Quantization and Cohomology (Week 21)
Posted by John Baez
This week in our course on Quantization and Cohomology we used Chen’s ‘smooth space’ technology to implement a new approach to Lagrangian mechanics, based on a smooth category equipped with an ‘action’ functor:
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Week 21 (Apr. 17) - Any quotient of a smooth space becomes a
smooth space. The category of smooth spaces has pushouts.
The category of smooth spaces is cartesian closed. The path groupoid of a smooth space . The path groupoid is a smooth category. Smooth functors.
Theorem: a smooth functor is the same as a 1-form
on X.
Supplementary reading:
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John Baez and Urs Schreiber,
Higher gauge theory II: 2-connections, draft version.
Section 6.1: proof that for any Lie group , smooth functors are the same as -valued 1-forms on
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John Baez and Urs Schreiber,
Higher gauge theory II: 2-connections, draft version.
Last week’s notes are here; next week’s notes are here.
In a bit more detail: we saw that any smooth space has a smooth groupoid of paths . The Lagrangian approach to classical mechanics involves a smooth groupoid where the objects are ‘configurations’ of our system and the morphisms are ‘processes’ or ‘paths’. The action should define a functor . So, it’s nice that in the special case when is a path groupoid, such functors turn out to be familiar entities! They’re just 1-forms on .
However, this isn’t quite general enough. What we really want is something that looks locally like a 1-form on , but not globally: a connection on a bundle over ! This, after all, is what people use in geometric quantization — usually in the special case where is a symplectic manifold.
To get this answer, we’ll need to generalize from smooth functors to smooth anafunctors, as defined by Toby Bartels. A smooth anafunctor is something that’s locally isomorphic to a smooth functor!