Higher stack

In mathematics, especially algebraic geometry and algebraic topology, a higher stack is a higher category generalization of a stack (a category-valued sheaf). The notion goes back to Grothendieck’s Pursuing Stacks.^[1]

Toën suggests the following principle:^[2]

As 1-stacks appear as soon as objects must be classified up to isomorphism, higher stacks appear as soon as objects must be classified up to a notion of equivalence which is weaker than the notion of isomorphism.

Sometimes a derived stack (or a spectral stack) is defined as a higher stack of some sort.