Mfcttrf

Let $(M,xi)$ be a transversally orientable contact manifold, that is, there exists a form $alpha in Omega^1(M)$ such that $xi = ker alpha$. Then we can associate to $(M,xi)$ its symplectisation $(mathbbR times M,d(e^talpha))$, a symplectic manifold. I wondered, if there is a categorical setting for this process. I mean, naively, we could consider symplectisation as a map on objects $$S colon mathsfTOCont to mathsfSymp$$ where $mathsfTOCont$ denotes the category of transversally orientable contact manifolds as objects and maps $F in C^infty(M,widetildeM)$ such that there exists a nowhere vanishing function $f in C^infty(M)$ with $F^* widetildealpha = falpha$ as morphisms $F colon (M,xi = ker alpha) to (widetildeM,widetildexi = ker widetildealpha)$. Likewise, $mathsfSymp$ denotes the category with objects symplectic manifolds and morphisms $F colon (M,omega) to (widetildeM,widetildeomega)$ such that $F in C^infty(M,widetildeM)$ with $F^*widetildeomega = omega$.

Now the problem I am facing is the following: I would define $S$ on morphisms
$$S(F) colon (mathbbR times M,d(e^talpha)) to (mathbbR times widetildeM,d(e^twidetildealpha))$$ by
$$S(F) := operatornameid_mathbbR times F.$$ But then, if $F^* widetildealpha = falpha$, we compute $$S(F)^* d(e^twidetildealpha) = d(e^tfalpha),$$ that is, $S(F)$ is not a morphism in $mathsfSymp$. If $f > 0$, we could use the definition
$$S(F)(t,x) := (t - log(f(x)),F(x))$$ and things would work out fine. However, this would impose a restriction on orientation.

I think everything boils down to the fact that if $(M,xi = ker alpha)$ is a contact manifold, then also $xi = ker falpha$ for every nowhere vanishing smooth function $f$. But I guess the symplectisations are not symplectomorphic in general in this case, that is, a single t.o. contact manifolds admits different non-symplectomorphic symplectisations. Is that right? Do you have any idea how to turn symplectisation into a functor between appropriate categories?

first of all I think your $S(F)$ can be modified into
beginalign*
S(F)(t,x)=(t-log(|f(x)|), F(x))
endalign*
since $f$ is non-vanishing, this is always smooth. Nevertheless, there is a more conceptual way to see the symplectization:
the symplectization $S$ is a functor from contact manifolds into homogeneous symplectic manifolds. The latter is the category of pairs $(P,omega)$ consisting of a $mathbbR^times$-principal bundle
$P$ and a symplectic structure $omegain Omega^2(P)$, such that
beginalign*
h_r^*omega=romega
endalign*
for the principal action $hcolon mathbbR^timestimes Pto P$.
The morphisms are equivariant symplectomorphisms. This functor is even an equivalence of categories and
does not work just for co-orientable contact structures. Everything what I said is (more or less) done in https://arxiv.org/abs/1507.05405.

HD