I have a specific question with a broader question attached to it.
Part 1
Homogeneous spaces are defined as smooth manifolds endowed with a transitive action from a Lie group. In the event that $G\times M \to M$ is a smooth transitive action, then $M\cong G/G_p$ where $G_p$ is the isotropy subgroup of the point $p\in M$. Remarkably though, we find that any time we find that a Lie group $G$ acts transitively on a set $X$, and as long as $G_p \subseteq G$ is closed for some $p \in X$ we find that $X$ is a smooth manifold with a homogeneous structure.
I'm new to category theory, but with the basic knowledge of functors, limits, and colimits, this theorem feels like it has a deeply categorical flavor to it. For instance, if we let $\mathsf{Lie}$ be the category with Lie groups and Lie group homomomorphisms, $\mathsf{Diff}$ the category of smooth manifolds and $C^\infty$ mappings between manifolds, and $\mathsf{Set}$ the category of sets and mappings, then to me the above theorem appears that it could be rephrased in the following way:
Given any functor $F:\mathsf{B}G \to \mathsf{Set}$ for $\mathsf{B}G$ the associated monoid of a Lie group $G \in \text{Ob}(\mathsf{Lie})$ such that the colimit colim $F$ is a singleton, then we have that this functor $F$ lifts to a functor $\widetilde{F}:\mathsf{B}G \to \mathsf{Diff}$ such that the commutative diagram holds:
Where $U$ here is the forgetful functor. The condition that colim $F$ be singleton is equivalent to the fact that functors from a monoid to $\mathsf{Set}$ essentially assign group actions from the underlying group, and the colimit of this functor is essentially the orbit space of the set being acted upon. Therefore transitivity of the action is equivalent to colim $F \cong \{*\}$. Is such a categorical proof possible and what machinery is needed to prove such a statement?
Part 2
To me a key part of the above observation (if it's valid) is learning (re-learning?) mathematical subjects from a fundamentally categorical view. There are obviously a lot of texts on category theory itself, and a lot of good topology books that view the subject from a strictly categorical perspective (e.g. Peter May's Concise Course). Are there any good books/articles for learning/re-learning classical differentiable geometry from the categorical perspective? Other books that attempt to revamp a part of mathematics from a categorical perspective, aside from topology? I'd be interested in learning of analysis, measure theory, Hilbert and Banach spaces from this perspective if literature exists on the topic. I know that algebraic geometry deeply employs categories, but I'm broadly unfamiliar with this subject (for now).