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Stream: learning: questions

Topic: holomorphic functions and monoidal closure

Cole Comfort (Dec 09 2024 at 17:17):

This question is probably very naive, but does anyone know if holomorphic functions can be regarded as morphisms in a monoidal closed category?

The kind of thing I want to be able to ``curry'' a holomorphic function $f(z):\mathbb{C}^n\to \mathbb{C}^m$ into something like a function $g(z, \bar w): \mathbb{C}^n\times\mathbb{C}^m \to \mathbb{C}$ holomorphic in $z$ and antiholomorphic in $w$. But I can't even figure out what the right types need to be.

If this question doesn't make sense to ask, does anyone have a good reference on complex analysis written more from the perspective of a category theorist?

Mike Shulman (Dec 09 2024 at 17:21):

I'm not by any means an expert, but the obvious thing that occurs to me is to use the Hermetian inner product on $\mathbb{C}^m$: $g(z,w) = \langle f(z), w \rangle$.

Mike Shulman (Dec 09 2024 at 17:21):

But that isn't using an actual internal-hom, only the "analogous" inner product.

Cole Comfort (Dec 09 2024 at 17:33):

That is essentially my motivation. I want to build a category of "holomorphic matrices" where the morphisms $n \to m$ are functions $g(z, \bar w) : \mathbb{C}^ n \times \mathbb{C}^m \to \mathbb{C}$ holomorphic in $z$ and antiholomorphic in $w$, but my lack of complex analysis knowledge is making this kind of hard to do.

Mike Shulman (Dec 09 2024 at 17:56):

Oh, so your question is how to make the inner product into an actual monoidal closed structure?

Cole Comfort (Dec 09 2024 at 18:06):

Yeah. But I assume something breaks if you want to get an actual monoidal closed structure.

David Michael Roberts (Dec 09 2024 at 20:47):

The space of homomorphic functions is infinite-dimensional, though...

John Baez (Dec 09 2024 at 21:36):

Cole Comfort said:

That is essentially my motivation. I want to build a category of "holomorphic matrices" where the morphisms $n \to m$ are functions $g(z, \bar w) : \mathbb{C}^ n \times \mathbb{C}^m \to \mathbb{C}$ holomorphic in $z$ and antiholomorphic in $w$, but my lack of complex analysis knowledge is making this kind of hard to do.

There should be a rather simple framework where you focus on mixed multi-linear and multi-antilinear maps between complex vector spaces, maybe using a 2-typed prop or something. But you seem eager to explore a bigger world, where you allow non-multilinear functions that are holomorphic in some variables and antiholomorphic variables. That might also give a 2-typed prop. But if you feel limited in complex analysis skills, maybe do the multilinear case first? It's already good for something.

John Baez (Dec 09 2024 at 21:41):

By the way, any complex vector space $V$ has a 'conjugate' $\overline{V}$ where multiplication by $i$ is defined to be the old multiplication by $-i$, if you know what I mean. Unless I'm losing my remaining marbles, this 'conjugation' thing is an automorphism of the symmetric monoidal category $\mathsf{FinVect}_{\mathbb{C}}$ with its usual $\otimes$. Then an antilinear map from $V$ to $W$ is the same as a linear map from $V$ to $\overline{W}$. Similarly, an antiholomorphic from $V$ to $W$ is a holomorphic map from $V$ to $\overline{W}$.

Thus, instead of working with a 2-typed prop as I was just suggesting, it may be easier to admit this 'conjugation' functor as part of the framework.

John Baez (Dec 09 2024 at 21:45):

Also by the way, we can generalize this 'conjugation' functor

$\overline{\phantom{a}} \colon \mathsf{FinVect}_{\mathbb{C}} \to \mathsf{FinVect}_{\mathbb{C}}$

to other fields. The group of all automorphisms of a field $k$ acts as automorphisms of the category $\mathsf{FinVect}_k$. And this sort of idea lurks behind the theory of 'Galois descent'. So if you want to go algebraic, you could work with linear (or polynomial!) maps between finite-dimensional vector spaces over any field $k$, and then think about how $\mathrm{Aut}(k)$ acts on whatever sort of category of those you want to study.

Jean-Baptiste Vienney (Dec 09 2024 at 22:00):

That's probably not what you want but in a coordinate free approach it would be simpler to obtain the cartesian closed structure. There is a definition of analytic map between Banach spaces in this paper by Grothendieck: La théorie de Fredholm. You could maybe replace "Banach space" by "finite dimensional $\mathbb{C}$-vector space" and add your antiholomorphic constraint in the story.

Jean-Baptiste Vienney (Dec 09 2024 at 22:01):

Hmm no sorry, the map spaces will not be finite dimensional.

Cole Comfort (Dec 09 2024 at 22:13):

I realize that a similar category already exists in the literature for smooth functions and Euclidean space in a paper by Abramsky et al. which I have been trying to read for a while. They construct a monoidal catgory $\sf DRel$ called the category of tame distributions. Instead of being monoidal closed, $\sf DRel$ has a nuclear ideal of "regular distributions." The structure of a nuclear ideal is a profunctorial version of rigidity.

I suppose I just need to see if I can carefuly replace "smooth" with "(anti)holomorphic" and see if everything works out, because in their paper they develop the general theory of nuclear ideals for monoidal categories with conjugation functors.

I care about holomorphic functions only because of the connection to the holomorphic Fock space. It feels to me like the holomorphic Fock space is more elegant than the symmetric algebra for what I need it for, so I should just read a complex analysis textbook.

Jean-Baptiste Vienney (Dec 09 2024 at 22:44):

I’m wondering whether the Hilbert-Schmidt maps they talk about aren’t already analytic for some reasonable definition of analytic. They don’t deal just with “smooth maps”.

Jean-Baptiste Vienney (Dec 09 2024 at 22:47):

Sorry I’m writing complete BS

Jean-Baptiste Vienney (Dec 09 2024 at 22:48):

These maps are linear maps so yes the are probably “analytic” for any reasonable definition of the word in this context.

Jean-Baptiste Vienney (Dec 09 2024 at 22:51):

This paper seems to talk about analytic maps (they have a definition of “Fréchet entire map”), Hilbert-Schmidt maps, Fock spaces and their relations: this

Cole Comfort (Dec 09 2024 at 23:08):

Jean-Baptiste Vienney said:

I’m wondering whether the Hilbert-Schmidt maps they talk about aren’t already analytic for some reasonable definition of analytic. They don’t deal just with “smooth maps”.

The nuclear ideal of Hilbert-Schmidt maps in the category of Hilbert spaces is closely related, but I think what I want to do is to generalize the aforementioned nuclear ideal structure of the category of regular distributions by adding complex structure to the mix.

A morphism in $\sf DRel$ is a particularly well-behaved smooth function (ie. a tame distribution) which induces a continuous linear map from distributions on the domain to distributions on codomain, as well as a continuous linear map in the other direction. But I don't think that the topological vector space of distributions on an open set is a Hilbert space.

Cole Comfort (Dec 09 2024 at 23:14):

Ideally, I would hope that there is a nuclear functor from the category of bounded linear maps between holomorphic fock spaces to a complex version of $\sf DRel$, but that is just wishful thinking.

Jean-Baptiste Vienney (Dec 09 2024 at 23:55):

Maybe you should look into the paper “Finiteness spaces”. More specifically the coKleisli category. It seems that what you want is close to a model of the differential lambda calculus, which has been invented later than the paper you linked.

Cole Comfort (Dec 09 2024 at 23:58):

Jean-Baptiste Vienney said:

But what are you trying to achieve exactly?

We are trying to extend the Gaussian ZX-calculus to a more expressive setting to accomodate for things like number states. So I am searching for a nice semanics which is inspired by the holomorphic fock space. For the semantics for the Gaussian ZX-calculus we don't use the category of Hilbert spaces because of its lack of nice closure properties, and instead use the category of positive definite affine Lagrangian relations between finite dimensional complex vector spaces which is dagger compact closed.

Cole Comfort (Dec 10 2024 at 00:02):

Jean-Baptiste Vienney said:

Maybe you should look into the paper “Finiteness spaces”. More specifically the coKleisli category. It seems that what you want is close to a model of the differential lambda calculus, which has been invented later than the paper you linked.

That is another good alternative semantics to Hilbert spaces, where you are *-autonomous instead of having the structure of a nuclear ideal.

David Michael Roberts (Dec 10 2024 at 01:17):

Cole Comfort said:

If this question doesn't make sense to ask, does anyone have a good reference on complex analysis written more from the perspective of a category theorist?

At the very least, I should recommend the work of my colleague Finnur Lárusson, who works in complex analysis, but is comfortable with category theory. It's not from the CT pov, but might lead somewhere.

David Michael Roberts (Dec 10 2024 at 01:18):

Consider the case n=m=1. Given a holomorphic function like sin(z) or exp(z), how would you write it as a two-variable function?