# What is the space of maps between superplane $\mathbb{R}^{0|2}$ and a smooth manifold $M$?

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)

• It seems that in your definition of $f^*$, the roles of $M$ and $N$ are swapped. Nov 4, 2022 at 12:34

The manifold $$\def\Hom{\mathop{\rm Hom}}\def\R{{\bf R}}\Hom(\R^{0|2},M)$$ is isomorphic to the pullback of the parity-reversed bundle vector bundle $$TM⊕TM$$ along the projection map $$TM→M$$.