A project which began in May 2006 to categorify the Erlangen Program.
Normally we think of maps from $B G$ (a group treated as a category) to $Set$ which are transitive actions. Then the set of cosets of $H$, the subgroup which fixes an element, is isomorphic to the carrier set of the action.
Atomic $G$-invariant relations between $X$ and $Y$, transitive $G$-sets, are the connected components of the action groupoid of $G$ acting on $X \times Y$.
We could make $G$ 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 $G$ there is the category of faithful functors to $G$. Each object corresponds to a closed subgroup of $G$ (vertex group of action groupoid), and to the $G$-set $G/H$.
If there is a morphism $G \to H$, what does this do to the associated Klein geometries? When we think about representations, $[G, Vect]$, there are induced representations. What difference does $Set$ 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 $P(A)= S^A$, where $S$ 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 $A$ having one object and all morphisms invertible, one sees that this hyperdoctrine includes the theory of permutation groups; in fact, such $A$ are groups and a “property” of $A$ is nothing but a representation of $A$ 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)
Given any group map $f:G \to H$ (and fix some base field $F$) you can define induction and restriction functors $f_*: Rep(G) \to Rep(H)$ and $f^*: Rep(H) \to Rep(G)$. Then induction is, erm, left adjoint to restriction. There is also a coinduction functor $f_!:Rep(G) \to Rep(H)$ 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 $G \to H$. 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 $Set$? What for infinite $G$? If $H$ is a closed subgroup of a Lie group $G$, then $H$ is a Lie subgroup, and $G/H$ is a smooth space. So representation here are to $Man$? If space $X$ with transitive $G$-action is not compact, then $G/H \to X$ is continuous bijection, but inverse need not be continuous. It is if $X$ is compact.
What difference does it make that we make $G$ 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?