Fermionic components of super-gauge transformations Consider $4d$ $\mathcal{N}=1$ super Yang-Mills with gauge group $U(N)$, but really any supersymmetric gauge theory in any dimension will do. People have developed a formalism to quantize such theories in superspace, adding fermionic coordinates $\theta_\alpha$, see https://arxiv.org/abs/hep-th/0108200, where the gauge transformation parameter is promoted to a chiral superfield $\chi$, valued in the complexified lie algebra. On a chiral superfield $\Phi$ in the fundamental representation of $U(N)$, for example, finite gauge transformations would act as $$\Phi\rightarrow e^{i\chi}\Phi.$$
My question is: what are the fermionic components of $e^{i\chi}$ valued in? It is clear that the bottom component $e^{i\chi}|_{\theta=0}$ will be valued in $GL(N,\mathbb{C})$, but what about the other components? Is $e^{i\chi}$ actually valued in a supermanifold or supergroup as mathematicians define it, or is this a completely different beast?
 A: It is hard to tell with certainty since you have not defined everything precisely enough (and your reference is more than 500 pages long, and you haven't specified a page/section). I am going to make some reasonable assumptions and it shouldn't be hard to correct if any of them is inaccurate.
If we assume that $\chi=\phi+\theta\psi$, with $\theta$ a Grassmann parameter, $\phi$ a real scalar and $\psi$ a Majorana fermion, then
$$
e^{i\chi}=e^{i\phi}(1+\theta\psi)
$$
Therefore, the bosonic component takes values in $e^{i\phi}\in G_\mathbb C$ and the fermionic component takes values in $e^{i\phi}\psi\in G_\mathbb C\otimes\mathfrak g$. Here $G$ is the gauge group and $\mathfrak g$ is its algebra.
If the superfunction $\chi$ has a more complicated structure, this will be inherited to the final answer. For example if $\theta$ carries any extra indices, then $\psi$ will carry dual indices, and the fermionic part will live in some extended space which carries these extra indices.
But anyway, the basic answer is that the fermonic component lives in $G_\mathbb C\otimes\mathfrak g$, perhaps with some extra factors if $\theta$ transforms under some other group. No need to introduce supermanifolds nor anything particularly fancy.
