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 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.
   • CommentAuthorTim_Porter
   • CommentTimeApr 21st 2014
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexVladimir Sotirov has asked a question at [[contravariant functor]].

   Vladimir Sotirov has asked a question at contravariant functor.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeApr 21st 2014
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI've thought about considering categories with covariant and contravariant functors as forming a category enriched over $Cat\times Cat$ with a slightly funny non-symmetric monoidal structure, but that doesn't allow for transformations that mix variance. Perhaps whoever wrote that page was thinking of a similar monoidal structure on $Cat/I$ (with $I$ the walking iso)?

   I’ve thought about considering categories with covariant and contravariant functors as forming a category enriched over $Cat\times Cat$ with a slightly funny non-symmetric monoidal structure, but that doesn’t allow for transformations that mix variance. Perhaps whoever wrote that page was thinking of a similar monoidal structure on $Cat/I$ (with $I$ the walking iso)?

3. • CommentRowNumber3.
   • CommentAuthorZhen Lin
   • CommentTimeApr 21st 2014
   • PermaLink
   Author: Zhen Lin
   Format: MarkdownItexThe underlying 1-category can be obtained as the Grothendieck construction applied to the obvious diagram of shape $\mathbb{B} (\mathbb{Z} / 2 \mathbb{Z})$, but I don't see how to get the 2-cells in a "natural" way.

   The underlying 1-category can be obtained as the Grothendieck construction applied to the obvious diagram of shape $\mathbb{B} (\mathbb{Z} / 2 \mathbb{Z})$, but I don’t see how to get the 2-cells in a “natural” way.