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.

Toën suggests the following principle:Bertrand Toën, Higher and Derived Stacks: A Global Overview, arXiv:math /0604504

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

References

Carlos Simpson, Algebraic (geometric) n-stacks, 1996, arXiv:alg-geom/9609014. André Hirschowitz, Carlos Simpson, Descente pour les n-champs (Descent for n-stacks), 1998, arXiv:math/9807049.

Further reading https://ncatlab.org/nlab/show/higher+stack David Carchedia, On the étale homotopy type of higher stacks, Higher Structures 5(1):121–185, 2021. https://math.stackexchange.com/questions/2493119/higher-stacks-and-bg