In algebraic geometry, an etale sheaf on a Noetherian scheme is called constructible if the scheme has a finite stratification by locally closed subschemes such that the pullback of the sheaf to each of the subschemes is lcc (locally constant constructible). This definition seems to make sense both for sheaves of abelian groups and sheaves of sets. There is a similar definition in algebraic topology and I have no doubt that analogues of my questions can be formulated in the algebraic topology world.
I don't really have such a good feeling for these constructible sheaves, and, when re-reading the definition recently, I realised that I could easily formulate simple questions about building such things, that I could not answer. Let me ask the most general question first, which might be too general to have a reasonable answer.
General question. Say I have a Noetherian scheme $X$ and I am given a stratification by finitely many locally closed subschemes $U_i$. Say I have lcc sheaves on each of the $U_i$. What additional data do I need to glue these sheaves together to get a constructible sheaf on $X$?
If I am lucky then there is some slick cocycle-related answer and we can all go home happy. But I am not a big fan of these general types of question, so here are a few completely specific special cases---the simplest non-trivial (as far as I am concerned) ones, which I am already a bit hesitant on.
Q2 Let $X$ be the affine line over the complexes. Let $P$ denote the origin and let $U$ be $X\backslash P$. Up to isomorphism, how many constructible sheaves of sets are there on $X$, whose pullback to $U$ is the constant sheaf of sets of size 1 (represented by the identity $U\to U$) and whose pullback to $P$ is the constant sheaf of sets of size 2 (represented by $P\coprod P\to P$)?
There are, I guess, two notions of isomorphism: one can imagine that the stalk above $P$ is a given set of size 2 and demand that isomorphisms must induce the identity on this set, or one can just demand that the isomorphism induces a bijection on the stalk at $P$. I've noticed that different people interpret the question in different ways so let me just ask that the isomorphism induces an arbitrary bijection, to fix ideas.
I am a little scared of sheaves of sets because the initial and the terminal objects in the category of sets are not isomorphic, and this tripped me up a bit when I was trying to answer Q2. We don't have such a problem in the category of abelian groups so perhaps that's a safer place to be:
Q3 $X$ the affine line, $P$ the origin and $U$ the rest, as before. How many isomorphism classes of constructible sheaves of abelian groups on $X$ are there, whose stalk at $P$ is cyclic of order 4 and whose pullback to $U$ is constant and cyclic of order 2?
What is the motivation for Q2 and Q3? There are several answers. The motivation is certainly not "I need to know the answers to these questions!". One possible answer is "I feel like I cannot say I understand constructible sheaves unless I can answer these basic questions". Another is "what I really want to know is the answer to the general question, but I am worried the general question is too vague so I am asking special cases of it in order to get a feeling for the general question".