Triangulating Baez and Rovelli

[marcus]

Baez and Rovelli have each tossed out two freshly minted mathematical ideas.
These appear to be new in the sense of not having been studied yet (new theorems available to prove, one assumes)
both ideas seem to have possibilities
we ought to be able to treat the two ideas as "surveyer stakes" and
triangulate from them where some future development might be

Baez idea is *-category
which is a category with paired morphisms
each morphism in the category has a mate
(the morphisms are like socks in the sock drawer ought to be but arent)

if f goes from X to Y, then the mate of f goes from Y to X
it is a symmetric relation----f** = f
it is reflexive on identities----if 1 is the identity morphism on an object X then
it is its own mate----1* = 1
the pairing respects composition---if f goes from X to Y and g goes from Y to Z so that gf is defined going from X to Z then the mate of gf is f*g*----you go back by g* from Z to Y and then back by f* from Y to X

the reason your eyes may be glazed is that it is so obvious---we used to call these things Columbus eggs for some reason, for some reason nobody thought of it till recently but now it seems like a really obvious thing to require of a category

the category of Hilbertspaces with linear operators between them which is the Home Town of quantum mechanics is a star-category
so is the home town of gen rel, Baez notes in his paper

but the category of sets is not a star-category (although it is like a paradigm for category theory, oh well)

---------------
Rovelli's idea, which plays a key role in his new book QG
and also is the subject of the recent Fairbairn Rovelli paper
is extending a certain gauge group
to be the extended diffeomorphisms which are
smooth except on a finite set

and this boils the quantum states of space (the pure states of the geometry of the universe or whatever) down to a countable list of colored knots

if we are going to triangulate from these two ideas we should
think if there is a likely category in which the objects are colored knots

and how do you get from one knot to another?

by cobordisms?
by "moves"?
by 2-knots?
by spinfoams...well?

so what are the morphisms that get you from one knot to another
and doubtless they are paired so you get a star-category a la Baez

 

[matt grime]

try looking up the definition of a contravariant functor (that is fully faithful)

 

[marcus]

try looking up the definition of a contravariant functor (that is fully faithful)

Hi Matt, I know the defn of a contravariant functor.

Probably also a bunch of other people who visit here at PF also know the term
(arivero had a category theory thread here a while ago)

You may be making an interesting and valid point but I do not see clearly what it is. Maybe you could elaborate.

Baez has defined a term "star-category" which as far as I know is a type of category (it is not a functor---a functor is a function between categories) and as far as I know it is a type of object which as such has not been studied a lot.

A *-category is clearly not the same thing as a "contravariant functor!
But perhaps you would like to rephrase Baez definition of *-category in
functorial language?

Why not spell it out, if you want. It would bring along more people with you.

 

[matt grime]

The star operation is a contravariant functor.

 

[marcus]

Yes!
and of a very special type.

since this is possibly interesting I have put some basic definitions
over in the math forum
in a thread which meteor started

https://www.physicsforums.com/showthread.php?p=187436#post187436

the question now is, what is so special about the * operator
it is not just any old contravariant functor.

 

[matt grime]

because it takes Hilb to Hilb and not Hilb to Hilb^{op}?

 

[marcus]

because it takes Hilb to Hilb and not Hilb to Hilb^{op}?

You are free here---doing constructive mathematics. Baez gives us the definition of a certain kind of category, he calls it "star-category" and in defining it he uses a pairing relation on the morphisms in that category.
this pairing is not, on the face of it, a functor
(functors take categories to categories)
but perhaps it is a functor of a certain restricted kind
(one with no effect on objects?)
he gives certain axioms that the pairing relation should satisfy that make the category be a star-category

but you may be able to use functorial language to paraphrase Baez
and give another, perhaps more general or suggestive, definition of
star-category.

at least somebody should try to do this, to see if it is a good idea to do it.

 

[matt grime]

do you want to do this here or in the hijacked thread in maths?

the star is a contravariant idempotent functor from Hilb to Hilb that is the identity on objects.

 

[marcus]

do you want to do this here or in the hijacked thread in maths?

the star is a contravariant idempotent functor from Hilb to Hilb that is the identity on objects.

I think the thread in maths would be the right place. Meteor called it "Category Theory" so it would be a good place for discussion of
things like this. Unless Meteor, who started it, objects in which case I shall start a "Category Theory 2" thread. Let's go there and check your definition.

https://www.physicsforums.com/showthread.php?p=187506#post187506

 

[matt grime]

I don't believe it is the right thread as your post does not address any of the issues raised in the previous post by others. I think that implies you ought to start a new thread if you've got new topics to raise, but my answer above contains all you need to know at the moment. Moreover the title does not indicate the type of subject inside the thread