# Higher stack

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Higher_stack
> Markdown URL: https://mediated.wiki/source/Higher_stack.md
> Source: https://en.wikipedia.org/wiki/Higher_stack
> Source revision: 1339170805
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

In mathematics, especially [algebraic geometry](/source/algebraic_geometry) and [algebraic topology](/source/algebraic_topology), a '''higher stack''' is a [higher category](/source/higher_category) generalization of a [stack](/source/stack_(mathematics)) (a category-valued sheaf). The notion goes back to Grothendieck’s ''[Pursuing Stacks](/source/Pursuing_Stacks)''.<ref>{{harvnb|Töen|2014|loc=§ 1. Selected pieces of history.}}</ref>

Toën suggests the following principle:<ref>Bertrand Toën, Higher and Derived Stacks: A Global Overview, arXiv:math /0604504</ref>
{{blockquote|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](/source/derived_stack) (or a [spectral stack](/source/spectral_stack)) is defined as a higher stack of some sort.

== References ==
{{reflist}}
*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.
*{{cite journal | first = Bertrand | last = Töen | author-link = Bertrand Toën | title = Derived algebraic geometry | journal = EMS Surveys in Mathematical Sciences | volume = 1 | issue = 2 | year = 2014 | pp = 153–240 | doi = 10.4171/EMSS/4 | url = https://ems.press/journals/emss/articles/12837 | arxiv = 1401.1044 }}

== 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://higher-structures.math.cas.cz/api/files/issues/Vol5Iss1/Carchedi]
* https://math.stackexchange.com/questions/2493119/higher-stacks-and-bg

{{categorytheory-stub}}

Category:Higher category theory
Category:Stacks (mathematics)

---
Adapted from the Wikipedia article [Higher stack](https://en.wikipedia.org/wiki/Higher_stack) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Higher_stack?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.