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).

The case of schemes is given a more general treatment in a fixed "geometric context", which is a category with a grothendieck topology and a fixed class of morphisms compatible with it. A scheme is then simply a "geometric variety" in the "algebro-geometric context", which is CRing^op equipped with the etale topology, where the fixed class of morphisms is the class of smooth morphisms of affines (morphisms corresponding to smooth morphisms in CRing).

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).

https://ncatlab.org/nlab/show/Master+course+on+algebraic+stacks

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).

The case of schemes is given a more general treatment in a fixed "geometric context", which is a category with a grothendieck topology and a fixed class of morphisms compatible with it. A scheme is then simply a "geometric variety" in the "algebro-geometric context", which is CRing^op equipped with the etale topology, where the fixed class of morphisms is the class of smooth morphisms of affines (morphisms corresponding to smooth morphisms in CRing).

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).

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).

The case of schemes is given a more general treatment in a fixed "geometric context", which is a category with a grothendieck topology and a fixed class of morphisms compatible with it. A scheme is then simply a "geometric variety" in the "algebro-geometric context", which is CRing^op equipped with the etale topology, where the fixed class of morphisms is the class of smooth morphisms of affines (morphisms corresponding to smooth morphisms in CRing).