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Let $\Delta$ be the standard cosimplicial space sending $[n]\mapsto \Delta^n\subset \mathbb R^{n+1}$.

Then the correspondence $\delta\colon [n]\mapsto {\rm Sh}(\Delta^n)$ defines a cosimplicial object in the category of toposes.

It follows from general nonsense that the pair of functors $${\rm Lan}_y\delta : \widehat{\Delta}\to {\bf Topoi} : X_\ast\mapsto \int^{n\in\Delta}X_n\cdot {\rm Sh}(\Delta^n)$$ where $y\colon\Delta\to\widehat{\Delta}$ is the Yoneda embedding, and $$N_\delta : {\bf Topoi} \to \widehat{\Delta} : {\cal E}\mapsto\Big([n]\mapsto {\bf Topoi}({\rm Sh}(\Delta^n), \cal E)\Big)$$ are mutually adjoint.

Has this construction a particular name? Has it a particular interest? Can it be characterized in a more hands-on way? What if I replace the easy cosimplicial topos $[n]\mapsto {\rm Sh}(\Delta^n)$ with a generic $[n]\mapsto {\rm Sh}(Y^n)$, for a generic cosimplicial (topological) space $Y^\ast$?

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  • $\begingroup$ 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. $\endgroup$
    – Todd Trimble
    Commented Sep 14, 2013 at 10:57
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    $\begingroup$ You should have a look at Moerdijk's Classifying spaces and classifying topoi. $\endgroup$
    – Zhen Lin
    Commented Sep 14, 2013 at 11:09
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    $\begingroup$ 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. $\endgroup$
    – Zhen Lin
    Commented Sep 14, 2013 at 11:33
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    $\begingroup$ 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.) $\endgroup$
    – Zhen Lin
    Commented Sep 14, 2013 at 13:22
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    $\begingroup$ 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. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Feb 23, 2014 at 8:10

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