Baez and Rovelli have each tossed out two freshly minted mathematical ideas.(adsbygoogle = window.adsbygoogle || []).push({});

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)

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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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# Triangulating Baez and Rovelli

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