Intuition for the need of generalizing from mappings to morphisms to functors in supermathematics?

Question

I am now 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 ...
 

Answer 1

It seems to me that at this point in the text the phrase "parity reversal has to be a functor" is meant simply in the sense that "it is clear that parity reversal is a functor" ("has to be" in the sense of "it just has to be true").

Because for that to be true the operation given on superrings (the objects) needs to extend to an operation on the maps between them (the morphisms) and in this case it is clear that they do, because the maps preserve the grading and the operation simply swaps the gradings, so that the swapped gradings are then still preserved.

This is a common step when passing to category theory: At the beginning one typically does not so much introduce sophisticated category theoretic constructions, instead one goes along and recognizes that a lot of traditional constructions secretly are already category theoretic, and one just makes this explicit. 

But one may still ask the question that you are asking, regarding the motivation to use tools from category theory. The answer is that looking at supergeometry via tools from category theory helps to organize concepts, avoid mistakes, and generalize appropriately.

For instance

• traditional approaches to supergeometry have difficulties with speaking about infinite-dimensional supermanifolds such as super mapping spaces, which are at the heart of physics. In the functorial perspective this generalization is immediate and transparent.
• traditional supergeometry has "principles" such as the one called "even rules", which are useful but remain a little mysterious. In the functorial perspective these principles gain the status of precise mathematical facts which leave no doubt about what is going on.