Formal Definitions
A stack is a category X over the étale site satisfying the following three properties.
- We can define restrictions of objects over a scheme S to objects in open coverings of S: The category X is fibered in groupoids over the étale site.
- We can patch isomorphisms: Isomorphisms are a sheaf for X.
- We can patch objects: Every descent datum is effective.
Note that the étale site is the name for the usual category of schemes considered together with the étale Grothendieck topology.
Technically an algebraic stack is a stack that can be suitably "covered" by algebraic spaces with respect to an appropriate Grothendieck topology.
Read more about this topic: Algebraic Stack
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