This question is meant both to really try to understand presheaves and sections as well as check for feasibility for an idea that I have. Forgive if the question misses the point of presheaves; if my example lacks the generality. I am wanting to both understand the concepts as well as see if my idea works.
Question: Is there a way to define a presheaf on prime number subsets of natural numbers and if so would the ideas below work?
Some work: Perhaps a presheaf is similar to various numbers and there factors e.g. $14 \subset 70$ and $14 \subset 42$ as the set of its factors $\{2,7\} \subset \{2,5,7\}$ and $\{2,7\} \subset \{2,3,7\}$ so although there are other numbers that are both less then 70 and 42 such as 15, the set of numbers of the prime decomposition of 15 is not a subset of that of 70 or 42 so it can not evolve to either.
Now I am investigating a topology on finite sets of natural numbers such as $\{2,5,7\}$ and am thinking it would be something such as the $\textit{Discrete Topology}$ in which each set is both open and closed i.e. $\{5\} \subset \{2,5,7\}$ as well as all $2^n$ subsets. If the subset $\{2,5,7\}$ were open we could look at the category of open sets $Op_X$ on $X$, the objects are open sets and the morphisms are inclusions. I am thinking that maybe Let $X = \bigcup U_i; U_i \in X$ a cover of $X$. If that were the case then:
A presheaf is a functor $F$ on $X$ which associates to each open set $U_i \subset X$, a set $F(U_i)$, and to an open set $U_h \subset U_i$ a map $\rho_{U_{i}U_{h}}: F(U_i) \rightarrow F(U_h)$
Now can I or should I be able to treat take say $F(70) \rightarrow F(14)$ as the reverse morphism which $F(70) \rightarrow \{2,5,7\}$ and $F(14) \rightarrow \{2,7\}$ where we would have the morphism $\{2,5,7\} \rightarrow \{2,7\}$ which would make the triangle commute (if it does)? I see that $5$ is lost and likewise $3$ is lost in the morphism from $G(42)$ where $G \in Op_X^{op} \rightarrow Set$.
I read on another stackexchange answer that "A presheaf is a set that evolves continuously over a topological space $X$" so it seems that I am looking in to how the evolution of numbers works for base sets that are common to particular numbers. I am thinking that they would be like a "Stalk".
I know that in Category theory one is not concerned with individual elements, or from my understanding is not supposed to be. I am not sure how to go about understanding the inclusions and do understand that there could be many such inclusions e.g. there could be several possible subsets of $F(70)$ and whatever is going on would cover a set of elements that fit the criteria. Picking one of those elements though (one of a possible several) I suppose that part of what I am stuck on is that I don't see what $F$ really does in this example, i.e. would $F(70) \rightarrow \{2,5,7\}$? I am associating the number 70 to a set and then doing mappings such as $F(70) \rightarrow F(14)$ which would then just project on to a subset. What does this do for me, why is it useful?
Thanks,
Brian