If X is a topological space then there is an equivalence between the category, π_¹(X)- Set, of actions of the fundamental group of X on sets, and the category of covering spaces on X. Moreover the latter is also equivalent to the category of locally constant sheaves on X.
Grothendieck has conjectured that this should be the 'n=1' case of a result which is true for all n, and it is the 'n=2' case we look at in this thesis.
The desired generalisation should replace actions of the group π_¹(X) (which is an
algebraic model for the 1-type of X) by actions of a crossed module (i.e., by an algebraic model for the 2-type) on groupoids; 'locally constant sheaves of sets' by 'locally constant stacks of groupoids'; and 'covering space' by a locally trivial object whose fibres are groupoids.
This last object we handle using the machinery of simplicial fibre bundles (twisted Cartesian products) and formal maps, building a simplicial object, Z(λ), where the fibre is now a (nerve of) a groupoid. To interpret Z(λ) as a stack, we show that just as sheaves on X are equivalent to étale spaces, we can define a notion of 2-étale space corresponding to stacks and show that from Z(λ) we can construct a locally constant stack on X.