Sorry, it seems to be silly-season for me at the moment; I'm reading up on Category Theory (following "Category Theory in Context", by Emily Riehl, which I like), and it inspires a lot of speculation - this time I came across the concept of point-free topology. The idea here is that you take topology spaces, throw away the underlying point-sets and study what is left. So, to the question:
Is there any theory for point-free differential geometry?
Apparently not, which may not be surprising, considering the inherent difficulty in talking about differentiation in such a context. But is it impossible? And what would it take? I imagine one would have to dream up some sort of algebraic structure that would work for certain parts of what is intuitively "open subsets" - eg. for a 'topology' $\tau$, there would be an operator $+ \colon \tau \times \tau \rightarrow \tau$, so for some $o_1, o_2 \in \tau$ $\exists o_3 \in \tau \colon o_3 = o_1 + o_2$ - and so on. (sorry for rambling on, but I think it is interesting). Is it crazy?