Defining a general structure of "Calculus"

Question

I've been thinking lately, is there a way to generalize the fundamental concepts of Calculus such as convergence, differentiability and integrability to it's "maximum potential"? That is, defining a really long $n$ - uple in the algebraic sense (like in group theory, $(G,\,\cdot\,)$) that would include all the other "cases of Calculus" (in the real line, in $\mathbb{R}^n$, in Banach Spaces, in a manifold, etc). I know we could generalize the notion of differentiability to normed spaces and manifolds, and the notion of integrability to any measurable space, including, again, manifolds, so I think that the most likely structure to harbor the strongest definition of what "Calculus" comes to be are the manifolds.
That put, I've came across the notion of Difeological Space, in which the manifolds (even the infinite-dimensional ones) are replaced by stronger counterparts; similarly, I've encountered some sources claiming that the language of Category Theory could make such a powerful generalization of Calculus, creating it's own "algebraic structure", in some sort of manner.
What do you guys think? Such a thing would be possible?
Thanks in advance!