I would like to ask for some reference textbooks or articles, or any information that you know about developing topological concepts and differential geometric concepts using as a foundation ETCS which is elementary theory of category of sets by Lawvere.
My motivation for it is that recently I have seen a reasonable discussion that ETCS could be a valid alternative foundation for set theory and being young I would like to take the risk and develop my machinery for differential geometry using this foundation with a hope that it could give me some useful intuition and an interesting point of view that those who used conventional set theory like ZFC/NBG don't have (interesting question is, is it really possible? I am pretty sure that both foundational systems can be shown to be equivalent if you equip ETCS with some additional axioms, but I would argue that the method of developing set theory differently makes you think about set theory differently as well).
As a sidenote, I would also be very interested in your opinions about developing topology and differential geometry using ETCS. Maybe the set theoretic equivalents are not possible, but it is possible to construct something equivalent or more general than topological spaces/manifolds etc.?