Editing Algebraic stack (section)

=== Motivation ===
One of the motivating examples of an algebraic stack is to consider a [[groupoid scheme]] <math>(R,U,s,t,m)</math> over a fixed scheme <math>S</math>. For example, if <math>R = \mu_n\times_S\mathbb{A}^n_S</math> (where <math>\mu_n</math> is the [[group scheme]] of roots of unity), <math>U = \mathbb{A}^n_S</math>, <math>s = \text{pr}_U</math> is the projection map, <math>t</math> is the group action<blockquote><math>\zeta_n \cdot (x_1,\ldots, x_n)=(\zeta_n  x_1,\ldots,\zeta_n x_n)</math></blockquote>and <math>m</math> is the multiplication map<blockquote><math>m: (\mu_n\times_S \mathbb{A}^n_S)\times_{\mu_n\times_S \mathbb{A}^n_S} (\mu_n\times_S \mathbb{A}^n_S) \to \mu_n\times_S \mathbb{A}^n_S</math></blockquote>on <math>\mu_n</math>. Then, given an <math>S</math>-scheme <math>\pi:X\to S</math>, the groupoid scheme <math>(R(X),U(X),s,t,m)</math> forms a groupoid (where <math>R,U</math> are their associated functors). Moreover, this construction is functorial on <math>(\mathrm{Sch}/S)</math> forming a contravariant [[2-functor]]<blockquote><math>(R(-),U(-),s,t,m): (\mathrm{Sch}/S)^\mathrm{op} \to \text{Cat}</math></blockquote>where <math>\text{Cat}</math> is the [[2 category|2-category]] of [[Small category|small categories]]. Another way to view this is as a [[fibred category]] <math>[U/R] \to (\mathrm{Sch}/S)</math> through the [[Grothendieck construction]]. Getting the correct technical conditions, such as the [[Grothendieck topology]] on <math>(\mathrm{Sch}/S)</math>, gives the definition of an algebraic stack. For instance, in the associated groupoid of <math>k</math>-points for a field <math>k</math>, over the origin object <math>0 \in \mathbb{A}^n_S(k)</math> there is the groupoid of automorphisms <math>\mu_n(k)</math>. However, in order to get an algebraic stack from <math>[U/R]</math>, and not just a stack, there are additional technical hypotheses required for <math>[U/R]</math>.<ref>{{Cite web|title=Section 92.16 (04T3): From an algebraic stack to a presentation—The Stacks project|url=https://stacks.math.columbia.edu/tag/04T3|access-date=2020-08-29|website=stacks.math.columbia.edu}}</ref>