# Is the global sections functor on smooth manifolds an embedding?

Is the functor $\Gamma:M \mapsto C^{\infty}(M)$ an embedding from the category of smooth manifolds to the (opposite) category of real algebras?

Or equivalently, one has a map of sets $C^{\infty}(M,N) \rightarrow \text{Hom}(C^{\infty}(M),C^{\infty}(N))$. Is this map a bijection?

• What is an embedding of categories? You mean a fully faithful functor? – Qiaochu Yuan Feb 11 '16 at 4:57