Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Forgot your password?
• Apply for membership

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 27th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexsomebody asked me for the proof of the claim at _[[canonical topology]]_ that for a Grothendieck topos $\mathbf{H}$ we have $\mathbf{H} \simeq Sh_{can}(\mathbf{H})$. I have added to the entry pointers to the proof in Johnstone's book, and to related discussion for $\infty$-toposes. Myself I don't have more time right now, but maybe somebody feels inspired to write out some details in the nLab entry itself?

   somebody asked me for the proof of the claim at canonical topology that for a Grothendieck topos we have .

   I have added to the entry pointers to the proof in Johnstone’s book, and to related discussion for -toposes. Myself I don’t have more time right now, but maybe somebody feels inspired to write out some details in the nLab entry itself?

2. • CommentRowNumber2.
   • CommentAuthorZhen Lin
   • CommentTimeNov 27th 2013
   • PermaLink
   Author: Zhen Lin
   Format: MarkdownItexYou could copy-paste from my answer [here](http://math.stackexchange.com/a/473614/5191). The proof in the Elephant is for a more general situation.

   You could copy-paste from my answer here. The proof in the Elephant is for a more general situation.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeNov 27th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexOkay, did that.

   Okay, did that.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeAug 26th 2021
   • (edited Aug 26th 2021)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Cynthia Lester]], *Covers in the Canonical Grothendieck Topology* ([arXiv:1909.03384](https://arxiv.org/abs/1909.03384)) * [[Cynthia Lester]], *The canonical Grothendieck topology and a homotopical analog*, 2019 ([uoregon:1794/24924](https://scholarsbank.uoregon.edu/xmlui/handle/1794/24924) [pdf](https://scholarsbank.uoregon.edu/xmlui/bitstream/handle/1794/24924/Lester_oregon_0171A_12522.pdf?sequence=1&isAllowed=y)) <a href="https://ncatlab.org/nlab/revision/diff/canonical+topology/12">diff</a>, <a href="https://ncatlab.org/nlab/revision/canonical+topology/12">v12</a>, <a href="https://ncatlab.org/nlab/show/canonical+topology">current</a>

   added pointer to:

   • Cynthia Lester, Covers in the Canonical Grothendieck Topology (arXiv:1909.03384)

   • Cynthia Lester, The canonical Grothendieck topology and a homotopical analog, 2019 (uoregon:1794/24924 pdf)

   diff, v12, current

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeApr 16th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded the arXiv link for this item: * [[Cynthia Lester]], *The canonical Grothendieck topology and a homotopical analog* (2019) &lbrack;[arXiv:1909.03188](https://arxiv.org/abs/1909.03188)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/canonical+topology/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/canonical+topology/13">v13</a>, <a href="https://ncatlab.org/nlab/show/canonical+topology">current</a>

   added the arXiv link for this item:

   • Cynthia Lester, The canonical Grothendieck topology and a homotopical analog (2019) [arXiv:1909.03188]

   diff, v13, current

6. • CommentRowNumber6.
   • CommentAuthorDean
   • CommentTimeApr 13th 2025
   • PermaLink
   Author: Dean
   Format: MarkdownItexMaybe we could include a section on the functor from copresheaves on locally compact spaces (some of which are sheaves with respect to the canonical coverage) into the category of topological spaces, and show that it is idempotent, similar to the theorem here: https://mathoverflow.net/questions/338423/cg-spaces-from-the-perspective-of-sheaves-over-compact-hausdorff-spaces This would entail "locally compactly generated" in the sense of "not necessarily hausdorff", and the same argument as the one there seems to work to show that a point-free convenient category of topological spaces is either sheaves or presheaves and has an idempotent adjunction with respect to Top.

   Maybe we could include a section on the functor from copresheaves on locally compact spaces (some of which are sheaves with respect to the canonical coverage) into the category of topological spaces, and show that it is idempotent, similar to the theorem here:

   https://mathoverflow.net/questions/338423/cg-spaces-from-the-perspective-of-sheaves-over-compact-hausdorff-spaces

   This would entail “locally compactly generated” in the sense of “not necessarily hausdorff”, and the same argument as the one there seems to work to show that a point-free convenient category of topological spaces is either sheaves or presheaves and has an idempotent adjunction with respect to Top.