categories: Re: Limits

[jdolan@math.ucr.edu]

|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.