David Corfield
Klein 2-Geometry

A project which began in May 2006 to categorify the Erlangen Program.

Notes

  • Normally we think of maps from BGB G (a group treated as a category) to SetSet which are transitive actions. Then the set of cosets of HH, the subgroup which fixes an element, is isomorphic to the carrier set of the action.

  • Atomic GG-invariant relations between XX and YY, transitive GG-sets, are the connected components of the action groupoid of GG acting on X×YX \times Y.

  • We could make GG any groupoid, and look at faithful functors to it. E.g., example in Weinstein’s paper.

  • Klein geometry as arising from the category of Lie groups. To each group GG there is the category of faithful functors to GG. Each object corresponds to a closed subgroup of GG (vertex group of action groupoid), and to the GG-set G/HG/H.

  • If there is a morphism GHG \to H, what does this do to the associated Klein geometries? When we think about representations, [G,Vect][G, Vect], there are induced representations. What difference does SetSet make?

As a final example of a hyperdoctrine, we mention the one in which types are finite categories and terms arbitrary functors between them, while P(A)=S AP(A)= S^A, where SS is the category of finite sets and mappings, with substitution as the special Godement multiplication. Quantification must then consist of generalized limits and colimits…By focusing on those AA having one object and all morphisms invertible, one sees that this hyperdoctrine includes the theory of permutation groups; in fact, such AA are groups and a “property” of AA is nothing but a representation of AA by permutations. Quantification yields “induced representations” and implication gives a kind of “intertwining representation”. Deductions are of course equivariant maps. (Adjointness in Foundations, p. 14)

  • Also note Simon Willerton’s comments for finite groups

Given any group map f:GHf:G \to H (and fix some base field FF) you can define induction and restriction functors f *:Rep(G)Rep(H)f_*: Rep(G) \to Rep(H) and f *:Rep(H)Rep(G)f^*: Rep(H) \to Rep(G). Then induction is, erm, left adjoint to restriction. There is also a coinduction functor f !:Rep(G)Rep(H)f_!:Rep(G) \to Rep(H) which is right adjoint to restriction.

…you can write down the formula for the candidate natural transformation between induction and coinduction and see that involves the reciprocal of the order of the kernel of the map GHG \to H. This means that you have “Frobenius Reciprocity” provided that the order of the kernel is invertible in the field over which you’re taking representations. In particular you get induction as a left and right adjoint if the map is an injection or if the base field is the complex numbers.

  • What can be said here for SetSet? What for infinite GG? If HH is a closed subgroup of a Lie group GG, then HH is a Lie subgroup, and G/HG/H is a smooth space. So representation here are to ManMan? If space XX with transitive GG-action is not compact, then G/HXG/H \to X is continuous bijection, but inverse need not be continuous. It is if XX is compact.

  • What difference does it make that we make GG act on a topological/geometric space? Recall Weinstein about how what makes groupoids sufficiently different from groups is interaction with topology.

  • In HDA VI, the categorification of a manifold is given as a Lie groupoid. So the coset space of two Lie 2-groups should be one. What are the symmetries of a Lie groupoid?

Revised on April 28, 2009 14:01:35 by David Corfield (129.12.18.22)