# How is a section viewed as a bundle map?

Suppose $\pi$ is the projection, $E$ total space, $B$ the base space, and $F$ the fiber.

A section of a fiber bundle is a continuous map $f\colon B \to E$ such that $\pi(f(x))=x$ for all $x \in B$.

Suppose that $M$ and $N$ are base spaces, and $\pi_E : E \to M$ and $\pi_F : F \to N$ are fiber bundles over $M$ and $N$, respectively. A bundle map consists of a pair of continuous functions

$\varphi\colon E\to F,\quad f\colon M\to N$

such that $\pi_F\circ \varphi = f\circ\pi_E$.

I wonder how the following is consistent with the definition of section above?

A bundle map from the base space itself (with the identity mapping as projection) to $E$ is called a section of $E$.

Specifically, how is the other fiber bundle like for the bundle map?

Added: I wonder if people think of a section more often as a "inverse" of projection, or as a bundle map? What is the purpose of viewing a section as a bundle map, which seems to me so indirect?

All quotes are from Wikipedia.

Thanks and regards!

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Isn't the instruction clear? Take $E = M$ and $\pi_{E}: E \to M$ to be the identity. A section $\sigma: M \to F$ of the bundle $\pi_{F}: F \to M$ is then the same thing as a bundle map $E \to F$ having $\sigma: E \to F$ as component "upstairs" and $\operatorname{id}_{M}: M \to M$ "downstairs" (upstairs means on the total space, downstairs on the base). – t.b. Aug 14 '11 at 23:12
@Tim: Certainly as a right inverse to the bundle projection. It is nice though that you don't have to leave that category of fiber bundles to speak of cross-sections, so this has some merits. However, it is good to think of a cross section as an "embedding" of the base into the total space. – t.b. Aug 14 '11 at 23:32
@Tim: With that argument everything is set theory. More seriously, it also depends on where bundles are used, and that's in algebraic and differential topology, predominantly. I don't think bundles play a comparably important role in general topology. – t.b. Aug 14 '11 at 23:34
I want to echo Theo's sentiments: you don't usually want to think about sections as maps of bundles (their importance and usage make them worth understanding in their own context). However, different perspectives give new insight, and it's useful to know that generalities about bundle maps can be specialized to give information about sections, or that things which are true for sections can be extended to results on general bundle maps. Here is one quick application: the functor sending a bundle to its sections can be viewed through the lens of the Yoneda embedding. – Aaron Aug 15 '11 at 0:21
(Re: purpose) In the context of vector bundles, viewing a nowhere-vanishing section as a trivial 1-dim subbundle is convenient sometimes, I believe (say, when thinking about what restrictions on char classes having such a section gives). – Grigory M Aug 15 '11 at 7:30