Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

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 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 nforum 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 sheaves 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
   • CommentTimeMar 10th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownAdded some discussion of other ways to define doubly-weak [[double categories]].

   Added some discussion of other ways to define doubly-weak double categories.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeSep 27th 2010
   • PermaLink
   Author: zskoda
   Format: MarkdownItexAt [[virtual double category]] (or fc-multicategory) there is a statement that virtual double category generalized weak double category. The same statement is kind of in Leinster book, the table after Example 5.1.1. Tom says that *most familiar 2d structures are degenerate fc-multicategories*. Then he gives a table with 3 rows and 3 columns. The top row is no degeneracy. The columns are *no horizontal composition*, *weak horizontal composition*, *strict horizontal composition* and entries at 11,12,13 are fc-multicategory, weak double category, strict double category. How can the case of weak horizontal composition be a special case of *no* horizontal composition ?? Horizontal composition is a structure, not a property. Am I missing something ?

   At virtual double category (or fc-multicategory) there is a statement that virtual double category generalized weak double category. The same statement is kind of in Leinster book, the table after Example 5.1.1. Tom says that most familiar 2d structures are degenerate fc-multicategories. Then he gives a table with 3 rows and 3 columns. The top row is no degeneracy. The columns are no horizontal composition, weak horizontal composition, strict horizontal composition and entries at 11,12,13 are fc-multicategory, weak double category, strict double category. How can the case of weak horizontal composition be a special case of no horizontal composition ?? Horizontal composition is a structure, not a property. Am I missing something ?

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeSep 27th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexIn the same way that a [[monoidal category]] is a special case of a [[multicategory]], and that a category with products is a special case of a category. A virtual double category does not have a composition operation itself, but it has sufficient structure enabling the further existence of composites to become a property.

   In the same way that a monoidal category is a special case of a multicategory, and that a category with products is a special case of a category. A virtual double category does not have a composition operation itself, but it has sufficient structure enabling the further existence of composites to become a property.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeSep 27th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexThe characterization of composites by a universal property can be found in section 5 of [this paper](http://arxiv.org/abs/0907.2460). One day it should make its way into the nLab.

   The characterization of composites by a universal property can be found in section 5 of this paper. One day it should make its way into the nLab.

5. • CommentRowNumber5.
   • CommentAuthorzskoda
   • CommentTimeSep 27th 2010
   • PermaLink
   Author: zskoda
   Format: MarkdownItexThat is a great answer. Thanks.

   That is a great answer. Thanks.