It is conjectured there is a final theory of higher stacks which model homotopy types. Grothendieck conjectured these would be modelled by globular groupoids, or a weak form of their definition. Simpson gives a useful definition in the spirit of Grothendieck's ideas. Recall that an algebraic stack (here a 1-stack) is called representable if the fiber product of any two schemes is isomorphic to a scheme.Which can be checked by looking at the diagonal morphism and checking if that itself is representable. Check out https://math.dartmouth.edu/~jvoight/notes/moduli-red-harvard.pdf for more information If we take the ansatz that a 0-stack is just an algebraic space and a 1-stack is just a stack, we can recursively define an n-stack as an object such that the fiber product along any two schemes is an (n-1)-stack. If we go back to the definition of an algebraic stack, this new definition agrees.
