This is my second question on supermanifolds, the previous one is at 


[Morphisms between supermanifolds R^{0|1}→R^{0|1}](https://mathoverflow.net/questions/54927/morphisms-between-supermanifolds-r01r01)


I've learn the difference between homomorphism and internal-hom of supermanifolds. Also, I know that the homomorphisms are generally easy to calculate, it is defined to be morphisms of superalgebras of functions of supermanifolds, or a bit correctly morphisms of ringed spaces; while the internal-hom is defined indirectly by adjoint functor. I thus want to know how to calculate the internal-hom of supermanifolds, in particularly $Map(\mathbb{R}^{0|n},M)$ for any supermanifold $M$. Thanks for your help!


with regards,


maming



How to caculate the internal hom of supermanifolds?
(Sorry for not answering this on the previous post, you asked this question before. By the way, you didn't say which paper you are reading...)


$Map(\mathbb R^{0|0},M)=M$. Trivial.


$Map(\mathbb R^{0|1},M)=\Pi T M$. ($TM$ is the tangent bundle of $M$, and $\Pi$ reverses grading of a vector bundle. So $\Pi T M$ denotes the total space of the degree-reversed tangent bundle of $M$. If $\dim M=d|\delta$, then $\dim \Pi T M=d+\delta|d+\delta$.)


Since


$Hom(S, Map(\mathbb R^{0|n},M))
= Hom(S\times \mathbb R^{0|n},M)$
$= Hom(S\times \mathbb R^{0|n-1}\times \mathbb R^{0|1},M)
= Hom(S\times \mathbb R^{0|n-1},\Pi T M)$,


we obtain inductively $Map(\mathbb R^{0|n},M)=(\Pi T)^n M$.
However, the interesting thing about $Map(\mathbb R^{0|n},M)$ is that it has an action by
$Diff(\mathbb R^{0|n})$, the supermanifold of invertible maps from $\mathbb R^{0|n}$ to itself.
I guess this is not so visible if you write $(\Pi T)^n M$, since this was obtained by destroying the symmetry in the odd coordinates.


For a description which keeps the symmetry:
$Map(\mathbb R^{0|2},M)$ is the pullback of $\Pi( T M\oplus T M) \to M$ along $TM\to M$,
which I learned from Dan Berwick Evans at Berkeley.
I would guess this is as explicit as it gets in general, and that probably more difficult pullbacks squares involving $\Pi T M$ exist for $Map(\mathbb R^{0|n},M)$ with bigger $n$.


One can find a description and discussion of $Map(\mathbb R^{0|n},M)$ in the paper with the nice title "Differential gorms, differential worms", Denis Kochan, Pavol Severa arXiv:math/0307303.


