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Stream: learning: questions

Topic: generalised points and states

Jules Hedges (Jan 24 2022 at 13:09):

nLab says that a [[generalized element]] of an object is any morphism into that object. I feel like it's also reasonable to say that any covariant functor $F : \mathcal C \to \mathbf{Set}$ is a "theory of generalised elements in $\mathcal C$", which generalises the usual definition further by allowing $F$ to not be representable. The idea being that if you have $x \in F(X)$ then you can apply any morphism to get $F(f)(x) \in F(Y)$. I also instinctively feel that this is exactly the situation where the word "copresheaf" is appropriate. Is this a right perspective?

What I actually want to do is for monoidal $\mathcal C$ to say that any lax monoidal functor $F : \mathcal C \to \mathbf{Set}$ is a "theory of generalised states in $\mathcal C$". That is, states $x \in F(X)$ and $y \in F(Y)$ can be composed into a joint state $x \nabla y \in F (X \otimes Y)$. This abstracts what seems to be the useful property of $\mathcal C (I, -)$. Is this a perspective that already exists?

I've taken this seriously up to the point of using the string diagram language for ordinary states for these "generalised states". I didn't prove this is sound, although I strongly suspect it is

Jules Hedges (Jan 24 2022 at 13:12):

We used this perspective quite heavily in https://arxiv.org/abs/2105.06332 and we're continuing building on it. Figure 8 in that paper is an example of a string diagram where generalised states are drawn as though they're actual states

Matteo Capucci (he/him) (Jan 24 2022 at 13:40):

I idly thought about it onece (since you suggested this already sometime ago), and realized drawing diagrams in the opposite of the category of representable copresheaves (i.e. $\mathrm{Copsh}(\mathcal C)^{op}$) does what you want.

Matteo Capucci (he/him) (Jan 24 2022 at 13:40):

The monoidal structure is [[Day convolution]]

Matteo Capucci (he/him) (Jan 24 2022 at 13:42):

Lax monoidal copresheaves are then comonoids (because there's an op) wrt to that monoidal product, which abstracts the fact that `state $\otimes$ state' is still a state

Matteo Capucci (he/him) (Jan 24 2022 at 13:44):

The reason you want an op is that the 'coYoneda embedding' is contravariant, so if you want your diagrams to not be mirrored when you promote them to diagrams of representable copresheaves, you need to op everything

Nathaniel Virgo (Jan 24 2022 at 13:55):

This sounds like it could be related to the [[category of elements]].

The string diagrams you mention sound like they could be related to the string diagrams David Spivak uses for lenses in Generalized Lens Categories via functors $\mathcal{C}^{\rm op}\to\mathsf{Cat}$. Those are for lenses derived from a functor $\mathcal{C}^\text{op}\to \mathbf{Cat}$ and look like costates, but if you only consider discrete categories then I think the Grothendieck construction gives you the category of elements, and if you lose the op you'll get string diagrams that look like states.

(Sorry if that ends up being irrelevant or not making sense, I'm firing it off before going to bed without thinking about it very much.)

I also remember thinking that Day convolution makes these behave nicely with the monoidal product, in some context vaguely similar to this at some point.

Paolo Perrone (Jan 24 2022 at 14:17):

Not quite the same, but we use a similar philosophy in this paper, section 3.2, when we talk about a category of "random elements" (probably "random states" would be a better name).

Martti Karvonen (Jan 24 2022 at 14:29):

A working hypothesis I have is that most/all resource theories arise by taking the category of elements of a lax monoidal functor. This is true for the resource theories constructed in that paper, and it's also true for resource theories of contextuality and of non-locality. As a result, I'm tempted to think of a lax monoidal functor into Set as something that gives for each object $A$ the set of resouces of type $A$. At times the set of resources has more structure (it's convex, or has a metric on it etc) so instead of sets we land somewhere else, but that's the basic idea.

Joe Moeller (Jan 24 2022 at 14:45):

In Network models, I consider lax monoidal functors into Mon (monoids) or Cat as specifying a type of network. This is what made me think about the monoidal Grothendieck construction. Then in Network models from Petri nets with catalysts, we consider the category structure as carrying information about agents and resources. So this sounds sorta related!

Martti Karvonen (Jan 24 2022 at 15:04):

Yeah I use your work on the monoidal Grothendieck construction to conclude for cheap that the resulting category of elements is monoidal (or symmetric monoidal etc) in various situations.

Jules Hedges (Jan 24 2022 at 15:48):

Hm, this seems to be more folkloric than I was expecting

John Baez (Jan 24 2022 at 17:38):

"More folkloric than I expected" meaning "already being done for longer than I expected" or "not being documented as well as I'd hoped"?

Jules Hedges (Jan 24 2022 at 17:55):

The second thing

Jules Hedges (Jan 24 2022 at 17:56):

I'm vaguely optimistic that the view of not-necessarily-representable copresheaves as generalised points might appear already in Grothendieck

John Baez (Jan 24 2022 at 17:57):

It seems like a completely standard idea to think of presheaves on a category as a generalization of representable presheaves, aka objects of the category. This is what the Yoneda embedding is about.

John Baez (Jan 24 2022 at 17:59):

Anyway, Grothendieck wouldn't have talked about copresheaves, rather presheaves. But I don't think the "op" in working with copresheaves is a big deal, except of course one must constantly struggle to get all the "ops" worked out correctly.

Jens Hemelaer (Jan 24 2022 at 18:26):

Jules Hedges said:

I'm vaguely optimistic that the view of not-necessarily-representable copresheaves as generalised points might appear already in Grothendieck

Points of the presheaf topos $\mathbf{PSh}(\mathcal{C})$ are the same as flat functors $\mathcal{C} \to \mathbf{Set}$, and these often include non-representable ones.

In topos theory, arbitrary functors $\mathcal{C} \to \mathbf{Set}$ agree with distributions on $\mathbf{PSh}(\mathcal{C})$ (this idea was introduced by Lawvere). The points of the topos can then be identified with Dirac delta distributions, via the work of Bunge and Funk. So in this sense distributions are literally generalized points. Maybe the words "state" and "distribution" refer to the same kind of intuition.

Nathaniel Virgo (Jan 25 2022 at 02:23):

Here's a closely related question. I think the string diagram picture ought to generalise to profunctors. Instead of being a state, an element of a profunctor $\mathcal{C}^\text{op}\times \mathcal{D}\to \mathbf{Set}$ ends up looking like a morphism that sits "between" categories $\mathcal{C}$ and $\mathcal{D}$, so that you can pre-compose morphisms of $\mathcal{C}$ and post-compose morphisms of $\mathcal{D}$.

image.png

If $\mathcal{C}$ is a monoidal category then there should be some structure we can put on a profunctor $P\colon \mathcal{C}^\text{op}\times \mathcal{C}\to \mathbf{Set}$ that makes such diagrams work nicely with the monoidal product, so that we can tensor elements of $P$ with objects of $\mathcal{C}$ and slide them around like morphisms. What would that structure be? Is it enough to demand that $P$ be a lax monoidal functor $\mathcal{C}^\text{op}\times \mathcal{C}\to \mathbf{Set}$?

Fabrizio Genovese (Jan 25 2022 at 10:47):

Wait isn't this idea of elements in a profunctor image as morphisms sitting between morphisms of C and D exactly captured by profunctor collage?

Jules Hedges (Jan 25 2022 at 13:39):

Yes, it's another instance of the same intuition, in this case I'd call a profunctor a "theory of heteromorphisms"

Matteo Capucci (he/him) (Jan 25 2022 at 15:01):

Jens Hemelaer said:

Jules Hedges said:

I'm vaguely optimistic that the view of not-necessarily-representable copresheaves as generalised points might appear already in Grothendieck

Points of the presheaf topos $\mathbf{PSh}(\mathcal{C})$ are the same as flat functors $\mathcal{C} \to \mathbf{Set}$, and these often include non-representable ones.

Indeed! And flat roughly means 'finite limits preserving', which is similar in spirit to 'lax monoidal' (I might be abusing handwaving here)

Mike Shulman (Jan 25 2022 at 16:40):

If $T$ is the 2-monad whose algebras are finite-limit categories, then a lax $T$-morphism is precisely a finite-limit-preserving functor (and is automatically a strong $T$-morphism). So there's a very precise analogy between finite-limit-preserving functors and lax monoidal functors.

Nathaniel Virgo (Jan 26 2022 at 00:08):

Fabrizio Genovese said:

Wait isn't this idea of elements in a profunctor image as morphisms sitting between morphisms of C and D exactly captured by profunctor collage?

For categories, yes, exactly - I was wondering what the monoidal version would be.