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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeAug 25th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded cross-links of this ancient entry with *[[higher topos theory]]* and with *[[2-topos]]* etc. Notice that there is an old unresolved discussion box here. <a href="https://ncatlab.org/nlab/revision/diff/1-topos/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/1-topos/5">v5</a>, <a href="https://ncatlab.org/nlab/show/1-topos">current</a>

   added cross-links of this ancient entry with higher topos theory and with 2-topos etc.

   Notice that there is an old unresolved discussion box here.

   diff, v5, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeSep 18th 2022
   • (edited Sep 18th 2022)
   • PermaLink
   Author: Urs
   Format: MarkdownItexi am removing the following old query box from the entry: *** +-- {: .query} When you say 'a locale is not identical to its topos of sheaves', you mean that the following square cannot be filled in with an equivalence in $2 Cat$? $$ \array { (0,1) Top & \overset{topos\:of\:sheaves}\longrightarrow & (1,1) Top \\ \downarrow & & \downarrow \\ (0,1) Cat & \longrightarrow & 1 Cat } $$ ---Toby =-- *** But the answer to the question is: "Yes"; and maybe it's really worth making this explicit: So I have kept the diagram in the entry, now re-rendered via `tikzcd`. <a href="https://ncatlab.org/nlab/revision/diff/1-topos/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/1-topos/6">v6</a>, <a href="https://ncatlab.org/nlab/show/1-topos">current</a>

   i am removing the following old query box from the entry:

   ---

   +– {: .query} When you say ’a locale is not identical to its topos of sheaves’, you mean that the following square cannot be filled in with an equivalence in $2 Cat$?

   $$\array { (0,1) Top & \overset{topos\:of\:sheaves}\longrightarrow & (1,1) Top \\ \downarrow & & \downarrow \\ (0,1) Cat & \longrightarrow & 1 Cat }$$

   —Toby =–

   ---

   But the answer to the question is: “Yes”; and maybe it’s really worth making this explicit: So I have kept the diagram in the entry, now re-rendered via tikzcd.

   diff, v6, current