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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 definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration 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 simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topological topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJan 19th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexI rewrote a good bit of the entry [[sheaf]], trying to polish and strengthen the exposition. The rewritten material is what now constituttes the section "Definition". This subsumes essentially everything that was there before, except for some scattered remarks which I removed and instad provided hyperlinks for, since they have meanwhile better discussions in other entries. I left the discussion of sheaves and the general notion of localization untouched (it is now in the section "Sheaves" and localization"). This would now need to be harmonized notationally a bit better. Maybe later.

   I rewrote a good bit of the entry sheaf, trying to polish and strengthen the exposition.

   The rewritten material is what now constituttes the section “Definition”. This subsumes essentially everything that was there before, except for some scattered remarks which I removed and instad provided hyperlinks for, since they have meanwhile better discussions in other entries.

   I left the discussion of sheaves and the general notion of localization untouched (it is now in the section “Sheaves” and localization”). This would now need to be harmonized notationally a bit better. Maybe later.

2. • CommentRowNumber2.
   • CommentAuthorsheaf
   • CommentTimeJan 10th 2025
   • PermaLink
   Author: sheaf
   Format: MarkdownItexI've been using SheaF as a quirky editor name, but I realize now its been cumulating at ['Pages contributed to'](https://ncatlab.org/nlab/author/sheaf ). Sorry about that, I'll cut that out.

   I’ve been using SheaF as a quirky editor name, but I realize now its been cumulating at ’Pages contributed to’. Sorry about that, I’ll cut that out.

3. • CommentRowNumber3.
   • CommentAuthorjonsterling
   • CommentTimeJul 2nd 2025
   • PermaLink
   Author: jonsterling
   Format: MarkdownItexDefinition 2.6 seems wrong to me unless I am misreading it. It claims that a presheaf $F$ is a sheaf when, for each covering family $\{f_i\colon U_i\to U\}$, the presheaf $F$ is right orthogonal to each $y(f_i)$. That can't be right. The correct definition involves taking the corresponding sieve $S \hookrightarrow y(U)$ and then checking that $F$ is orthogonal to that.

   Definition 2.6 seems wrong to me unless I am misreading it. It claims that a presheaf $F$ is a sheaf when, for each covering family $\{f_i\colon U_i\to U\}$, the presheaf $F$ is right orthogonal to each $y(f_i)$. That can’t be right. The correct definition involves taking the corresponding sieve $S \hookrightarrow y(U)$ and then checking that $F$ is orthogonal to that.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJul 2nd 2025
   • (edited Jul 2nd 2025)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThe preceding Remark 2.5 introduces that sieve $S(\{U_i\})$, doesn't it, and Def. 2.6 refers to that notion, it seems to me. (?) (In either case, there is much room to polish this entry further.)

   The preceding Remark 2.5 introduces that sieve $S(\{U_i\})$, doesn’t it, and Def. 2.6 refers to that notion, it seems to me. (?)

   (In either case, there is much room to polish this entry further.)