What is a section?

Question

This question comes out of the answers to Ho Chung Siu's question about vector bundles. Based on my reading, it seems that the definition of the term "section" went through several phases of generality, starting with vector bundles and ending with any right inverse. So admittedly I'm a little confused about which level of generality is the most useful.

Some specific questions:

• Why can we think of sections of a bundle on a space as generalized functions on the space? (I'm being intentionally vague about the kind of bundle and the kind of space.)
• What's the relationship between sections of a bundle and sections of a sheaf?
• How should I think about right inverses in general? I essentially only have intuition for the set-theoretic right inverse.

Pointers to resources instead of answers would also be great.

Answer 1

To your first question, "function on a space" $X$ usually means a morphism from $X$ to one of several "ground spaces" of choice, for example the reals if you work with smooth manifolds, Spec(A) if you work with schemes over a ring, etc. (This is a fairly selective use of the word "function" which used to confuse me.) A section $\gamma$ of a (some-kind-of) bundle $E\to X$ is thought of as a "generalized function" on $X$ by thinking of it as a funcion with "varying codomain", i.e. at each point $x\in X$, it takes value in the fibre
$E_x\to x$. If one is talking about locally free / locally trivial bundles, meaning $E$ is locally (over open sets
$U\subset X$) isomorphic to some product $U\times T$, then we can locally identify the fibres with $T$. Thus locally a section just looks like a function with codomain $T$, which is often required to be nice.

To your second question, I generally take the "right-inverse" or "pre-inverse" definition from category theory, because it relates back to others in the following precise way:

Say $\pi: Y\to X$ is a space over $X$ (intentionaly vague). The word "over" is used to activate the tradition of suppressing reference to the map $\pi$ and refering instead to the domain $Y$. For $U\subseteq X$ open, the notation $\Gamma(U,Y)$ denotes sections of the map $\pi$ over $U$, i.e. maps $U\to Y$ such that the composition $U \to Y\to X$ is the identity (thus necessarily landing back in $U$). It's not hard to see that
$\Gamma(-,Y)$ actually forms a sheaf of sets on $X$.

Conversely, given any sheaf of sets $F$ on a space $X$, one can form its espace étalé, a topological space over $X$, say $\pi: \acute{E}t(F) \to X$. Then for an open $U\subseteq X$, the elements of $F(U)$ correspond precisely to sections of the map $\pi$, which by the above notation is written $\Gamma(U,\acute{E}t(F)$. That is to say,
$F(-)\simeq\Gamma(-,\acute{E}t(F))$ as sheaves on $X$. This explains why people often refer to sheaf elements as "sections" of the sheaf.

Moreover, what we now denote by $\acute{E}t(F)$ actually used to be the definition of a sheaf, so people tend to identify the two and write $\Gamma(-,F)$ a instead of $\Gamma(-,\acute{E}t(F))$. This explains the otherwise bizarre tradition of writing $\Gamma(U,F)$ instead of the the more compact notation $F(U)$.

$\Big($Unfortunate linguistic warning: Many people incorrectly use the term "étale space". However, the French word "étalé" means "spread out", whereas "étale" (without the second accent) means "calm", and they were not intended to be used interchangeably in mathematics. This is unfortunate, because the espace étalé has very little to with with étale cohomology. More unfortunate is the annoying coincidence that when dealing with schemes the projection map from the espace étalé happens to be an étale morphism, because it is locally on its domain an isomorphism of schemes, a much stronger condition.$\Big)$

To your third question, I think the observation that $\Gamma(-,Y)$ forms a sheaf on $X$ gives a nice context in which to think of sections $X$ to $Y$: they "live in" the sheaf $\Gamma(-,Y)$ as its globally defined elements.

Answer 2

A section of a bundle $B$ over a space $S$ is just a map $\sigma: S \rightarrow B$ such that for each $x \in S$, $\sigma(x) \in B_x$, where $B_x$ is the fiber over $x$.

Functions correspond to the case where the base is the domain $D$ and the bundle is $D \times R$, where $R$ is the range. In that sense, sections are generalized functions.

The notion of a section of a sheaf is essentially the same, but has to be adapted appropriately.

It seems to me that any reference that defines a vector bundle or a sheaf would explain this. I guess I would recommend any modern introduction to Riemann surfaces, where there is a minimum of machinery and complication. Although I don't work with Riemann surfaces, I found them fun to learn about, because they have the advantages of being 1-dimensional but have the rich structure of a 2-dimensional object.

Answer 3

Bundles are usually defined as being locally trival thingamajigs. A trivial bundle with fiber $F$ looks like the projection map $U \times F \to U$. A section of a trivial bundle is just a function $U \to F$. A global function on a manifold is the same as a bunch of local functions that literally agree on the overlaps. Similarly a global section of a bundle is the same as a bunch of local sections (which, again, are just functions) that "agree" on the overlaps, where now "agree" does not mean literally agree, but "agree after a twist", where the "twists" comes from the transition functions of the vector bundle.

Whenever we have a bundle, we can form a sheaf out of it. The corresponding sheaf is the one which maps open sets $U$ to the set of sections of the bundle over $U$. Conversely, if we have a sheaf on a space $X$, it is possible to construct a space $Y$ and a map $Y \to X$ such that the "sections" of the sheaf correspond to the actual sections of the map $Y \to X$. This is called the espace étalé and is discussed here and somewhere in Hartshorne chapter II section 1.

You may also be interested in looking at Hartshorne chapter II exercise 5.18.

Answer 4

If $\pi: E \to M$ is a bundle over a topological space $M$, you can define a sheaf on $M$ that associates to each open set $U \subseteq M$ the set of sections over it, i.e., maps $\sigma: U \to E$ such that $\pi \circ \sigma = \mathrm{id}_{U}$. Conversely, given a sheaf $\mathcal{F}$ on $M$ you can construct a topological space such that your $\mathcal{F}$ is its sheaf of sections. This Wikipedia page has some information on it. You will also be able to find information on any introductory book an algebraic geometry (e.g., Hartshorne).