Universal property of the product of two smooth manifolds

Question

I'm reading Ravi Vakil's book Foundations of Algebraic Geometry, and in the first chapter, which discusses category theory, he starts the discussion by showing that one can define the Cartesian product of two sets using a universal property. He then says the following:

This definition has the advantage that it works in many circumstances... such as the category of vector spaces, where the maps are taken to be linear maps; or the category of differentiable manifolds, where the maps are taken to be submersions, i.e., differentiable maps whose differential is everywhere surjective

My question is about the case of the category of smooth manifolds. Why do the maps are restricted to be submersions? Does something goes wrong when we allow the morphisms to be arbitrary smooth maps?

Answer 1

The product of two smooth manifolds certainly exists, in the category of arbitrary smooth maps (or $C^n$ maps, or whatever.) The products of $\mathbb{R}$ with itself in this category are, after all, the subject of multivariable calculus, and one knows from that course that smoothness of a function into $\mathbb{R}^n$ is determined on coordinates. I'm not sure whether Vakil is giving a more interesting example, or just thinking ahead to fiber products.

Answer 2

In general, the intersection of two submanifolds $N,N'$ of $M$ is not a submanifold, and $N\cap N'$ is the fiber product of $i_N:N\rightarrow M$ and $i_{N'}:N'\rightarrow M$.