Differential Geometry and Categories

Question

Is there such a thing as a category of manifolds? If so what is the functor from that category to the one of vector spaces? (It seems natural that this would correspond to the push-forwards), sounds like a free functor somehow but I have not been able to find sufficiently good treatments on that. Also any good references combining differential geometry and category theory would be appreciated.

Answer 1

Indeed, (smooth) manifolds form a category, with the morphisms being the (smooth) maps between manifolds.

If $M$ is a (smooth) manifold and $C(M)$ is the space of (real or complex) (smooth) functions on $M$, then $C$ is a contravariant functor from $M$ to $C(M)$. Notice that $C(M)$ naturally sits in several categories: sets, commutative groups, vector spaces, algebras, rings - you choose which one is of interest to you by dropping some structure off $C(M)$.

(Notice that $C$ takes the morphism $f : M \to N$ into the morphism $C(f) : C(N) \to C(M)$ given by $C(f) (\phi) = \phi \circ f$, i.e. $C(f) = f^*$, the pull-back.)

A very well known category-flavoured book on differential geometry is "Natural Operations In Differential Geometry" by Ivan Kolár, Peter W. Michor, Jan Slovák (freely available online) - but don't expect the same level of category theory as in algebraic topology or algebraic geometry.

Another, more recent book, that presents differential geometry using an approach heavily inspired by algebraic geometry is "Manifolds, Sheaves and Cohomolgy" by Torsten Wedhorn.