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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorzskoda
   • CommentTimeJul 31st 2023
   • PermaLink
   Author: zskoda
   Format: MarkdownItexStub <a href="https://ncatlab.org/nlab/revision/schemes+as+sheaves+on+affine+schemes/1">v1</a>, <a href="https://ncatlab.org/nlab/show/schemes+as+sheaves+on+affine+schemes">current</a>

   Stub

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJul 31st 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexThis has some overlap with the existing entry *[[functorial geometry]]*.

   This has some overlap with the existing entry functorial geometry.

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeJul 31st 2023
   • (edited Jul 31st 2023)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexYes, I will add a link. We need again both entries as the [[functorial geometry]] is a general approach to spaces due Grothendieck, while the fact that schemes (as locally ringed spaces) are equivalent to a category of spaces on $Aff$ in Zariski Grothendieck topology is a nontrivial theorem which deserves its own entry (if schemes are defined in a standard way). The fundamental theorem of morphisms of schemes is just a part of that theorem which claims embedding into a presheaves, the rest is about characterizing the essential image via local representability + sheaf condition. A similar statement (with another topology) should be somewhere stated for [[algebraic space]]s.

   Yes, I will add a link. We need again both entries as the functorial geometry is a general approach to spaces due Grothendieck, while the fact that schemes (as locally ringed spaces) are equivalent to a category of spaces on $Aff$ in Zariski Grothendieck topology is a nontrivial theorem which deserves its own entry (if schemes are defined in a standard way). The fundamental theorem of morphisms of schemes is just a part of that theorem which claims embedding into a presheaves, the rest is about characterizing the essential image via local representability + sheaf condition. A similar statement (with another topology) should be somewhere stated for algebraic spaces.