# What is the geometric significance of the definition of supermanifold?

We know that a supermanifold $M$ is a locally ringed space $(M,O_M)$ which is locally isomorphic to $(U,C^\infty(U) \otimes \wedge W^\ast)$, where $U$ is an open subset of $\mathbb{R}^n$, $W$ is a finite dimensional real vector space and the above isomorphism defined in the category of $\mathbb{Z}_2$ graded algebra i.e. the parity $\bigoplus_{k \geq 0}C^\infty(U) \otimes \wedge^k W^\ast \rightarrow \mathbb{Z}_2$ defined by $f \otimes x \rightarrow |f \otimes x|:=|x|=k \mod 2$. I would like to know how can we geometrically think of this. For example we know that the algebra of differential forms $\Omega(M)$ on a manifold $M$ which is locally isomorphic to $C^\infty(U) \otimes T^\ast_x M$ for some $x \in U$, therefore the sheaf of differential forms on a manifold corresponds to a supermanifold. How can we geometrically visualize this? Moreover what is the significance of defining a supermanifold structure for the sheaf of differential forms for a manifold $M$.

Forgetting about the "super" part, already an ordinary manifold can be described as a ringed space $$(M,O_M)$$, locally isomorphic to $$(U,C^\infty(U))$$. When $$U = \mathbb{R}^n$$, with standard coordinates $$x^i$$, the coordinates $$x^i$$ play the role of commuting generators (up to taking limits of polynomial expressions) of $$C^\infty(\mathbb{R}^n)$$. The point of this description is that in the pair $$(M,O_M)$$, the sheaf of algebras $$O_M$$ should be interpreted as the sheaf of algebras of functions on open subsets of $$M$$. But since you are looking into ringed spaces at all, I suspect that you have already seen this interpretation.
Supergeometry allows the sheaf $$O_M$$ to be a supercommutative algebra (not just a commutative one). The heuristic is that the supermanifold $$M$$ should be covered by charts of the form $$\mathbb{R}^n_{even} \times \mathbb{R}^m_{odd}$$ with corresponding algebras of functions $$C^\infty(\mathbb{R}^n_{even} \times \mathbb{R}^m_{odd}) := C^\infty(\mathbb{R}^n_{even}) \otimes \bigwedge^\bullet (\mathbb{R}^m_{odd})^*$$, where the second tensor factor is the algebra of functions generated by the odd (hence supercommuting) "coordinates" $$\theta^j$$ on $$\mathbb{R}^m_{odd}$$. Now, since $$\mathbb{R}^n_{even} \times \mathbb{R}^m_{odd}$$ does not exist as a topological space, we simply consider the algebra $$C^\infty(\mathbb{R}^n_{even} \times \mathbb{R}^m_{odd})$$ as assigned to the $$\mathbb{R}^n_{even}$$ factor, which does exist as a topological space.
This is exactly analogous to how the sheaf of functions on a fibered manifold $$N \to M$$, locally modeled on charts of of the form $$V\times U \to U$$, gives rise to the ringed space $$(M,O_M)$$ where which locally looks like $$(U, C^\infty(V\times U))$$. Batchelor's theorem makes this precise, by identifying any supermanifold $$(M,O_M)$$ with the total space of a fibered supermanifold $$M \to M_0$$, where $$M_0$$ is an ordinary manifold and the fibers of $$M \to M_0$$ are purely odd.
In this sense, the example that you gave, $$(M,\Omega^\bullet)$$, corresponds to the total space of the fibered supermanifold $$(\Pi T)M \to M$$, where $$\Pi$$ denotes the parity shift (from even to odd) of the cotangent fibers. There might be a common name for this supermanifold, but I can't recall it at the moment.