I am interested in generalizing the notion of Lie groups and Lie algebras to higher categories. One way to do this is to replace the category of vector spaces with a more general category of modules over rings, and then consider the tangent spaces of the objects in this category.
However, I want to go even further and consider the category of supermodules over superrings, which are $\mathbb{Z}_2$-graded modules over $\mathbb{Z}_2$-graded rings. A superring is a ring that has both even and odd elements, and a supermodule is a module that has both even and odd components. The even elements and components behave like usual, but the odd ones satisfy the relation $xy = -yx$ for any two odd elements or components.
My question is: what kind of mathematical objects have supermodules over superrings as their tangent spaces? What kind of mathematical objects are, in their tangent, supermodules over superrings ?
Any insights are welcome.
