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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 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 internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure 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 topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJan 6th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexhave made [[locally infinity-connected (infinity,1)-site]]

   have made locally infinity-connected (infinity,1)-site

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeNov 5th 2017
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI don't understand the definition at [[locally infinity-connected (infinity,1)-site]]. Why does it refer to hypercompletion? It seems to me that it should clearly be about the ordinary $\infty$-topos of sheaves, not its hypercompletion. Also, the definition should really be an explicit one in terms of coverages, like that of a [[locally connected site]]. The obvious corresponding condition is that all covering sieves have contractible nerves; is that sufficient?

   I don’t understand the definition at locally infinity-connected (infinity,1)-site. Why does it refer to hypercompletion? It seems to me that it should clearly be about the ordinary $\infty$-topos of sheaves, not its hypercompletion.

   Also, the definition should really be an explicit one in terms of coverages, like that of a locally connected site. The obvious corresponding condition is that all covering sieves have contractible nerves; is that sufficient?

3. • CommentRowNumber3.
   • CommentAuthorMarc Hoyois
   • CommentTimeNov 5th 2017
   • PermaLink
   Author: Marc Hoyois
   Format: MarkdownItex@Mike It is indeed necessary and sufficient, since a constant presheaf with value $X$ is a sheaf iff $X=lim_{R} X=Map(N(R),X)$ for every covering sieve R.

   @Mike It is indeed necessary and sufficient, since a constant presheaf with value $X$ is a sheaf iff $X=lim_{R} X=Map(N(R),X)$ for every covering sieve R.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeNov 5th 2017
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI'm having trouble typechecking your equation; can you elaborate it for me?

   I’m having trouble typechecking your equation; can you elaborate it for me?

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeNov 5th 2017
   • (edited Nov 5th 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks for the alert. I forget why I mentioned hypercompleteness when I created the entry. I suppose one may as well mention it, if it is what one is after, but since the remainder of the entry does not refer back to it anyway, we can as well leave it open. So I have removed the adjective. Then I noticed that in prop. 3.1 the "$X$" suddenly turned into a "$C$". I have fixed that now, too. In #3, the first equivalence is the sheaf condition, and the second its unwinding, where due to the constancy of the presheaf, on the left "$X$" denotes the presheaf while on the right it denotes the $\infty$-groupoid that it is constant on.

   Thanks for the alert. I forget why I mentioned hypercompleteness when I created the entry. I suppose one may as well mention it, if it is what one is after, but since the remainder of the entry does not refer back to it anyway, we can as well leave it open. So I have removed the adjective.

   Then I noticed that in prop. 3.1 the “$X$” suddenly turned into a “$C$”. I have fixed that now, too.

   In #3, the first equivalence is the sheaf condition, and the second its unwinding, where due to the constancy of the presheaf, on the left “$X$” denotes the presheaf while on the right it denotes the $\infty$-groupoid that it is constant on.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeNov 6th 2017
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexThanks, I have added this other characterization to the page.

   Thanks, I have added this other characterization to the page.