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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 22nd 2010
   • PermaLink
   Author: Urs
   Format: Markdowncreated [[topological localization]]

   created topological localization

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeFeb 22nd 2010
   • (edited Feb 22nd 2010)
   • PermaLink
   Author: Urs
   Format: Markdownadded Definition-section to [[(infinity,1)-category of (infinity,1)-sheaves]] HELP: the floating TOC here and at other entries is still broken. Andrew said this was due to some software update. I forget what I need to do to fix it....

   added Definition-section to (infinity,1)-category of (infinity,1)-sheaves

   HELP: the floating TOC here and at other entries is still broken. Andrew said this was due to some software update. I forget what I need to do to fix it....

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 13th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexhave polished and expanded the entry [[topological localization]]. Will try to add details on proofs later.

   have polished and expanded the entry topological localization. Will try to add details on proofs later.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeNov 2nd 2014
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI have added to [[topological localization]] a warning that left exactness has to be required as part of the definition; a general localization at a set of monos need not be left exact.

   I have added to topological localization a warning that left exactness has to be required as part of the definition; a general localization at a set of monos need not be left exact.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeNov 2nd 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexOh, thanks for catching this.

   Oh, thanks for catching this.

6. • CommentRowNumber6.
   • CommentAuthorDean
   • CommentTimeApr 29th 2025
   • (edited Apr 29th 2025)
   • PermaLink
   Author: Dean
   Format: MarkdownItexA topological localization is related to stability under pullbacks, but does anything change when we consider stability under homotopy pullbacks instead? I was hoping that there would be a modification to the concept of a "left exact left adjoint up to homotopy" (sheafification defined up to homotopy defined by iterating the plus construction n times on (n,1)-skeleta) would produce a notion of topology involving a homotopy-pullback based notion of "infinity effective epimorphism". That could be similar to the condition that the left adjoint preserve path space path : X^I -> X x X instead of being topological.

   A topological localization is related to stability under pullbacks, but does anything change when we consider stability under homotopy pullbacks instead?

   I was hoping that there would be a modification to the concept of a “left exact left adjoint up to homotopy” (sheafification defined up to homotopy defined by iterating the plus construction n times on (n,1)-skeleta) would produce a notion of topology involving a homotopy-pullback based notion of “infinity effective epimorphism”. That could be similar to the condition that the left adjoint preserve path space path : X^I -> X x X instead of being topological.