Editing Algebraic stack (section)

===== Representable morphisms of categories fibered in groupoids =====
To say this morphism <math>\mathcal{U} \to \mathcal{X}</math> is smooth or surjective, we have to introduce representable morphisms.<ref>{{Cite web|title=Section 92.6 (04ST): Representable morphisms of categories fibred in groupoids—The Stacks project|url=https://stacks.math.columbia.edu/tag/04ST|access-date=2020-10-03|website=stacks.math.columbia.edu}}</ref> A morphism <math>p:\mathcal{X} \to \mathcal{Y}</math> of categories fibered in groupoids over <math>(Sch/S)_{fppf}</math> is said to be representable if given an object <math>T \to S</math> in <math>(Sch/S)_{fppf}</math> and an object <math>t \in \text{Ob}(\mathcal{Y}_T)</math> the [[2-fibered product]] <blockquote>'''<math>(Sch/T)_{fppf}\times_{t,\mathcal{Y}} \mathcal{X}_T</math>'''</blockquote>is representable by a scheme. Then, we can say the morphism of categories fibered in groupoids <math>p</math> is '''smooth and surjective''' if the associated morphism<blockquote>'''<math>(Sch/T)_{fppf}\times_{t,\mathcal{Y}} \mathcal{X}_T \to (Sch/T)_{fppf}</math>'''</blockquote>of schemes is smooth and surjective.