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.


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.

  • $\begingroup$ This is fascinating. Thanks. $\endgroup$ – nigel Feb 20 '13 at 6:11
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    $\begingroup$ 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 $F_*:T_P(M)\to T_{F(P)}(N)$. $\endgroup$ – lkat Feb 20 '13 at 6:17

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