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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
   • CommentTimeNov 6th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexI got annoyed with the fact that these links did not exist, and so I created now stubs for them: * [[2-topos theory]] * [[(infinity,1)-topos theory]] To the latter entry I moved the references on $(\infty,1)$-topos theory that had been lsited at [[higher topos theory]].

   I got annoyed with the fact that these links did not exist, and so I created now stubs for them:

   • 2-topos theory

   • (infinity,1)-topos theory

   To the latter entry I moved the references on $(\infty,1)$-topos theory that had been lsited at higher topos theory.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeNov 7th 2010
   • PermaLink
   Author: zskoda
   Format: MarkdownItexMike has in his pages much about his thinking about 2-topoi...

   Mike has in his pages much about his thinking about 2-topoi…

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeNov 7th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexYeah, and I've been planning to move it to the main nLab for a while. But for now, a link will have to do.

   Yeah, and I’ve been planning to move it to the main nLab for a while. But for now, a link will have to do.

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 24th 2021
   • (edited Aug 24th 2021)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexSince Joel Hamkins is looking at partial orders at his [blog](http://jdh.hamkins.org/an-algebra-of-orders/), I was wondering about [[Pos]] and its structure. Presumably it should be the archetypal (1,2)-topos, whatever that is. Those (non-commutative) ordered sum and product presumably are explicable category-theoretically.

   Since Joel Hamkins is looking at partial orders at his blog, I was wondering about Pos and its structure. Presumably it should be the archetypal (1,2)-topos, whatever that is.

   Those (non-commutative) ordered sum and product presumably are explicable category-theoretically.

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeAug 24th 2021
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexThe ordered sum is the collage of the terminal profunctor, and also the cocomma object of the product projections $A \leftarrow A\times B \to B$ (a.k.a. the join). The best I can do right now for the ordered product is that it's the (generalized) Grothendieck construction of the lax functor $B \to Prof$ that sends all objects to $A$ and all nonidentity morphisms to the terminal profunctor. Note that well-definedness of the latter depends on antisymmetry of $B$; it's not clear to me that it has a generalization to non-posets, or even to other (1,2)-toposes (since the definition as I gave it uses excluded middle).

   The ordered sum is the collage of the terminal profunctor, and also the cocomma object of the product projections $A \leftarrow A\times B \to B$ (a.k.a. the join).

   The best I can do right now for the ordered product is that it’s the (generalized) Grothendieck construction of the lax functor $B \to Prof$ that sends all objects to $A$ and all nonidentity morphisms to the terminal profunctor. Note that well-definedness of the latter depends on antisymmetry of $B$; it’s not clear to me that it has a generalization to non-posets, or even to other (1,2)-toposes (since the definition as I gave it uses excluded middle).

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 24th 2021
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexOk, thanks! So is there anything to learn from it being a $(1,2)$-topos, a topic we have absolutely nothing about on nLab.

   Ok, thanks! So is there anything to learn from it being a $(1,2)$-topos, a topic we have absolutely nothing about on nLab.

7. • CommentRowNumber7.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 25th 2021
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexTalking with John Baez elsewhere, seeing these poset composition operations in terms of operads might shed some light.

   Talking with John Baez elsewhere, seeing these poset composition operations in terms of operads might shed some light.