Higher Abstraction than Manifold Theory I am a physicists so please excuse me if I get something wrong. What I was wondering about is whether there is a mathematical topic that is a higher generalization than manifold theory. What I mean is, first we study Analysis of one variable, then Analysis of many variables then manifold theory. Is there something more general than manifold theory?
 A: Well it probably depends what you mean by abstraction. From the examples and the progression it seems to me that you are searching for more general spaces on which some Version of calculus and Differential geometry makes sense (the other generalisation direction would be to give up on calculus and study general spaces e.g. in the sense of topological spaces).
So with that being said here are some generalisations of settings with calculus tools available:

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*Orbifolds: These can be defined similar to manifolds but are locally  quotients of euclidean spaces Modulo a finite group. Think for example two dimensional space modulo a rotation group generated by a rational angle. This gives you a manifold at all points except the origin where you get a cone shaped singularity. For this reason, orbifolds are often called manifolds with mild singularities. These objects appear naturally in physics (there are articles by Vafa and Dixon and Witten from the 80s). The moderne way to treat this structure is via certain Lie groupoids (see  e.g. https://mathoverflow.net/questions/1861/looking-for-an-introduction-to-orbifolds/2403)


*Infinite dimensional calculus: remove the restriction that your manifolds need to be modeled on finite dimensional spaces. This allows one to think of diffeomorphism groups as Lie groups. Many naturally occuring examples from physics fall into this framework (the Bondi-Metzner-Sachs group from general relativity; spaces and groups of use in Connes-Kreimers approach to the renormalisation of Quantum field theory to showcase just a few examples). Note that there are several flavours of calculus beyond the familiar setting of Banach manifolds. To name the most popular: Convenient calculus (a la Kriegl and Michor), Bastiani calculus or diffeological spaces have all claims at being good generalisations for various reasons. If you Google these keywords you will hit upon suitable introductions.
