In his 1982 book Statistical Decision Rules and Optimal Inference, N. N. Cencov studies statistical models (parametrized families of probability distributions) from an unconventional category-theoretic and geometric perspective. While reading it, I came across an interesting passage where Cencov proposes a variation on Klein's Erlangen program, according to which groups are replaced by groupoids:
One of the central problems here is to determine invariants of figures—real-valued funtions which assume equal values on figures—and to construct a complete system of invariants. It is easy to give this problem a rigorous formulation in terms of an appropriate category of figures. The objects of this category are the figures; the morphisms for any pair of figures are all motions mapping the first into the second... Thus invariants of figures are invariants of the category of figures, if (as is indeed natural) one means by an invariant of the category any characteristic of an object which assumes equal values on equivalent [isomorphic] objects. (p. 53)
Cencov attributes this viewpoint to himself. In the book, he studies the differential geometry of statistical models.
I'm curious whether this viewpoint has been applied to more conventional geometric problems. I understand that Ronnie Brown and others have pushed groupoids in algebraic topology, but I haven't seen them advocated in Klein geometry.
