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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
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
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