I am considering moduli spaces of sheaves on irreducible holomorphic symplectic manifolds. I haven't seen a general theory to describe the fundamental group of moduli spaces of sheaves yet. Is there such a theory? What does it mean to have a loop in such a space? Is there any connection to nonabelian Hodge theory?
I also was thinking about homotopy groups of stacks of sheaves. There are two ways I could think of to approach this.
The first is to take the algebraic stack or space and associate to it a topological stack or space. If its a topological space then you can take its homotopy groups. Then homotopy groups of a topological stack are also defined (see papers of Noohi).
The second way is more vague. One could say a map from any topological space $T$ to the stack of sheaves on $X$ is a sheaf of vector spaces on $X \times T$ such that over each point in $T$ it is a coherent sheaf on $X$ with the appropriate restrictions. But then one wants it to be flat over $T$ somehow. Once this is defined you can define homotopy groups in the usual way. They are not going to be sets though unless you mod out by some isomorphisms between sheaves.