Ytukyg

Hartshorne (in his book on algebraic geometry) constructs, for a presheaf $mathscr{F}$ over a topological space $X$, the sheafification of $mathscr{F}$, $mathscr{F}^+$, as
$$ mathscr{F}^+(U) = { s!: Uto bigsqcup_{pin U}mathscr{F}_p text{ such that }(*) } $$
with $(*)$ being the following conditions:

1. for each $pin U$, $s(p)inmathscr{F}_p$.


2. for each $pin U$, there is a neighbourhood $Vsubset U$ of $p$, and some $tinmathscr{F}(V)$, such that for any $qin V$, the germ $t_q = s(q)$.

My question essentially boils down to the following (and I'll expand on this):

What, explicitly, are the elements of $mathscr{F}^+(U)$?

What I mean is that, while the description above is in some ways explicit, it leaves too many details out (at least for me), and so I want some "classification" of the various kinds of sections we can get.

For example, it is clear that any element $sinmathscr{F}^+(U)$ is of the form $s!: pmapsto t_p$, with $t$ not necessarily being fixed. The case where it is fixed, of course, is the image of the natural morphism $theta!:mathscr{F}tomathscr{F}^+$, since this is given by $theta(s)mapsto (pmapsto s_p)$. This gives some of the elements of $mathscr{F}^+(U)$, but not all of them.

As far as I've been able to tell, any section $ainmathscr{F}^+(U)$ will be determined by considering an open cover ${U_i}_{iin I}$ of $U$, with classes of sections $[t^i]subsetmathscr{F}(U_i)$, such that all $sin[t^i]$ agree on germs (i.e. for any $s,vin[t^i]$ and $pin U_i$ we have $s_p = v_p$) and, similarly, for any $sin[t^i]$, $vin[t^j]$, and $pin U_icap U_j$ we have $s_p = v_p$ (that is, the germs agree on intersections).

So, a secondary question is:

Is the above characterization of the sections of $mathscr{F}^+$ correct?

So, basically the sections of the associated sheaf are functions which are locally sections of the presheaf. Maybe a clear way to see this is the following example: take a smooth manifold $M$ and define a presheaf $mathscr{F}$ by $mathscr{F}(U)={f:Uto mathbb{R}:f:text{is constant}}$. If $M$ is not connected, then write $M_1sqcup M_2$ for instance, both clopen. We can write a function $f_1:M_1to mathbb{R}$ by $f_1(x)=1$ and $f_2:M_2to mathbb{R}$ by $f_2(x)=2$. Because $M_1cap M_2=varnothing$, if $mathscr{F}$ were a sheaf, then the piecewise function $f:Mto mathbb{R}$ given by
$$f(x)= begin{cases}
f_1(x)&xin M_1\
f_2(x)& xin M_2.
end{cases}$$
would have to be in the sheaf. However, it is not because $f$ is not constant globally, only locally. Once we pass to the sheafification, $mathscr{F}^+$, our sections on $Usubseteq M$ are
$$ mathscr{F}^+(U)={f:Uto mathbb{R}:(1):&:(2)}.$$
$(1)$ is the property that $f(P)in mathscr{F}_P$. $mathscr{F}_P=mathbb{R}$ so this is fine.

$(2)$ is the property that around each $Pin U$, there exists a neighborhood $Vsubseteq U$ with $f|_V(P)=s_P$ for $sin mathscr{F}(U)$. That is, $f$ is locally a member of $mathscr{F}$. That is, $f$ is locally constant.

So, the sections of $mathscr{F}^+(U)$ are locally constant functions valued in $mathbb{R}$. Morally, a sheaf is supposed to be an object that locally assigns functions on the space, and the sheafification formalism says that indeed this is the case.