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Stream: deprecated: mathematics

Topic: Rel versus Hilb

John Baez (Mar 04 2023 at 01:37):

There's a new paper that compares the category of sets and relations to the category of Hilbert spaces and bounded linear operators:

• Andre Kornell, Axioms for the category of sets and relations.

This paper lists a bunch of axioms that these categories both obey, and then give extra axioms that uniquely characterize the category of sets and relations, on the one hand, and the category of (real or complex) Hilbert spaces and bounded linear operators, on the other.

John Baez (Mar 04 2023 at 01:38):

So, we're getting better at seeing how quantum mechanics is and is not like "possibilistic" mechanics where operators are replaced by relations!

Morgan Rogers (he/him) (Mar 04 2023 at 07:07):

I contacted Andre about that paper, since he didn't mention (and apparently didn't know about) allegories, the axiomatization of categories of relations for regular categories. His characterization looks simpler than the definition of power allegory, which is the axiomatization of the category of relations of a topos (of which Rel is, of course, a special case), so I'm curious to see whether this is just because Rel is very special (quite likely) or because that characterization can be simplified somewhat.

David Michael Roberts (Mar 04 2023 at 08:31):

I was also put in mind of @Mike Shulman's [[SEAR]], but that's a first-order axiomatisation, not a category-theoretic one.

David Michael Roberts (Mar 04 2023 at 08:31):

I do like Kornell's paper very much, though!

Fabrizio Genovese (Mar 04 2023 at 13:37):

Lol, I thought "Wait didn't Chris Heunen do something similar in the past?" and then it turns out that they indeed coauthored the paper about the axiomatization of Hilb

John Baez (Mar 04 2023 at 16:03):

Right, this is sort of an extension of those ideas to Rel.

Cole Comfort (Mar 05 2023 at 01:11):

John Baez said:

So, we're getting better at seeing how quantum mechanics is and is not like "possibilistic" mechanics where operators are replaced by relations!

In odd prime dimensions qupit stabilizer quantum mechanics (modulo scalars) is a subcategory of affine relations over $\mathbb{F}_p$. I wonder if some mixture of the axioms for relations and hilb would reproduce this.

Jean-Baptiste Vienney (Mar 05 2023 at 01:16):

Can I ask what is an affine relation? :)

Cole Comfort (Mar 05 2023 at 01:25):

Given a field $k$ an affine relation from $k^n \to k^m$ is an affine subspace of $k^n \oplus k^m$. The composition of affine relations is relational composition because it is a category of relations (and thus embeds into $\sf Rel$ ). Very underrated category imo

Jean-Baptiste Vienney (Mar 05 2023 at 02:03):

Note that a similarity between $Rel$, $Hilb_{\mathbb{R}}$ and $Hilb_{\mathbb{C}}$ is that they are all categories enriched over $\mathbb{Q}^{+}$-modules. In $Rel$, you can define $\frac{n}{p}.R = (n.1_{\mathbb{B}})(p.1_{\mathbb{B}})^{-1}.R = R$ if $\frac{n}{p} \neq 0$ and $0$ if $\frac{n}{p}=0$ where $\mathbb{B}=\{0,1\} \cong Rel(*,*)$ is the semi-ring such that $1+1=1$, whereas the category of affine relations over $\mathbb{F}_{p}$ is not. Thus, I guess that your characterization could be something with more axioms and more complicated ones rather than a mixture of these two characterizations because losing the ability to multiply your morphisms by rational scalars always make things more difficult and divide some concepts into several nonequivalent ones (for instance the equalizer and the coequalizer of the $n!$ permutations $A^{\otimes n} \rightarrow A^{\otimes n}$ have the same underlying object in a symmetric monoidal category enriched over $\mathbb{Q}^{+}$-modules but have two non isomorphic underlying objects in $Vec_{\mathbb{F}_{p}}$).

Jean-Baptiste Vienney (Mar 05 2023 at 02:25):

Maybe it could be easier to see if we can remove some axioms from the characterization of $\{Hilb_{\mathbb{R}},Hilb_{\mathbb{C}}\}$ to obtain a characterization of the three categories $\{Hilb_{\mathbb{R}},Hilb_{\mathbb{C}}, Hilb_{\mathbb{H}}\}$ of Hilbert spaces over a division algebra or the ten categories of super Hilbert spaces over a super division algebra, if I have understood the tenfold way :)

Jean-Baptiste Vienney (Mar 05 2023 at 02:29):

We should first answer this question: which axiom in the characterization of $\{Hilb_{\mathbb{R}},Hilb_{\mathbb{C}}\}$ is not verified by $Hilb_{\mathbb{H}}$?

John Baez (Mar 05 2023 at 18:08):

Cole Comfort said:

Given a field $k$ an affine relation from $k^n \to k^m$ is an affine subspace of $k^n \oplus k^m$. The composition of affine relations is relational composition because it is a category of relations (and thus embeds into $\sf Rel$ ). Very underrated category imo

I guess Jean-Baptiste understood this, but in case anyone out there didn't: an affine subspace of a vector space $V$ is a subset of that's a translate of a linear subspace: that is, a subset of the form

$\{ v + p : v \in L \}$

where $L \subseteq V$ in a linear subspace and $p \in V$ is some point.

John Baez (Mar 05 2023 at 18:09):

Equivalently but more conceptually, it's a subset of $V$ closed under affine linear combinations

$(v, w) \mapsto c v + (1-c) w$

where $c$ is any element of the field $k$.

John Baez (Mar 05 2023 at 18:13):

Jean-Baptiste Vienney said:

We should first answer this question: which axiom in the characterization of $\{Hilb_{\mathbb{R}},Hilb_{\mathbb{C}}\}$ is not verified by $Hilb_{\mathbb{H}}$?

Because $\mathbb{H}$ is not commutative, $Hilb_{\mathbb{H}}$ is not a symmetric monoidal category in the same way $\{Hilb_{\mathbb{R}}$ and $Hilb_{\mathbb{C}}\}$ are: the tensor product of quaternionic Hilbert spaces is not a quaternionic Hilbert space.

John Baez (Mar 05 2023 at 18:19):

You can define an 'affine space' to be a set equipped with operations

$(v, w) \mapsto c v + (1-c) w$

obeying some axioms. Just like vector spaces over a fixed field, affine spaces over a fixed field are algebras of a monad. And this is a [[commutative monad]], meaning that all the operations 'commute' with each other in a certain sense. And this implies that the category of algebras of the monad is a symmetric monoidal category with a 'tensor product' a bit like that of vector spaces.

John Baez (Mar 05 2023 at 18:21):

Nikolai Durov has generalized a huge amount of algebra from modules of commutative rings to algebras of finitary commutative monads - see [[generalized ring]].

John Baez (Mar 05 2023 at 18:21):

Another nice example of this theory is 'convex spaces'.

Keith Elliott Peterson (Mar 16 2023 at 23:09):

Jean-Baptiste Vienney said:

Note that a similarity between $Rel$, $Hilb_{\mathbb{R}}$ and $Hilb_{\mathbb{C}}$ is that they are all categories enriched over $\mathbb{Q}^{+}$-modules. [...]

Is there I paper I can read more on this?

Jean-Baptiste Vienney (Mar 16 2023 at 23:21):

Not yet but we will talk of that in our paper "Graded differential categories and graded differential linear logic" with @JS PL (he/him) and I'll talk of that in a paper on string diagrams for symmetric powers in symmetric monoidal $\mathbb{Q}^{+}$-linear categories (which is a name for symmetric monoidal categories enriched over $\mathbb{Q}^{+}$-modules).

Jean-Baptiste Vienney (Mar 16 2023 at 23:22):

And I'll make posts on Zulip when they are published.