Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 7 down vote accepted

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.

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.