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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 definitions 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 k-theory lie-theory limits linear linear-algebra locale localization logic mathematics 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 object 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
   • CommentTimeMar 16th 2012
   • (edited Mar 16th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexat [[2-adjunction]] I would like to list a bunch of 2-category theoretic analogs of standard facts about ordinary adjunctions. Such as: a right adjoint is a full and faithful 2-functor precisely if the counit of the 2-adjunction is an equivalence, etc. But I haven't really thought deeply about 2-adjunctions myself yet. Is there some reference where we could take such a list of properties from?

   at 2-adjunction I would like to list a bunch of 2-category theoretic analogs of standard facts about ordinary adjunctions. Such as: a right adjoint is a full and faithful 2-functor precisely if the counit of the 2-adjunction is an equivalence, etc.

   But I haven’t really thought deeply about 2-adjunctions myself yet. Is there some reference where we could take such a list of properties from?

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeMar 16th 2012
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexIs there a nice pictorial representation of the equations for 2-adjunctions, just as in the case of ordinary adjunctions ([week 174](http://math.ucr.edu/home/baez/week174.html))?

   Is there a nice pictorial representation of the equations for 2-adjunctions, just as in the case of ordinary adjunctions (week 174)?

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeMar 16th 2012
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexDavid, the triangulators satisfy a swallowtail coherence condition, for which there's a nice picture...

   David, the triangulators satisfy a swallowtail coherence condition, for which there’s a nice picture…

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeMar 16th 2012
   • (edited Mar 16th 2012)
   • PermaLink
   Author: zskoda
   Format: MarkdownItex1: Some of these "obvious" categorifications are not worked out, many are known. For example there are few papers of Marmolejo and of Vitale which work out quite much including Beck-type monadicity/descent theorems. Some relevant stuff should also be in Pronk's papers on bicategorical localization and some in older papers of Austrakian school, e.g. Kelly.

   1: Some of these “obvious” categorifications are not worked out, many are known. For example there are few papers of Marmolejo and of Vitale which work out quite much including Beck-type monadicity/descent theorems. Some relevant stuff should also be in Pronk’s papers on bicategorical localization and some in older papers of Austrakian school, e.g. Kelly.

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeMar 16th 2012
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexThat reminds me of the beginning of a long [conversation](http://golem.ph.utexas.edu/category/2006/11/this_weeks_finds_in_mathematic_2.html#c006153) at the Cafe, which ended up looking at generalizing the tangle hypothesis. I remember another interesting conversation we all had once about how from thinking about one dimensional things in the plane, one could be led to important category theoretic structures, and how this might happen with surfaces. Ah, here it [is](http://golem.ph.utexas.edu/category/2007/10/this_weeks_finds_in_mathematic_18.html#c012613).

   That reminds me of the beginning of a long conversation at the Cafe, which ended up looking at generalizing the tangle hypothesis.

   I remember another interesting conversation we all had once about how from thinking about one dimensional things in the plane, one could be led to important category theoretic structures, and how this might happen with surfaces. Ah, here it is.