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 $${\rm Lan}_y\delta : \widehat{\Delta}\to {\bf Topoi} : X_*\mapsto \int^{n\in\Delta}X_n\cdot {\rm Sh}(\Delta^n)$$ 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 $Y^*$?

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 $Y^\ast$?

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 simplicial object in the category of toposes.

It follows from general nonsense that the pair of functors $${\rm Lan}_y\delta : \widehat{\Delta}\to {\bf Topoi} : X_*\mapsto \int^{n\in\Delta}X_n\cdot {\rm Sh}(\Delta^n)$$ 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 simplicial topos $[n]\mapsto {\rm Sh}(\Delta^n)$ with a generic $[n]\mapsto {\rm Sh}(Y^n)$, for a generic cosimplicial (topological) space $Y^*$?

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 $${\rm Lan}_y\delta : \widehat{\Delta}\to {\bf Topoi} : X_*\mapsto \int^{n\in\Delta}X_n\cdot {\rm Sh}(\Delta^n)$$ 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 $Y^*$?