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Stream: theory: category theory

Topic: inner products work a bit like homs

Antoine Van Muylder (Aug 11 2021 at 19:13):

Hello!
Let us work in the category of finite dimensional Hilbert spaces $\mathrm{FHilb}$. By definition every such space has an inner product map $x y \mapsto \langle x | y \rangle$ wich is sesquilinear among other things. This means that it has type
$\qquad\bar{H} \otimes H \to \mathbb{C}$
where $\bar{H}$ is the complex conjugate vector space.

Now in what extent is this situation comparable to categories and functors? In the later case we have
$\qquad hom: \mathcal{C}^{op}\times \mathcal{C} \to \mathrm{Set}$.
I am seeking for a formal correspondence preferably.

I can already say that an equivalent of the Yoneda lemma seems to hold. To state Yoneda in $\mathrm{FHilb}$ we need some amount of self enrichment/closedness for $\mathrm{FHilb}$. We somehow need to turn $\mathrm{FHilb}(H,K)$ into a Hilbert space. In particular we need an inner product. Here is a proposition ($f,g\in\mathrm{FHilb}(H,K))$
$\qquad \langle f | g \rangle := \frac{1}{C}\int_{x\in S(H)} \langle fx | gx \rangle_K$
where $S(H)$ is the hypersphere included in $H$, $C$ is a smartly chosen constant, and we integrate with Lebesgue measure. Now the wannabe Yoneda embedding has type $H \to \mathrm{FHilb}(\bar{H},\mathbb{C})$ and the wannabe Yoneda lemma looks like this
$\qquad\langle \langle \cdot | y \rangle | s \rangle = sy$
for $s\in \mathrm{FHilb}(\bar{H},\mathbb{C})$ a wannabe presehave.

The computation goes
$\qquad\langle\langle\cdot | y \rangle | s \rangle =$
$\qquad\quad\frac{1}{C}\int_{x\in S(H)}\langle \langle x | y \rangle | sx \rangle_{\mathbb{C}} =$
$\qquad\quad \frac{1}{C}\int_{x\in S(H)} \langle y | x \rangle sx =$
$\qquad\quad \frac{1}{C}\int_{x\in S(H)} s (\langle x | y \rangle x) =$
$\qquad\quad \frac{1}{C}s( \quad \int_{x\in S(H)} \langle x | y \rangle x \quad) =$
because s is antilinear and thanks to antisymmetry of inner product.
And finally by some geometric reasoning it is possible to show that the integral above must be proportional to $y$, with proportionality constant $C$, say.

Is this a mere coincidence or is it derivable from a more general correspondance? thanks:)

John Baez (Aug 11 2021 at 19:17):

It's not a coincidence. I think it's very important. But I think it's still somewhat mysterious. I don't think anyone has figured out a general framework in which the common features of FHilb and Cat both emerge as special cases. One thing I did was invent "2-Hilbert spaces", which are like Hilbert spaces, but categories instead of sets:

• John Baez, Higher-dimensional algebra II: 2-Hilbert spaces, Adv. Math. 127 (1997), 125-189.

The hom of a 2-Hilbert space plays the role of its inner product. There are lots of nice examples of 2-Hilbert spaces in physics and mathematics, like categories of unitary representations of groups.

John Baez (Aug 11 2021 at 19:19):

Bruce Bartlett has done some nice work on 2-Hilbert spaces, improving my definition a bit.

Antoine Van Muylder (Aug 11 2021 at 19:26):

Thanks! exactly what I am looking for it seems

Naso (Aug 12 2021 at 05:21):

there is this table in the representation theory book from Etingof et al: unknown.png

Mike Shulman (Aug 12 2021 at 13:02):

The closest thing I know of to a general framework that includes both FHilb and Cat is the Chu construction.

John Baez (Aug 12 2021 at 15:05):

Ah! Chu!

John Baez (Aug 12 2021 at 15:06):

I'm finally interested in the Chu construction. Nobody ever said that sentence to me before. Oh - but why do you say "the closest thing"? Does it not quite capture those two examples?

Mike Shulman (Aug 12 2021 at 18:26):

Well, it depends on exactly what aspects of those two examples you want to capture, and what you mean by "capture". As I sketched in this MO answer, the main point is that the two Chu-constructions Chu(Vect,k) and Chu(Cat,Set) contain as full subcategories the category of Hilbert spaces and adjoint pairs of transformations, and the category of categories and adjoint pars of functors, respectively. This doesn't directly address the Yoneda-type question of the OP, though.

dusko (Aug 21 2021 at 11:46):

Mike Shulman said:

Well, it depends on exactly what aspects of those two examples you want to capture, and what you mean by "capture". As I sketched in this MO answer, the main point is that the two Chu-constructions Chu(Vect,k) and Chu(Cat,Set) contain as full subcategories the category of Hilbert spaces and adjoint pairs of transformations, and the category of categories and adjoint pars of functors, respectively. This doesn't directly address the Yoneda-type question of the OP, though.

as evidence of [PS nietzsche's eternally returning idea of] eternal return, i copy a conversation (started by jdolan) from the categories mailing list from 5/9/2001:

Peter Freyd wrote:

A good question. I have no answer, only a similar (and ancient)
question: is there a setting in which adjoint operators on Hilbert
spaces can be seen to be examples of adjoint functors between
categories?

probably not, but the they seem to be instances of the same general
structure. (it is simple, pretty old, and i am sure many have noticed it,
but since no one mentioned it, here it goes.)

let U : Cat ---> CAT be the embedding of small categories in all,
and
let Y: Cat^op ---> CAT map each small category A to the presheaves
Psh(A).

now look at the (pseudo)comma category U/Y. each category A is
represented in it by the yoneda embedding A-->Psh(A). the morphisms
between A-->Psh(A) and B --> Psh(B) are exactly the pairs of adjoint
functors between A and B.

on the other hand, let I: Vec---> Vec be the identity functor,
and let * : Vec^op ---> Vec take a vector space V to its dual V*.

look at the comma category I/*. each hilbert space V is represented in it
by the obvious linear map V-->V. the morphisms between V-->V and
W-->W* are exactly the adjoint pairs of operators between V and W.

playing around a bit, these two comma categories can be thought of as
Chu(CAT,Set) and Chu(Vec,R) respectively. so both sorts of adjunctions
are the instances of the chu morphisms. they are the chu morphisms on the
"representation" objects, in the form X --> R^X, where R is the dualizing
object.

-- dusko

PS infact, one could start from Chu(SET,Set), and define categories as
the profunctors A-->Set^A which form a monoid with respect to the
profunctor composition. you'd get only the object part of the adjoint
functors as the morphisms of this chu, but the arrow part follows from
the adjunction (i think).

now can we characterize hilbert spaces in a similar way within
Chu(Vec,R)? this seems to be a completely different kind of question. in
particular, it is possible to define "profunctors" with respect to R or
C, like we did with respect to Set, and we can compose them, but hilbert
spaces do not seem to be monoids with respect to this composition, at
least the way it occurs to me. if there is no such composition that they
are, then hilbert spaces are like R-enriched graphs, rather than
categories.

dusko (Aug 21 2021 at 11:51):

oh, but in the meantime i know that adjoint operators over real hilbert spaces with bases are the kan extensions of their matrix form as an enriched profunctor over the category of reals. so my first "probably not" to freyd's question was wrong.

Mike Shulman (Aug 21 2021 at 13:56):

Thanks for finding that. I certainly didn't mean to claim any originality for the observation!

dusko (Aug 22 2021 at 15:57):

Mike Shulman said:

Thanks for finding that. I certainly didn't mean to claim any originality for the observation!

me neither :) it is exactly what you say: an observation. there is a planet and anyone can observe it if they look.