The differential complex of smooth forms

Question

I'm learning my self about differential forms from the book by Loring Tu, and I've come across this sentence which seems important by I didn't understand it: " The differential complex of smooth forms on a manifold can be pulled back under a smooth map, making the complex into a contravariant functor called the de Rham complex of the manifold"

I was thinking that the de Rham complex is the same as the complex of differential forms !

Could you please enlight the difference between de Rham complex and the complex of differential forms and what it is meant by "contravariant functor" and say more about the above sentence .

Thanks a lot!

Answer 1

Terminology aside, the idea is that we can define a contravariant functor $$\Omega^*:(\text{Smooth Manifolds})\to (\text{Chain Complexes of Vector Spaces})$$ by assigning to a manifold $M$ its de Rham Complex $$ \Omega^0(M)\xrightarrow{d}\Omega^1(M)\xrightarrow{d}\Omega^2(M)\xrightarrow{d}\cdots\xrightarrow{d} \Omega^n(M). $$ You can find the definition of a functor between categories formally defined all over (like on wikipedia) but I'll sketch the idea here. Being a (contravariant) functor means that $\Omega^*$ assigns to each manifold a chain complex like above, and to each smooth map of manifolds $f:M\to N$ an associated morphism of chain complexes $$\Omega^*(f):\Omega^*(N)\to \Omega^*(M).$$ The contravariance indicates the above flipping of directionality of the maps, i.e. $\Omega^*(f)$ goes from $\Omega^*(N)$ to $\Omega^*(M)$ rather than the other way around. More has to be true: $\Omega^*$ should respect compositions in that $\Omega^*(f\circ g)=\Omega^*(g)\circ \Omega^*(f)$ and have $\Omega^*(\Bbb{1}_M)=\Bbb{1}_{\Omega^*(M)}$. Anyway, so much for the formalities. The picture here is quite simple: we have specified $\Omega^*(M)$ explicitly already, and associated to $f:M\to N$ we define $\Omega^*(f)$ to be the morphism of chain complexes $f^*$ as $\require{AMScd}$ \begin{CD} \Omega^0(N) @>d>> \Omega^1(N)@>d>>\cdots@>d>>\Omega^n(N)\\ @V f^* V V @V f^*V V @VV V @V f^* VV\\ \Omega^0(M) @>>d> \Omega^1(M)@>>d>\cdots@>>d>\Omega^n(M). \end{CD} Note that the diagram commutes by $f^*\circ d=d\circ f^*$. You can check the functoriality properties listed above.