To my budding understanding, it looks like there’s a growing “categorical viewpoint” on differential geometry, using groupoids and especially Lie groupoids to reimagine a lot of ideas. Here’s an informal thread on what I know. If any experts read this please chime in.

The center of this that I’ve seen is the work of Crainic and Fernandez on the “integrability problem.” Some instances of this problem: integrating a Lie algebra, integrating a distribution or vector field, integrating an almost complex structure. I’m sure there are more

All of these are unified by considering the problem of integrating a lie algebroid (which seems to be some kind of vector bundle equipped with a Lie algebra structure and a homomorphism to the tangent bundle) to a lie groupoid (a small category in which every morphism...

Is invertible and the sets of objects and morphisms are both manifolds). How this integration works I don’t know. But it seems like a very unified perspective that has some interesting results.

But now I’m reading about gerbes and differentiable stacks - two things which I barely understanding but both seem to use groupoids in a critical way, stacks to describe quotients/moduli spaces and gerbes to describe...something related to obstruction and also bundles.

I’m taking a class next semester which looks to be a differential geometry class, but advertises that it’s going to focus on the “functor of points” perspective, which seems to mean that geometric objects are determined by the possible sheaves on them. But “functor of points”...

when I google it seems to mostly involve scheme theory. So here again we see these categorical ideas to prevalent in algebraic geometry being used to understand differential geometry.

This is a little different than above because the above examples focus on groupoids as the key unification tool. I wonder if these categorical perspectives are related or if they’re completely different but both happen to sue category theory

The idea of lie groupoids dates back to at least the 80s with Weinstein being a central figure, but it seems only in the last twenty years has this stuff really taken off, and from talking to my advisor who works in the area he seems to think that lie groupoids are the future