I am recently studying Differential Topology from the Milnor's book. I have some questions regarding smooth manifolds and smooth maps. Can I say the followings:
Are smooth manifolds equivalent to the ringed spaces which are locally identical to the ringed space with smooth functions ?
Please help me. Thanking you in advanced.
The category of smoothly embedded submanifolds of Euclidean space is equivalent to the category of abstract smooth manifolds because every smooth manifold can be embedded (this uses Hausdorff+paracompactness).
The category of abstract smooth manifolds and smooth maps is equivalent to the full subcategory of locally $\mathbb R$-ringed spaces spanned by the following objects: locally $\mathbb R$-ringed spaces which admit an open cover on which they are isomorphic (in the category of locally $\mathbb R$-ringed spaces) to an open subset of Euclidean space equipped with its sheaf of smooth real valued functions.