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1. • CommentRowNumber1.
   • CommentAuthorThomas Holder
   • CommentTimeJul 10th 2014
   • PermaLink
   Author: Thomas Holder
   Format: MarkdownItexAs I've added some material to [[classifying topos of the theory of objects]], I've done some rewriting as well. Feel free to rectify!

   As I’ve added some material to classifying topos of the theory of objects, I’ve done some rewriting as well. Feel free to rectify!

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 10th 2014
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   Author: Todd_Trimble
   Format: MarkdownItexClicking on the link, that hasn't been created yet. What was it supposed to be?

   Clicking on the link, that hasn’t been created yet. What was it supposed to be?

3. • CommentRowNumber3.
   • CommentAuthorThomas Holder
   • CommentTimeJul 10th 2014
   • PermaLink
   Author: Thomas Holder
   Format: MarkdownItexback in the old days, they knew why they'd called it the [[classifying topos for the theory of objects|object classifier]] -much less prone to typos!

   back in the old days, they knew why they’d called it the object classifier -much less prone to typos!

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeJul 10th 2014
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   Author: David_Corfield
   Format: MarkdownItexTodd, it's [[classifying topos for the theory of objects]].

   Todd, it’s classifying topos for the theory of objects.

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeJul 10th 2014
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   Author: Mike Shulman
   Format: MarkdownItexI guess they didn't have redirects in the old days either.

   I guess they didn’t have redirects in the old days either.

6. • CommentRowNumber6.
   • CommentAuthorThomas Holder
   • CommentTimeJul 10th 2014
   • PermaLink
   Author: Thomas Holder
   Format: MarkdownItexSorry, for the practical joke! Must have been quite a shock not to find the $\infty$-object classifier under the link. Rest assured I wouldn't dare to lay hands on the $\infty$-stuff, (though I intend to have a look at the $\infty$-Sierpinski topos one of these days and see whether I can grasp enough to get an idea if the entry on [[etendue]] needs a section on $\infty$-etendues.)

   Sorry, for the practical joke! Must have been quite a shock not to find the $\infty$-object classifier under the link. Rest assured I wouldn’t dare to lay hands on the $\infty$-stuff, (though I intend to have a look at the $\infty$-Sierpinski topos one of these days and see whether I can grasp enough to get an idea if the entry on etendue needs a section on $\infty$-etendues.)

7. • CommentRowNumber7.
   • CommentAuthorThomas Holder
   • CommentTimeJul 12th 2014
   • PermaLink
   Author: Thomas Holder
   Format: MarkdownItexto the $\Gamma$-set example in [[classifying topos for the theory of objects]] I've added Blass' general remark on classifiers for Horn theories from the MO-link. I hope that I get the 'op' s right in my rewiring of the $\Gamma$-stuff: Similarly, the _theory of pointed objects_ $O_\ast$ is classified by the presheaf topos $[FinSet_\ast, Set]$ on the opposite $FinSet_\ast^{op}$ of the category of finite [[pointed sets]] whose skeleton is [[Segal's category]], hence $[FinSet_\ast, Set]$ is equivalent to the topos of "$\Gamma$-sets" (cf. [[Gamma-space]] and for its role as a classifying space the following MO-discussion: [link](http://mathoverflow.net/questions/85600/what-do-gamma-sets-classify)) . More generally, classifying toposes of [[Horn theory|universal Horn theories]] $T$ correspond to the respective toposes of covariant set-valued functors on the category of finitely presentable models of $T$ (Blass&Scedrov (1984)). Preferably, I would put the general remark about Horn theories into a footnote, but the entry has already one and I don't know how to stack them. added also some remarks about theories of pointed and inhabited objects at [[theory of objects]].

   to the $\Gamma$-set example in classifying topos for the theory of objects I’ve added Blass’ general remark on classifiers for Horn theories from the MO-link. I hope that I get the ’op’ s right in my rewiring of the $\Gamma$-stuff:

   Similarly, the theory of pointed objects $O_\ast$ is classified by the presheaf topos $[FinSet_\ast, Set]$ on the opposite $FinSet_\ast^{op}$ of the category of finite pointed sets whose skeleton is Segal’s category, hence $[FinSet_\ast, Set]$ is equivalent to the topos of “$\Gamma$-sets” (cf. Gamma-space and for its role as a classifying space the following MO-discussion: link) . More generally, classifying toposes of universal Horn theories $T$ correspond to the respective toposes of covariant set-valued functors on the category of finitely presentable models of $T$ (Blass&Scedrov (1984)).

   Preferably, I would put the general remark about Horn theories into a footnote, but the entry has already one and I don’t know how to stack them.

   added also some remarks about theories of pointed and inhabited objects at theory of objects.