Timeline for A topos-theoretic thickening of the nerve functor
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Feb 23, 2014 at 8:10 | comment | added | მამუკა ჯიბლაძე | One benefit of working with general toposes instead of only sheaves over spaces is that you have much more economic models for various things. For example, you can use Set$^G$ instead of Sh(B$G$), etc. Similarly, Sh($\Delta^n$) may be replaced with Set$^{\{0<1<...<n\}}$. Once using the mentioned book of Moerdijk I worked out a description of a similar "economic realization" but with $\widehat\Gamma$ rather than $\widehat\Delta$. I believe that paper contains enough information to do it for $\widehat\Delta$ too. | |
| Sep 14, 2013 at 13:22 | comment | added | Zhen Lin | You could restrict your attention to locales instead of toposes. Then the usual abstract nonsense goes through, and you don't have to worry about the difference between 1-categories and 2-categories. (Incidentally, the embedding of locales into toposes does not preserve bicategorical colimits.) | |
| Sep 14, 2013 at 12:22 | history | edited | Ricardo Andrade | CC BY-SA 3.0 |
added top level tag
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| Sep 14, 2013 at 11:53 | comment | added | fosco | Understood. What would you propose to solve these issues? | |
| Sep 14, 2013 at 11:33 | comment | added | Zhen Lin | Well, your question contains some incorrect assertions. Firstly, $N_\delta$ is generally not a small simplicial set, if by $\mathbf{Topoi}(-, -)$ you mean the collection of geometric morphisms (even if you quotient out by natural isomorphism) – for instance, consider the classifying topos of any algebraic theory. Secondly, the category of Grothendieck toposes and geometric morphisms is not cocomplete as a 1-category (again, even if you quotient out by natural isomorphism), so you can't just say that the left adjoint exists by abstract nonsense. | |
| Sep 14, 2013 at 11:17 | comment | added | fosco | Does Moerdijk's book address my specific question? I saw that he defines precisely the same toposic-realization I had in mind, but I would like to have a couple of simple examples (the standard cosimplicial space seems perfect) | |
| Sep 14, 2013 at 11:09 | comment | added | Zhen Lin | You should have a look at Moerdijk's Classifying spaces and classifying topoi. | |
| Sep 14, 2013 at 11:08 | history | edited | fosco | CC BY-SA 3.0 |
Corrected a minor typo
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| Sep 14, 2013 at 10:57 | comment | added | Todd Trimble | Minor comment: I'd say that $\delta$ is a cosimplicial topos, because it is a composite of functors $\mathbf{\Delta} \to \textbf{Top} \to \textbf{Topoi}$, both of which are covariant. | |
| Sep 14, 2013 at 10:43 | history | asked | fosco | CC BY-SA 3.0 |