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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeDec 16th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexbrushed up [[ringed topos]] a little, added the version over any Lawvere theory and linked it to "related concepts" (for use at [[Tannaka duality for geometric stacks]]).

   brushed up ringed topos a little, added the version over any Lawvere theory and linked it to “related concepts” (for use at Tannaka duality for geometric stacks).

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeDec 16th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItextouched also [[ringed space]], [[locally ringed space]], [[ringed site]], [[structured (infinity,1)-topos]] and made sure they all link to each other

   touched also ringed space, locally ringed space, ringed site, structured (infinity,1)-topos and made sure they all link to each other

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMar 1st 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have made the definition of morphisms of ringed toposes explicit at [[ringed topos]].

   I have made the definition of morphisms of ringed toposes explicit at ringed topos.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJul 5th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexI was going to type something on limits of ringed toposes, but maybe give me a sanity check: the limit of a diagram of ringed toposes $(\mathcal{X}_i, \mathcal{O}_{\mathcal{X_i}})$ ( _not_ locally ringed, mind you) should be formed by 1. forming the limit of the underlying toposes, ${\lim_\leftarrow}_i (\mathcal{X}_i, \mathcal{O}_{\mathcal{X}_i}) \stackrel{p_i}{\to} (\mathcal{X}_i, \mathcal{O}_{\mathcal{X}_i})$ 1. then forming the colimit inside the limiting topos of the inverse images of all the internal ring objects ${\lim_\to} p_i^* \mathcal{O}_{\mathcal{X}_i} \in {\lim_\leftarrow}_i (\mathcal{X}_i, \mathcal{O}_{\mathcal{X}_i})$. Because with this the universal property is immediately checked. Unless I am mixed up.

   I was going to type something on limits of ringed toposes, but maybe give me a sanity check:

   the limit of a diagram of ringed toposes ( not locally ringed, mind you) should be formed by

   1. forming the limit of the underlying toposes, ${\lim_\leftarrow}_i (\mathcal{X}_i, \mathcal{O}_{\mathcal{X}_i}) \stackrel{p_i}{\to} (\mathcal{X}_i, \mathcal{O}_{\mathcal{X}_i})$

   2. then forming the colimit inside the limiting topos of the inverse images of all the internal ring objects ${\lim_\to} p_i^* \mathcal{O}_{\mathcal{X}_i} \in {\lim_\leftarrow}_i (\mathcal{X}_i, \mathcal{O}_{\mathcal{X}_i})$.

   Because with this the universal property is immediately checked. Unless I am mixed up.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeJul 5th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have tried to spell out details in a new subsection [Properties -- Limits and colimits](http://ncatlab.org/nlab/show/ringed+topos#LimitsAndColimits).

   I have tried to spell out details in a new subsection Properties – Limits and colimits.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeJul 5th 2011
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexThat does seem to work. I wonder what the generality of that is.

   That does seem to work. I wonder what the generality of that is.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJul 5th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItex> That does seem to work. Thanks for checking. > I wonder what the generality of that is. I was wondering, too. But for the moment I am content with this simple case. What I have found in the literature on this has been surprisingly disappointing. But I haven't looked at Grothendieck's original notes yet.

   That does seem to work.

   Thanks for checking.

   I wonder what the generality of that is.

   I was wondering, too. But for the moment I am content with this simple case.

   What I have found in the literature on this has been surprisingly disappointing. But I haven’t looked at Grothendieck’s original notes yet.

8. • CommentRowNumber8.
   • CommentAuthorzskoda
   • CommentTimeJul 5th 2011
   • PermaLink
   Author: zskoda
   Format: MarkdownItexIn addition to the related material in SGA4, let me mention that the [[relative scheme]]s over ringed toposes were developed by M. Hakim in her thesis under Grothendieck's guidance. That thesis also has a tiny bit of 2-categorical descent, I think. * Monique Hakim, _Topos annelés et schémas relatifs_, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 64. Springer 1972. vi+160 pp. By the way, interesting historical reminiscences from 2007 (Drinfel'd, Bloch, Illusie) about Grothendieck and his work with students, which I have not seen before: [pdf](http://www.math.uchicago.edu/~mitya/langlands/reminiscences1.pdf). I have added this to [[Alexander Grothendieck]].

   In addition to the related material in SGA4, let me mention that the relative schemes over ringed toposes were developed by M. Hakim in her thesis under Grothendieck’s guidance. That thesis also has a tiny bit of 2-categorical descent, I think.

   • Monique Hakim, Topos annelés et schémas relatifs, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 64. Springer 1972. vi+160 pp.

   By the way, interesting historical reminiscences from 2007 (Drinfel’d, Bloch, Illusie) about Grothendieck and his work with students, which I have not seen before: pdf.

   I have added this to Alexander Grothendieck.

9. • CommentRowNumber9.
   • CommentAuthoradhalanay
   • CommentTimeSep 4th 2018
   • PermaLink
   Author: adhalanay
   Format: MarkdownItexCorrected a small misprint in references. <a href="https://ncatlab.org/nlab/revision/diff/ringed+topos/17">diff</a>, <a href="https://ncatlab.org/nlab/revision/ringed+topos/17">v17</a>, <a href="https://ncatlab.org/nlab/show/ringed+topos">current</a>

   Corrected a small misprint in references.

   diff, v17, current

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeApr 16th 2023
    • (edited Apr 16th 2023)
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded DOI-link to: * [[Monique Hakim]], *Topos annelés et schémas relatifs*, Ergebnisse der Mathematik und ihrer Grenzgebiete **64**, Springer (1972) &lbrack;[doi:10.1007/978-3-662-59155-0](https://doi.org/10.1007/978-3-662-59155-0)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/ringed+topos/18">diff</a>, <a href="https://ncatlab.org/nlab/revision/ringed+topos/18">v18</a>, <a href="https://ncatlab.org/nlab/show/ringed+topos">current</a>

    added DOI-link to:

    • Monique Hakim, Topos annelés et schémas relatifs, Ergebnisse der Mathematik und ihrer Grenzgebiete 64, Springer (1972) [doi:10.1007/978-3-662-59155-0]

    diff, v18, current

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeApr 16th 2023
    • PermaLink
    Author: Urs
    Format: MarkdownItexfixed the previously broken link for * [[eom]], _[Ringed spaces](https://encyclopediaofmath.org/wiki/Ringed_space)_ <a href="https://ncatlab.org/nlab/revision/diff/ringed+topos/18">diff</a>, <a href="https://ncatlab.org/nlab/revision/ringed+topos/18">v18</a>, <a href="https://ncatlab.org/nlab/show/ringed+topos">current</a>

    fixed the previously broken link for

    • eom, Ringed spaces

    diff, v18, current

12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeApr 16th 2023
    • (edited Apr 16th 2023)
    • PermaLink
    Author: Urs
    Format: MarkdownItexexpanded out the previous reference item "[[SGA]] IV" to: * [[Alexander Grothendieck]], [[Jean-Louis Verdier]], *Topos annelés, localisation dans les topos annelés*, Section 11 in Exposé iv in: [[M. Artin]] et al., *Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos*, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 ([[SGA 4]]), Lecture Notes in Mathematics __269__, Springer (1972) &lbrack;[doi:10.1007/BFb0081551](https://link.springer.com/book/10.1007/BFb0081551), [pdf](http://www.cmls.polytechnique.fr/perso/laszlo/sga4/SGA4-1/sga41.pdf)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/ringed+topos/18">diff</a>, <a href="https://ncatlab.org/nlab/revision/ringed+topos/18">v18</a>, <a href="https://ncatlab.org/nlab/show/ringed+topos">current</a>

    expanded out the previous reference item “SGA IV” to:

    • Alexander Grothendieck, Jean-Louis Verdier, Topos annelés, localisation dans les topos annelés, Section 11 in Exposé iv in: M. Artin et al., Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Lecture Notes in Mathematics 269, Springer (1972) [doi:10.1007/BFb0081551, pdf]

    diff, v18, current