Synthetic Differential Supergeometry

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Three branches of modern mathematics make something of infinitesimals per se, namely, nonstandard analysis, synthetic differential geometry, and supergeometry. The first is concerned exclusively with invertible infinitesimals, whereas the second deals mainly with nilpotent ones. Both of the former two are engaged exclusively in commuting or bosonic infinitesimals, while the third treats anticommuting or fermionic ones, leading to so-called noncommutative mathematics. the unification of the first two approaches was nicely discussed by Moerdijk and Reyes, but the unification of the second and the third seems to remain open. The principal objective of this paper is to fill the gap, arguing that a super version of microlinear space, dubbed “supermicrolinear space,” is a natural generalization of supermanifold, just as the synthetic concept of microlinear space is replacing the classical concept of smooth manifold. The central result of the paper is that the graded tangency of a supermicrolinear space forms a Lie superalgebra, while it is well known that the tangency of a microlinear space (i.e., its totality of vector fields) forms a Lie algebra.

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