If I take just the definition of locally constant sheaves for algebraic varieties I get something pretty trivial (basically due to irreducibility); so how can one make a set up similar to what happen for manifolds? So here are my questions:
What is for algebraic varieties the analogue of what for manifold is locally constant sheaves?
Is there an analogue of the fact that they are basically equivalent to representations of the fundamental group? (Subquestion: What is the analogue of the fundamental group?)
Is there something like Poincaré duality about their cohomology? That is, can I relate their cohomology to some cohomology induced by the subvarieties? If yes, subquestion: What is this cohomology in the algebraic case? (In the analytic I have in mind the one coming from simplicial maps.)
Is their cohomology related to some kind of de Rham cohomology as in the manifold case? (Subquestion: What is the analogue of de Rham cohomology in the algebraic case?)
If yes: Where can I learn these things?
Thanks in advance!