# One-to-one correspondence between super morphisms $\varphi:S\to TX$ and pairs $(f:C^\infty(X)\to C^\infty(S) ,F\in Der_f(C^\infty(X),C^\infty(S))$

I'm trying to show that for an ordinary manifold $$X$$ and a supermanifold $$S$$, supermanifold morphisms $$\varphi:S\to TX$$ are one-to-one to the pairs $$(f,F)$$ where $$f:C^\infty(X)\to C^\infty(S)$$ is a super $$\mathbb{R}$$-algebra homomorphism and $$F:C^\infty(X)\to C^\infty(S)$$ is an even derivation with respect to $$f$$, i.e. $$F$$ is a parity-preserving $$\mathbb{R}$$-linear map and $$F(ab)=F(a)f(b)+f(a)F(b)$$ for any $$a,b\in C^\infty(X)$$.

Intuitively, $$f$$ is the pullback of smooth maps from the base manifold $$X$$ to $$S$$ and $$F$$ is a global section to the pullback vector bundle $$f^*TX$$ on $$S$$. Hence if we let $$p:TX\to X$$ be the usual projection, then $$f= \varphi^*\circ p^*$$. The remaining data of $$\varphi$$ should give $$F$$, but I cannot see how.

If $$S$$ is also an ordinary manifold, then $$F$$ can be defined point-wisely. We can express $$\varphi$$ point-wisely by $$\varphi(p)=(f(p),V_{f(p)})$$ for each $$p\in S$$, where $$f(p)\in X$$ and $$V_{f(p)}\in T_{f(p)}X$$. For any $$a\in C^\infty(X)$$, $$F(a)\in C^\infty(S)$$ is then the function that sends $$p\in S$$ to $$V_{f(p)}a\in \mathbb{R}$$.

However, if $$S$$ is a supermanifold, then the point-wise definition no longer works. I tried to formulate the point-wise definition in a non-point-wise way but failed.

I guess that $$F$$ should be a composition $$C^\infty(X)\xrightarrow{D}C^\infty(TX)\xrightarrow{\varphi^*}C^\infty(S)$$ for some natural derivation $$D:C^\infty(X)\to C^\infty(TX)$$ with respect to $$p^*$$, but I cannot see any derivation that arises "naturally".

Thanks in advance for any help.

The morphism $$\def\T{{\rm T}} φ:S→\T X$$ can be identified with the homomorphism of algebras $$\def\Ci{{\rm C}^∞} \Ci(\T X)→\Ci(S).$$ The algebra $$\Ci(\T X)$$ can be identified with the $$\Ci$$-symmetric algebra $$\def\CiSym{\mathop{\rm\Ci Sym}\nolimits} \CiSym_{\Ci(X)}(\Gamma(\T^*X))$$ of the $$\Ci(X)$$-module $$\Gamma(\T^* X)$$ of smooth sections of the cotangent bundle of $$X$$. Here $$\Ci$$-symmetric algebras are defined using exactly the same universal property as symmetric algebras, but working in the category of $$\Ci$$-rings instead of the category of commutative real algebras.
By the universal property of $$\Ci$$-symmetric algebras, the homomorphism $$\CiSym(\Gamma(\T^*X))→\Ci(S)$$ can be identified with a homomorphism of $$\Ci$$-algebras (equivalently, commutative real algebras) $$f:\Ci(X)→\Ci(S)$$ together with a morphism of $$\Ci(X)$$-modules $$Ψ:\Gamma(\T^*X)→\Ci(S).$$ The latter morphism can be identified with a derivation $$F:\Ci(X)→\Ci(S)$$ with respect to $$f$$: we set $$\def\d{{\rm d}} F(g)=Ψ(\d g)$$, where $$\d g∈Γ(\T^* X)$$ is the differential of $$g$$.
This approach works equally well for ordinary manifolds, supermanifolds, $${\bf Z}$$-graded manifolds, derived manifolds, etc.
• Thanks. Is it true that this $\Psi$ is the composition $\Gamma(T^*X)\hookrightarrow C^\infty(TX)\xrightarrow{\varphi^*} C^\infty(S)$, where $\Gamma(T^*X)\hookrightarrow C^\infty(TX)$ sends a global section $s\in \Gamma(T^*X)$, which is a $C^\infty(X)$-linear map $s:\Gamma(TX)\to C^\infty(X)$, to the smooth map on $TX$ that sends $(p,v_p)\in TX$ to $s(V)(p)\in \mathbb{R}$ for an arbitrarily chosen $V\in \Gamma(TX)$ with $V_p=v_p$? To be honest, I know nothing about $C^\infty$-rings, and the introduction on nlab is not quite clear to me. Apr 14, 2022 at 6:02
• Also, if my construction of $\Psi$ is correct, then I get one side of the one-to-one correspondence. But how can I tell the other side, without taking the universal property mentioned above for granted? Or equivalently, how can I show that the universal property is satisfied by $C^\infty(TX)$? I'd appreciate that if you could please give me some detailed materials about this fact. Apr 14, 2022 at 6:30