# What mathematical structure describes superspace and superfields?

In every book related to supersymmetry I have encountered at some point the idea of superspace is introduced. Superspace is presented as a space spanned by 4 "normal" directions and 4 Grassmannian ones (in the simplest, 4D case). So, it looks like $\mathbb{R}^8$ on this level. But then, some additional algebraical structure is added to distinguish between the standard and Grassmannian directions: the Grassmannians are equipped with a skew-symmetric product. On the other hand the "normal" variables treated as vectors form $\mathbb{R}^4$ do not have any natural multiplicative structure. Than it gets even worse to deal with: one introduces superfield, which can be written down in terms of variables of superspace (lets call them $(x^\mu,\theta^\alpha, \bar{\theta^\beta}$) as: $$F(x^\mu,\theta, \bar{\theta}) = f(x^\mu)+c(x^\mu)\theta +d(x^\mu) \bar{\theta}+g(x^\mu)\theta\theta+...$$ where $\theta\theta = \theta_1\theta_2$ and $+...$ stands for the rest of non-zero products of Grassmann variables. If I understand correctly, $f, x, d, g$ etc are functions $\mathbb{R}^4 \rightarrow \mathbb{R}$. But what is the codomain of $F$ function? What is the relation between domain of this function (the superspace) and codomain? Another doubt that I have on this topic is the meaning of coordinate changes on superspace. In many textbooks one can find coordinate transformations of such kind: $$x^{\prime \mu} = x^\mu + a^\mu + i(\theta\sigma^\mu\bar{\xi}-\xi\sigma^\mu\bar{\sigma})$$ with $\xi, \bar{\xi}$ being some spinor. How should be this formula thought of? As a affine transformation in an affine space with $a^\mu$ and $i(\theta\sigma^\mu\bar{\xi}-\xi\sigma^\mu\bar{\sigma})$ being some vectors or rather in component-manner, with $a^\mu \in \mathbb{R}$ being component of vector and $i(\theta\sigma^\mu\bar{\xi}-\xi\sigma^\mu\bar{\sigma})$ being some function whose arguments are a Grassmann and a spinor and value is a real number?

I would greatly appreciate any remarks on those structures as the feel rather confusing for me.