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(0 1)-category-theory 2-category 2-category-theory 2-monad abelian-categories accessible adjoint algebra algebraic algebraic-geometry analysis arithmetic beauty book bug bundle categories category category-theory chern-simons-theory cohesion cohesive-homotopy-type-theory cohomology combinatorics complex-geometry conference connection constructive constructive-mathematics cosmology database deformation-theory descent differential differential-cohomology differential-geometry duality enriched enriched-category-theory enrichment examples factorization-system fiber fibration forms foundation foundations functional-analysis functor galois-theory gauge-theory gebra general topology geometric geometric-quantization geometry gravity group-theory higher higher-algebra higher-category-theory higher-geometry higher-lie-theory higher-topos-theory history homological homological-algebra homotopy homotopy-theory homotopy-type-theory homtopy-type-theory infinity-groupoid infinity-topos integration-theory internal-categories kan lie-algebras lie-theory limit limits linear linear-algebra locale localization localization-theory logic manifolds mathematics measure measure-theory mechanics meta modal-logic model model-category-theory monoidal monoidal-category monoidal-category-theory morphism n-groups newpage noncommutative noncommutative-geometry object operator operator-algebra order-theory philosophy physics predicative pretopology pro-object probability-theory quantum quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry set set-theory sheaf simplicial site space stable-homotopy-theory stack string string-theory subobject supergeometry symplectic-geometry tannaka tensor terminology theory topologica-quantum-field-theory topology topos topos-theory torsor tqft type type-theory universal weighted-limit

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 16th 2012
   • (edited Mar 16th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexGiven a topos $\mathcal{T}$, and an [[internal site]] $\mathbb{C} \in Cat(\mathcal{T})$, there should be a nice way to describe the topos of internal sheaves on $\mathbb{C}$ in terms of morphisms in $\hat Cat(\mathcal{T})$ (the 2-category of large categories in $\mathcal{T}$, being some sub-2-category of [[2-sheaves]] in $Func(\mathcal{T}^op, \hat Cat)$) from $\mathbb{C}^{op}$ to the [[codomain fibration]] / self-indexing $\mathbb{T}$ of $\mathcal{T}$, i.e. regarding an internal sheaf as a morphism $\mathbb{C}^{op} \to \mathbb{T}$ in $\hat Cat(\mathcal{T})$. This should be obvious. But is there a nice discussion from this point of view somewhere in the literature?

   Given a topos $𝒯$, and an internal site $ℂ\in \mathrm{Cat}(𝒯)$, there should be a nice way to describe the topos of internal sheaves on $ℂ$ in terms of morphisms in $\widehat{\mathrm{Cat}}(𝒯)$ (the 2-category of large categories in $𝒯$, being some sub-2-category of 2-sheaves in $\mathrm{Func}({𝒯}^{\mathrm{op}},\widehat{\mathrm{Cat}})$) from ${ℂ}^{\mathrm{op}}$ to the codomain fibration / self-indexing $𝕋$ of $𝒯$, i.e. regarding an internal sheaf as a morphism ${ℂ}^{\mathrm{op}}\to 𝕋$ in $\widehat{\mathrm{Cat}}(𝒯)$.

   This should be obvious. But is there a nice discussion from this point of view somewhere in the literature?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeMar 16th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexB2.3.13?

   B2.3.13?

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMar 16th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexAh! :-)

   Ah! :-)

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMar 16th 2012
   • (edited Mar 16th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexOkay, so I was looking for such a pointer to include in the new entry [[internal sheaf]]. But I'll announce this in another thread... [here](http://nforum.mathforge.org/discussion/3688/internal-sheaf/).

   Okay, so I was looking for such a pointer to include in the new entry internal sheaf. But I’ll announce this in another thread… here.