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Given maifolds $M,N$ and a smooth map $\phi:M \to N$, and a smooth function $f:N \to \mathbb{R}$, we have the pullback of $\phi$ by $f$ to be the function $\phi^* f = f \circ \phi : M \to \mathbb{R}$. Similarly, given a linear map $T:V \to W$, we get a transpose map $T^*: W^* \to V^*$ such that $T^*g = g \circ T$. I just noticed that these ideas are really similar. Is there something more "going on"? It can't be a coincidence, since mathematicians have decided to use basically the same notation.

My question is: is there a general way to encapsulate this concept? I imagine (perhaps incorrectly) that such an answer would involve category theory (I am aware of the so-called "dual functor" associated to vector spaces, which I understand is related to this topic) If possible, could someone point me to a reference without too much category theory? Thanks.

p.s. any references would be good, so even if they do contain tons of category theory, that's OK.

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Both of these are examples of contravariant hom functors. Given a category $\mathcal{C}$ and an object $X$ one can define a functor $\text{Hom}(\bullet,X):\mathcal{C}\to\mathbf{Set}$ defined on objects by $\text{Hom}(\bullet,X)=\text{Hom}(Y,X)$ and defined on maps by $Y\xrightarrow{f}Z$ goes to $f^\ast:\text{Hom}(Z,X)\to\text{Hom}(Y,X)$ given by $f^\ast(g)=g\circ f$.

In your example with manifolds you are dealing the category $\mathbf{Man}$ of smooth manifolds and you're object $X$ is $\mathbb{R}$--so pullback functor is $\text{Hom}(\bullet,\mathbb{R})$.

In fact, in your vector space example (assuming you mean real vector spaces) you are also dealing with the contravariant hom functor associated to $\mathbb{R}$ (now thought of as a vector space instead of a smooth manifold). In particular, you are working with the category $\mathbf{Vect}_\mathbb{R}$ of $\mathbb{R}$-spaces and your object $X$ is $\mathbb{R}$.

Note that in BOTH cases your functor actually goes not from a category to $\mathbf{Set}$ but to $\mathbf{Vect}_\mathbb{R}$ (the dual space $V^\ast$ is a vector space as well as $C^\infty(M)$). This is a phenomenon of these particular examples and doesn't always happen.

A good reference for basic category theory is Awodey. Since this is really a categorical concept I don't know what non-category theory reference would make sense.

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This is fascinating. Thanks. –  nigelvr Feb 20 '13 at 6:11
Just to add a remark: contravariant is a type of functor that 'reverses' arrows between objects. The other type of functor, covariant functor, preserves the direction of arrows. For example, the pushforward is covariant: $F: M\to N$ induces $F_*:T_P(M)\to T_{F(P)}(N)$. –  lkat Feb 20 '13 at 6:17

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