I am doing some work on the topic "Adjunction" and my current interest is to give some examples. I've wanted to provide some examples connected to some basic concepts from analysis on manifolds if such exists. What I thought maybe would work is the following:
- Tangent space functor and cotangent space functor
- Tensor product functor and multilinear form functor
- Pushforward and pullback of differential form
I am not sure if any of these are indeed adjoint functors, so if they are I would appreciate any help in proving such a thing.
For an adjunction you need a pair of functors pointing in opposite direction. I don't see this happening in your examples.
Let me write $Man_\ast$ for the catgory of pointed smooth manifolds. The tangent space can be seen as a functor $T:Man_\ast \to Vect_\mathbf R$. It sends $(M,p)$ to the vector space $T_pM$. The tangent bundle can also be seen as a functor $T:Man \to Bundle$ where the second category is that of smooth vector bundles. Or you can see it as a functor $T:Man\to Man$ together with a natural transformation $T\Rightarrow Id$. But in any case I do not see how the cotangent space construction can be an adjoint, because it would have to point in the opposite direction which it does not. It also is contravariant.
The cotangent space construction can be seen as a functor $T^\vee:Man_\ast^{op}\to Vect_\mathbf R$. Unfortunately it is not possible to turn extend the construction $T^\vee M\to M$ of the cotangent bundle to maps, and there is no good way to turn $T^\vee$ into a functor on $Man^{op}$.
Tensor product and multilinear forms are related in so far as that the tensor product of two vector spaces represents multilinear forms. you have that \begin{align} Vect_\mathbf R(A\otimes_\mathbf R B,C) \cong Bilin_\mathbf R(A,B;C) \end{align} naturally in $A,B,C$ for example. But that isn't really an adjunction. The pushforward and pullbacks of differential forms are not really functors themselve, instead they are the result of applying a functor to a smooth map. When $f:(M,p)\to (N,q)$ is a map of pointed manifolds, then the pullback and pushforward operations are the result of applying $T:Man_\ast \to Vect_\mathbf R$ and $T^\vee:Man^{op}_\ast \to Vect_\mathbf R$ to that map. I am sorry, but most adjunctions that I am aware of come from algebra, algebraic geometry or logic. Many of the construction of scheme theory work for general ringed spaces, and you can get a lot of adjunctions this way, but I believe they are not used that often in differential geometry.
Side tangent: This is a bit esoteric, but the tangent bundle construction in synthetic differential geometry is actually an adjoint, since there $TM$ is the exponential $M^D$ where $D$ is the space of "first-order infinitesimals". You have an adjunction $-\times D\dashv (-)^D$. But the space $D$ does not exist in standard differential geometry, and I do not think that SDG is a good way to get comfortable with adjunctions.
If you want to see many examples of adjunctions in algebra and topology, then you should look into Emily Riehl's book "Categories in Context".