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I am currently reading this paper about the categorical formulation of superalgebras and supergeometry, where in definition 2.3 it says that to change the parity of a right supermodule a morphism will not do the job as morphisms have to preserve parity and a functor has to be used instead (and the right supermodules are then also objects of the corresponding category).

This makes me wanting to (intuitively at first) really know what changes when ramping up the level of abstraction when going from

  1. Vector spaces and mappings between them to
  2. Modules and morhismes to
  3. Objects in a category (?) and functors

I would like to get a rather intuitive overview that explains what morphisms can do that ordinary mappings can not and what additional superpowers (pun intended) functors have apart from changing parity compared to morphisms?

Reading the definition 2.3 in the paper I was also wondering if supersymmetry transformations should then strictly speaking be functors to ...

Even thoug I like the answer I got elsewhere, I would like to learn what the somewhat larger community here has to say too.

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