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Harry Gindi
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Toen's notes on stacks construct the category of schemes as the category of etale sheaves (presheaves satisfying descent in the etale topology) on CRing^op with a jointly surjective cover by smooth monomorphisms (exercise: show that smooth monomorphisms of affines are etale) of representable functors (i.e. affines).

http://www.math.univ-toulouse.fr/~toen/m2.html

He constructs algebraic spaces in a similar way, then constructs algebraic stacks using the same approach after a digression into homotopical descent theory (which generalizes readily to the approach taken in Toen-Vezzosi (Homotopical Algebraic Geometry).

Harry Gindi
  • 19.6k
  • 16
  • 123
  • 215