I have come across things such as that a Presheaf $\mathcal{F}$ associates data (such as rings, groups, other sets etc.) to open sets $U$ of $X$. That the Presheaf $\mathcal{F}$ becomes a Sheaf if each of these objects (rings, groups, etc.) are separated and can be glued together. That kind of makes sense.
How would one demonstrate a presheaf on, for example a Topological space
$(X,\mathcal{T})$ where $X = \{a,b,c\}$ and $\mathcal{T} = \{\emptyset,X,\{a\},\{b\},\{a,b\},\{b,c\}\}$
The proper open sets in $X$ are $\{\{a\},\{b\},\{a,b\},\{b,c\}\}$ and there are inclusions $\{a\} \subset \{a,b\}$, $\{b\} \subset \{a,b\}$ and $\{b\} \subset \{b,c\}$ and furthermore we have a couple different open coverings for $X$ such as $\{\{a\} \cup \{b,c\}\}$ and $\{\{a,b\} \cup \{b,c\}\}$ ?
By definition; A presheaf of $\mathcal{F}$ of sets on $X$ consists of the data;
a) for every open subset $U \subseteq X$ a set $\mathcal{F}(U)$
b) for every inclusion $V \subseteq U$ of open subsets of $X$, a morphism of sets $\rho_{UV}:\mathcal{F}(U) \rightarrow \mathcal{F}(V)$,
with the conditions of $\mathcal{F}(\emptyset) = 0$, $\rho_{UU}$ is the identity map, and $\rho_{UW} = \rho_{VW} \circ \rho_{UV}$
What can we say that $\mathcal{F}(U)$ are according to the Topological space given above? - What does $\mathcal{F}(U)$ look like? What could be possible sections and can they be glued to create a sheaf? I imagine that since we have an open covering $\{\{a,b\} \cup \{b,c\}\}$, can we just glue along $b$ to obtain $X$?
Lastly I wonder, the more intuitive part - I know that $\mathcal{F}$ is a functor, thus when this functor is applied to some open set we get a bunch of sections (morphisms)? But then if have $\rho_{UV}: \mathcal{F}(U) \rightarrow \mathcal{F}(V)$, then we are sending these sections/morphisms concerning some sort of structure (the Presheaf data $\mathcal{F}(U)$ on $U$) built out of $U$ on to the presheaf data built out of $V$ - but $V \subseteq U$! Is there no loss of information? I see this as saying $\mathcal{F}(\{a,b\}) \rightarrow \mathcal{F}(\{a\})$ I of course want to understand what kind of "data" these presheaves are associating, but furthermore wonder if there would be a loss of information - and whats the point?
Thanks,
Brian
I dont know if this will be helpful to you but it is the intuition that I use to think about things. A sheaf is (intuitively) the set of function of some type, continuous, differentiable, analytic, algebraic etc.. So $\mathcal{F}(U)$ is all such function defined on the open set $U$. The maps $r_{UV}:\mathcal{F}(U)\rightarrow \mathcal{F}(V)$ are restriction, sending a function on $U$ to its restriction to the smaller set $V$. Viewed in this light all of the machinery of sheaf theory makes perfect sense. And this was the original motivation.