Subject: categories: Re: Limits

From: [email-masked]
Date: Thu, 3 May 2001 16:38:18 -0700 (PDT)

|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?

i may as well state the obvious (not necesarily right) answer to this:
no, not quite.  rather, what seems to be going on is that the
phenomenon of adjoint linear operators is, in yetter's terminology, a
sort of decategorification of the phenomenon of adjoint functors.
decategorification is generally a somewhat destructive process,
destroying the morphisms between objects, and since the morphisms are
so intrinsic to the definition of adjoint functor it seems too much to
hope for that the decategorified phenomenon of adjoint linear
operators could actually qualify as a special case of adjoint
functors.  there are suggestive indications, though, that all of the
really interesting special cases of adjoint linear operators in
physics, for example, are decategorifications of interesting pairs of
adjoint functors.  (for example so-called "creation and annihilation
operators on fock space" have categorified analogs that live on a
categorified analog of fock space whose objects/vectors are something
like joyal's "species of structure".)

so roughly: the general phenomenon of adjoint linear operators is
technically probably not quite a genuine special case of adjoint
functors.  the actual interesting special cases of adjoint linear
operators, however, are often seen to be mere shadows of more
interesting cases of adjoint functors.

---

From: Dusko Pavlovic <[email-masked]>
Date: Wed, 09 May 2001 19:18:32 -0700

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.