If string theory/M-theory is a revolution in line with the Newtonian and then Einsteinian revolutions, then there needs to be a change in the language geometry to allow the expression of new physical principles.
String theory at its finest is, or should be, a new branch of geometry. (p. 95)
I would consider trying to elucidate this proper generalization of geometry as the central problem of physics, certainly the central problem of string theory. (p. 96)
What should have happened, by rights, is that the correct mathematical structures should have been developed in the twenty-first or twenty-second century, and then finally physicists should have invented string theory as a physical theory that is made possible by these structures. (p. 102)
we are paying the price for the fact that we didn’t come by this thing in the usual way. (p. 103)
Interview of Edward Witten (pp. 90-107) in P C W Davis & J Brown (eds.), Superstrings: A theory of everything?, Cambridge University Press 1988, 1991, Canto 1992 (gbooks):
Geometry as revolutionised by category and higher category theory:
In the classical paradigm, sets can be thought of as the most primitive notion of space – collecting things together in a minimalist way – from which other notions of space are formally derived. In the new paradigm, categories, and in particular higher categories, are the new primitive spatial notion from which the others are derived. Nowadays, categories are everywhere in topology and geometry, from the definition of the basic objects to the problems and methods of study. ((Anel and Catren (eds.) 2021a, 10)
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