Given a topological space $X$, a presheaf $F$ of some algebraic structure on $X$ is defined such that
$(1)$ for each open set $U \subseteq X$, there is an algebraic structure $F(U)$,
$(2)$ for every inclusion $V \subseteq U$ of open subsets of $X$, there is a morphism $\rho_{U,V}:F(U) \to F(V)$ called the restriction map and it satisfies few properties.
I am often confused about the definition of presheaf. Sometimes it looks like a functor on a category and sometimes it seems to be a collection of objects or a category.
How should one think about presheaf or a sheaf ?
Next, in Wikipedia, it says $F(U)$ is the $sections$ of the presheaf $F$ over the open set $U$, which is defined as $\Gamma(U,F)$ sometimes. What are these $sections$ ?
Are these one-way inverse function that we know in algebraic topology or algebraic geometry ?
For example, let $R$ be a commutative ring and $R$-Mod be the category of $R$-module, then $\text{Hom}_R(-,M)$ is a contravariant functor from the category $R$-Mod to $Ab$, the category of abelian groups. Can we think a presheaf $F$ to be $F:=\text{Hom}_R(-,M)$ in this case ? and then the sections are really functions $s \in F(N)=\text{Hom}_R(N,M)$.
Any comments and explanations are welcome
Best Answer
They're related but much more general as the comments alluded to.
I think the term 'sections' comes from an alternative definition of a sheaf (which can be recovered from arbitrary sheaves using the Etale Space I think). It more resembles the construction of a vector bundle. Notice that the sections of a vector bundle do form a sheaf.
There are a number of ways to define sheaves. Let's say we define an Etale Space on $X$ to be a topological space $F$ and a local homomorphism $\pi: F \to X$ so that the stalks $F_p :=\pi^{-1}(x)$ are abelian groups and the group operations are compatible with the topology on $F$. This is the definition of a sheaf from Ahlfor's 'Complex Analysis' but I think this kind of definition was used in Godement's 'Topologie Algébrique et Théorie des Faisceaux' according to Hartshorne.
From an Etale Space $\pi: F \to X$ we define a sheaf $\mathscr{F}$ on $X$. Let $$\mathscr{F}(U) = \{f: U \to F\;|\; \pi \circ f = \operatorname{Id}_U\}.$$ That is, $\mathscr{F}(U)$ are the sections of $\pi$ on $U$. It will then take on the abelian group structure from the stalks to form a sheaf of abelian groups on $X$ so that $\mathscr{F}_p \cong F_p$. Similarly, the restriction map is just restriction of functions.
I tend to prefer the usual definition since it is much simpler but I'm sure there are situations where the alternative definition is preferable. I like to think of sheaf or presheaf sections figuratively as functions on open sets, although the function intuition does not work as well for presheaves.