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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeAug 8th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have splitt off from _[[classifying topos]]_ an entry _[[classifying topos for the theory of objects]]_ and added the statement about the relation to finitary monads.

   I have splitt off from classifying topos an entry classifying topos for the theory of objects and added the statement about the relation to finitary monads.

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 8th 2013
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   Author: Todd_Trimble
   Format: MarkdownItexI added to your remark a related point of view, and linked to some notes of mine.

   I added to your remark a related point of view, and linked to some notes of mine.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeAug 8th 2013
   • (edited Aug 8th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexAh, thanks, that's nice!! We should add some kind of remark concerning $[\mathbb{P}^{op}, Set]$ also as an Example at _[[monoidal topos]]_. And I guess we should still add a pointer to an explanation of $\mathbb{P}$ to the entry. I can't right now, though, am in a rush here...

   Ah, thanks, that’s nice!!

   We should add some kind of remark concerning $[\mathbb{P}^{op}, Set]$ also as an Example at monoidal topos. And I guess we should still add a pointer to an explanation of $\mathbb{P}$ to the entry. I can’t right now, though, am in a rush here…

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeAug 8th 2013
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   Author: Mike Shulman
   Format: MarkdownItexNice page, thanks.

   Nice page, thanks.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeAug 9th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added a stub for _[[permutation category]]_, just for completeness. In the course of this I noticed that we already have _[[braid category]]_! I have now cross-linked that a bit more such as to make it easier to find.

   I have added a stub for permutation category, just for completeness. In the course of this I noticed that we already have braid category! I have now cross-linked that a bit more such as to make it easier to find.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeNov 26th 2014
   • (edited Nov 26th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexSpelled out [here](http://ncatlab.org/nlab/show/classifying+topos+for+the+theory+of+objects#Idea) the argument for why $PSh(FinSet^{op})$ is the classifying topos for objects by pointing to [this fact](http://ncatlab.org/nlab/show/morphism+of+sites#LemmaForClassifyingToposes). (Just for completeness.) Also added the remark that similarly $PSh((FinSet_\ast)^{op})$ is the classifying topos for pointed objects.

   Spelled out here the argument for why $PSh(FinSet^{op})$ is the classifying topos for objects by pointing to this fact.

   (Just for completeness.)

   Also added the remark that similarly $PSh((FinSet_\ast)^{op})$ is the classifying topos for pointed objects.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeNov 26th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded also at _[[classifying topos for the theory of objects]]_ remarks on the $\infty$-case: *** Similarly * $PSh(FinSet_\ast^{op})$ is the classifying topos for [[pointed objects]]. * write $Fin\infty Grpd$ for the [[full sub-(∞,1)-category]] on [[∞Grpd]] which is generated under [[finite (∞,1)-colimits]] from the point $\ast$ ([[Higher Algebra|HA, def. 1.4.2.8]]), then the [[(∞,1)-presheaf (∞,1)-topos]] $PSh_\infty(Fin\infty Grpd^{op})$ is the [[classifying (∞,1)-topos]] for objects; * write $Fin\infty Grpd_\ast$ for pointed finite $\infty$-groupoids in this sense, then $PSh_\infty((Fin\infty Grpd_\ast)^{op})$ is the classifying $(\infty,1)$-topos for pointed objects. See also at _[[spectrum object]]_ _[via excisive functors](spectrum%20object#ViaExcisiveFunctors)_.

   added also at classifying topos for the theory of objects remarks on the $\infty$-case:

   ---

   Similarly

   • $PSh(FinSet_\ast^{op})$ is the classifying topos for pointed objects.

   • write $Fin\infty Grpd$ for the full sub-(∞,1)-category on ∞Grpd which is generated under finite (∞,1)-colimits from the point $\ast$ (HA, def. 1.4.2.8), then the (∞,1)-presheaf (∞,1)-topos $PSh_\infty(Fin\infty Grpd^{op})$ is the classifying (∞,1)-topos for objects;

   • write $Fin\infty Grpd_\ast$ for pointed finite $\infty$-groupoids in this sense, then $PSh_\infty((Fin\infty Grpd_\ast)^{op})$ is the classifying $(\infty,1)$-topos for pointed objects. See also at spectrum object via excisive functors.

8. • CommentRowNumber8.
   • CommentAuthornLab edit announcer
   • CommentTimeNov 27th 2018
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItex* Added remarks on "finite" in FinSet. * Added remarks on object classifier as generalized space of "sets" Steve Vickers <a href="https://ncatlab.org/nlab/revision/diff/classifying+topos+for+the+theory+of+objects/21">diff</a>, <a href="https://ncatlab.org/nlab/revision/classifying+topos+for+the+theory+of+objects/21">v21</a>, <a href="https://ncatlab.org/nlab/show/classifying+topos+for+the+theory+of+objects">current</a>

   • Added remarks on “finite” in FinSet.
   • Added remarks on object classifier as generalized space of “sets”

   Steve Vickers

   diff, v21, current

9. • CommentRowNumber9.
   • CommentAuthornLab edit announcer
   • CommentTimeNov 27th 2018
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexAdded more concrete construction of the pointed set classifier. Steve Vickers <a href="https://ncatlab.org/nlab/revision/diff/classifying+topos+for+the+theory+of+objects/21">diff</a>, <a href="https://ncatlab.org/nlab/revision/classifying+topos+for+the+theory+of+objects/21">v21</a>, <a href="https://ncatlab.org/nlab/show/classifying+topos+for+the+theory+of+objects">current</a>

   Added more concrete construction of the pointed set classifier.

   Steve Vickers

   diff, v21, current

10. • CommentRowNumber10.
    • CommentAuthorThomas Holder
    • CommentTimeNov 27th 2018
    • PermaLink
    Author: Thomas Holder
    Format: MarkdownItexFixed some typos and highlighted (hopefully in a correct way) the role of the category of elements in this. It would be nice if one could bring the description of the generic object in section 2 notationally and conceptually in line with the description given in the current section 3 ! Thanks anyway for polishing the entries on 'geometric' logics and adding clarifications ! <a href="https://ncatlab.org/nlab/revision/diff/classifying+topos+for+the+theory+of+objects/22">diff</a>, <a href="https://ncatlab.org/nlab/revision/classifying+topos+for+the+theory+of+objects/22">v22</a>, <a href="https://ncatlab.org/nlab/show/classifying+topos+for+the+theory+of+objects">current</a>

    Fixed some typos and highlighted (hopefully in a correct way) the role of the category of elements in this. It would be nice if one could bring the description of the generic object in section 2 notationally and conceptually in line with the description given in the current section 3 ! Thanks anyway for polishing the entries on ’geometric’ logics and adding clarifications !

    diff, v22, current

11. • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 27th 2018
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexLinked to the page [[finite set]] in the comments about what finite sets are meant for $[FinSet, Set]$. <a href="https://ncatlab.org/nlab/revision/diff/classifying+topos+for+the+theory+of+objects/24">diff</a>, <a href="https://ncatlab.org/nlab/revision/classifying+topos+for+the+theory+of+objects/24">v24</a>, <a href="https://ncatlab.org/nlab/show/classifying+topos+for+the+theory+of+objects">current</a>

    Linked to the page finite set in the comments about what finite sets are meant for $[FinSet, Set]$.

    diff, v24, current