten's master thesis in 5.2.2). Perhaps from the fact that every Poisson manifold is already the infinitesimal part of a (possibly higher) symplectic Lie groupoid (apparently even an atlas for the underlying stack), we can think like everything goes like the quantization of $C^{\infty} (\mathfrak{g}^*)$. But I'm just spilling things I was told several years ago, so I'm not sure of anything.

John Baez (Jan 30 2025 at 23:35):

I would like to see how the approaches you're talking about play out for a simple case of a symplectic manifold, namely $S^2$ or $\mathbb{R}^2$. What deformation of the algebra of functions on these manifolds do you get in the end?

Fernando Yamauti (Jan 31 2025 at 00:20):

John Baez said:

I would like to see how the approaches you're talking about play out for a simple case of a symplectic manifold, namely $S^2$ or $\mathbb{R}^2$. What deformation of the algebra of functions on these manifolds do you get in the end?

I don't know enough about operator K-theory in order to translate back and forth. But, now, I've just realised that your first example, $n \times n$ matrices, is actually the convolution algebra of the pair groupoid of the discrete $n$-point space... so perhaps being a Hopf algebroid is not enough at all and we really need to require everything to come from a group.

But now I've got curious on how to complete the table on the nlab (if possible at all). If Hopf algebra corresponds to a rigid monoidal categories with a fiber functor, then what should correspond to Hopf algebroids?

The stupid guess would be that given a Hopf algebroid corresponding to the algebra of functions of a groupoid $G$, modules over that should be the same thing as quasicoherent sheaves over the stack $BG$. But that has a monoidal structure, so I'm confused...

Fernando Yamauti (Jan 31 2025 at 02:34):

Fernando Yamauti said:

The stupid guess would be that given a Hopf algebroid corresponding to the algebra of functions of a groupoid $G$, modules over that should be the same thing as quasicoherent sheaves over the stack $BG$. But that has a monoidal structure, so I'm confused...

I was being silly. Quasicoherent sheaves are comodules because representations of an algebraic group are usually comodules and not modules (unless the group is discrete). I keep confusing that.

Also the groupoid convolution algebra is naturally a cocommutative coalgebra over the algebra of functions of the space of objects. In particular $n \times n$-matrices is a coalgebra over $\mathbf{C}^n$ (is that a $\mathbf{C}$-bialgebra though?). It seems I was confusing two different algebras of functions: one is a Hopf algebroid, the other is an algebra that is also a coalgebra (bialgebra? Hopf?).

What's confusing is that a bibundle over groupoids (or a map between the respective stacks) should give a map between comodule categories of the respective Hopf algebroids and, yet, when we see from the other point of view, it gives us a map between module categories of the respective (convolution) algebras.

That completely breaks my intuition for operator K-theory. I always thought Hilbert modules over the convolution algebra of some groupoid $G$ could be seen as an incarnation of the cat of vector bundles (or maybe quasi-coherent sheaves) over the stack $BG$. I'm just getting more confused... :melting_face:

John Baez (Jan 31 2025 at 06:10):

Here's a good early paper on the deformation quantization of the algebra of functions on the 2-sphere:

• F. A. Berezin, General concept of quantization, Comm. Math. Phys. 40 (1975), 153--174.

John Baez (Jan 31 2025 at 06:11):

It gets the result I remembered: you get a well-behaved (not merely formal) quantization only when $\hbar = 1/n$, and then you get the algebra of $n \times n$ matrices!

John Baez (Jan 31 2025 at 06:15):

I don't think a matrix algebra is a bialgebra, at least not in any useful way.

You can make this algebra into a coalgebra. If you write $\mathbb{C}^n = V$ then you can think of the matrix algebra as $V \otimes V^\ast$, and this gets a coassociative but noncocommutative comultiplication

$V \otimes V^\ast \cong V \otimes \mathbb{C} \otimes V^\ast \to V \otimes V^\ast \otimes V \otimes V^\ast$

coming from the counit

$\mathbb{C} \to V \otimes V^\ast \cong V^\ast \otimes V$

of the adjunction (duality) between $V$ and $V^\ast$, together with the symmetry $V \otimes V^\ast \cong V^\ast \otimes V$.

John Baez (Jan 31 2025 at 06:16):

However, this comultiplication does not combine with the usual multiplication of matrices to give a bialgebra: instead it gives a [[Frobenius algebra]].

John Baez (Jan 31 2025 at 06:27):

However, I realize I'm talking at cross-purposes to you in this conversation, since you're talking about a noncommutative geometry analogue of vector bundles. Vector bundles give modules of the commutative algebra of functions on a space, and there's a perfectly nice tensor product of vector bundles - or of modules of any commutative algebra - but this does not arise from a bialgebra structure on that commutative algebra. It's simply this:

Given a commutative algebra $A$ over a field $k$ and modules $M$ and $N$, these modules naturally become $(A,A)$-bimodules where the left and right actions agree, and $M \otimes_A N$, the tensor product of bimodules, becomes another such bimodule.

It looks like to generalize this to noncommutative algebras we should be working with bimodules.

John Baez (Jan 31 2025 at 06:37):

It also feels like I'm talking about ideas that are roughly 20-50 years older than you... but I don't feel too bad about it, because the old and new ideas should connect, and it's fun to understand simple concrete examples like deformation quantization of the 2-sphere from many viewpoints.

By the way, I think people have studied "vector bundles on the fuzzy sphere" (the deformation quantization of the 2-sphere).

Fernando Yamauti (Jan 31 2025 at 15:12):

John Baez said:

It also feels like I'm talking about ideas that are roughly 20-50 years older than you... but I don't feel too bad about it, because the old and new ideas should connect, and it's fun to understand simple concrete examples like deformation quantization of the 2-sphere from many viewpoints.

By the way, I think people have studied "vector bundles on the fuzzy sphere" (the deformation quantization of the 2-sphere).

Yes. I agree. Everyone should do some computations with the simple examples in order to understand the general concepts, but it's tempting to get mesmerised by the abstract non-sense.

Thanks for the paper. I will check that with more care later. Your clarification that we only get a Frobenius algebra seems to suggest monoidal structures might not be present at all.

On a side note, btw, there's of course always a monoidal structure on ${}_{A} \text{Mod}$ given by the Day convolution, which I think is $A \otimes A \otimes (M \otimes N)$ by seeing $M \otimes N$ as a left $A \otimes A$-module. But that\s not the correct one unless everything is cartesian since it doesn't coincide with the desired one when $A$ is commutative.

Now, regarding considering bimodules, from one side that's seems not harmful since we still get the usual Morita category. So perhaps the correct category to assign to a convolution algebra is a cat of certain Hilbert bimodules.

On the other side, from the point of view of bivariant K-theories (like KK and E) that seems incorrect since $K (\text{stack})$ is supposed to coincide with $K (\text{convolution alg of that stack})$ and, then, if that equivalence lifts to the cats of modules, we get something like $\text{vector bundles over stack} \cong\text{right Kasparov modules over conv. alg.}$, so we get back to one sided modules. The difficult is the usual formulation of operator K-theory uses projections as the substitute of vector bundles and Kasparov description of K-theory uses right modules, so it's not clear at all if bimodules are implicitly somewhere.

The point is that I would like a cat of modules for algebras that ends up being compatible with algebraic K-theory (of commutative things) and also operator K-theory. A good starting point would be knowing whether the usual description of the K-theory of a differentiable stack using convolution algebras can be reinterpreted as the algebraic K-theory of some cat of vector bundles (actually, "perfect complexes" should be everywhere I'd written "vector bundles", but let's ignore that for now).

John Baez (Jan 31 2025 at 17:07):

This stuff is really interesting! The idea of trying to understand not just the K-theory of a differentiable stack but actually vector bundles (or perfect complexes) on a differentiable stack as (certain) modules of a convolution algebra makes a lot of intuitive sense - but also raises lots of interesting questions.

What's a "right Kasparov module"?

John Baez (Jan 31 2025 at 17:12):

I can't resist a digression: James Dolan has developed a new definition of algebraic stacks over a field k where what they actually are is symmetric monoidal locally presentable k-linear categories. These categories are what we'd usually call the categories of quasicoherent sheaves on the algebraic stacks. But James is saying we might as well decree those categories are the algebraic stacks. The relevant point here is that in this approach, we automatically know how to take tensor products of quasicoherent sheaves on an algebraic stack - it's built into the definition.

Fernando Yamauti (Feb 01 2025 at 02:46):

John Baez said:

I can't resist a digression: James Dolan has developed a new definition of algebraic stacks over a field k where what they actually are is symmetric monoidal locally presentable k-linear categories. These categories are what we'd usually call the categories of quasicoherent sheaves on the algebraic stacks. But James is saying we might as well decree those categories are the algebraic stacks. The relevant point here is that in this approach, we automatically know how to take tensor products of quasicoherent sheaves on an algebraic stack - it's built into the definition.

Yes, that does work well because of Lurie's version of the Tannakian reconstruction theorem (Part III of the SAG is the most general version). That essentially says that the copresentable of quasicoherent sheaves over the stack is the functor of points of the stack. For schemes, we can even do without the monoidal structure if one just wants to recover a scheme as an individual locally ringed space (but, then, the reconstruction is not functorial anymore).

That's probably the main idea in algebraic noncommutative geometry. But it's not clear at all how to make C* noncommutative geometry compatible with that. I mean there several analogies between algebraic bivariant K-theory, and operator KK-theory and E-theory. However, the first one takes those module categories as input (actually, some subcat like dualisable or compact objects), while the other two just take C* algebras.

In the case mentioned before of differentiable stacks, everything is compatible to the extent that $K_G (X) \cong K (\text{Conv.Alg} (X // G))$ for sufficiently nice spaces and groups, and the former can be described using $G$-equivariant bundles.

Doing things algebraically doesn't help much, because I don't have any canonical notion of measure on a say affine groupoid scheme unless everything is discrete. But, still, supposing that I could define a convolution algebra, that would mean some cat of comodules over a Hopf algebroid would correspond to some cat of modules over a Frobenius algebra (supposing that the other cases are also Frobenius, not sure though). Perhaps such a correspondence was already proven, I don't know...

Fernando Yamauti (Feb 01 2025 at 02:51):

John Baez said:

What's a "right Kasparov module"?

That's probably very idiosyncratic notation. People would usually say it's a $(\mathbf{C}, A)$-Kasparov bimodule or perhaps an $A$-parametrised Fredholm operator.