Within the algebrogeometric approach to supergeometry, a supermanifold of dimension $m|n$ is an ordinary $m$ dimensional smooth manifold $M$ and a sheaf of supercommutative super algebras $\mathbf{C}^\infty$ on $M$ (thought of as the sheaf of functions on the supermanifold) which is locally isomorphic to $C^\infty_M\otimes \bigwedge^* \mathbb{R}^n$.
The space underlying $\mathbb{R}^{m|n}$ is $\mathbb{R}^m$ and the sheaf of functions is $U\mapsto C^\infty(U)\otimes \bigwedge^* \mathbb{R}^n$. This means that $\mathbb{R}^{0|2}=(\mathbb{R}^0, C^\infty_{\mathbb{R}^0}\otimes \bigwedge^{*}\mathbb{R}^2)\cong (\ast, \bigwedge^*\mathbb{R}^2)$.
We can consider a trivial supermanifold structure on any smooth manifold by taking $\mathbf{C}_M^\infty=C_M^\infty\oplus 0$.
We can define a map between supermanifolds $(M, \mathbf{C}^\infty_M)\to (N, \mathbf{C}^\infty_N)$ as a smooth map $f:M\to N$ and a morphism $f^{*}: \mathbf{C}^\infty_N\to f_{*}\mathbf{C}^\infty_M$ where $f_{*}\mathbf{C}^\infty_M$ denotes the pushforward sheaf of $\mathbf{C}^\infty_M$ with respect to $f$, $(f_*\mathbf{C}^\infty_M)(U):=\mathbf{C}^\infty_M(f^{-1}(U))$.
As defined, a map $f:\mathbb{R}^{0|2}\to M$ between the superplane $\mathbb{R}^{0|2}$ and a smooth manifold $M$ consists of a map $\ast\to M$ and a morphism of sheaves of superalgebras $C^\infty_M\to f_*\bigwedge^{*}\mathbb{R}^2$ where $f_{*}\bigwedge^{*}\mathbb{R}^2(U)=0$ if $f(*)\notin U$ and $f_{*}\bigwedge^{*}\mathbb{R}^2(U)=\bigwedge^{*}\mathbb{R}^2$.
Since the sheaf $f_{*}\bigwedge^{*}\mathbb{R}^2$ is concentrated at $f(\ast)$ this map is uniquely determined by the map between stalks $C^\infty_{f(p)}\to \bigwedge^{*}\mathbb{R}^2$. Is there a nice way of understanding this space of maps?
(Migrated from Math Stack Exchange)
