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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.
   • CommentAuthorMike Shulman
   • CommentTimeApr 6th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownAdded some comments about the possibility of 2-dimensional unbiased composites in [[double categories]].

   Added some comments about the possibility of 2-dimensional unbiased composites in double categories.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeApr 7th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownCreated [[tileorder]] with the two characterizations due to Dawson-Pare.

   Created tileorder with the two characterizations due to Dawson-Pare.

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeApr 7th 2010
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownI've put a query under the definition section of [[tileorder]].

   I've put a query under the definition section of tileorder.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeApr 8th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI replied. If this answers your question, feel free to remove the query box and clarify the entry so that other people don't have the same question.

   I replied. If this answers your question, feel free to remove the query box and clarify the entry so that other people don’t have the same question.

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeApr 8th 2010
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownThanks Mike -- it did answer the question (and I should have read what you wrote more carefully). Probably it's not necessary to change anything, but I'll take another look in a moment.

   Thanks Mike -- it did answer the question (and I should have read what you wrote more carefully). Probably it's not necessary to change anything, but I'll take another look in a moment.

6. • CommentRowNumber6.
   • CommentAuthorEric
   • CommentTimeApr 8th 2010
   • (edited Apr 8th 2010)
   • PermaLink
   Author: Eric
   Format: MarkdownItexInteresting. I started typing out a question the other day, but don't remember if I posted it, along the lines, "Is there a term for the analogue of preorder in a double category?" I guess this term would be [[tileorder]]? I think you are overlapping with my dream of nPOV'ing (I like using nPOV as a verb these days) our discrete geometry stuff on directed n-cubes (or n-diamonds). The stuff on [[tileorder]] makes me think of a 2-diamond complex. Oh I wish I was smarter... Edit: I feel like my brain is going to explode. Urs and Domenico are blowing my mind. Mike and Todd are both doing stuff that is getting close to my heart. I've said it before, but I often feel like Salieri's character in Amadeus. I know just enough to appreciate the beauty of what you guys are doing, but tortured by not having the ability to create it myself.

   Interesting. I started typing out a question the other day, but don’t remember if I posted it, along the lines, “Is there a term for the analogue of preorder in a double category?” I guess this term would be tileorder?

   I think you are overlapping with my dream of nPOV’ing (I like using nPOV as a verb these days) our discrete geometry stuff on directed n-cubes (or n-diamonds). The stuff on tileorder makes me think of a 2-diamond complex.

   Oh I wish I was smarter…

   Edit: I feel like my brain is going to explode. Urs and Domenico are blowing my mind. Mike and Todd are both doing stuff that is getting close to my heart. I’ve said it before, but I often feel like Salieri’s character in Amadeus. I know just enough to appreciate the beauty of what you guys are doing, but tortured by not having the ability to create it myself.