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 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 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 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.
   • CommentAuthorChris SchommerPries
   • CommentTimeMar 12th 2014
   • PermaLink
   Author: Chris SchommerPries
   Format: MarkdownItexLet B and C be bicategories. Does there exist a "cylinder object", a bicategory Cyl(C) with a strict homomorphism $Cyl(C) \to C \times C$, with the following properties: (1) homomorphisms $B \to Cyl(C)$ are the same as pseudo-natural transformations between homomorphisms and (2) strict homomorphisms $B \to Cyl(C)$ are the same as strict natural transformations between strict homomorphisms? If so where could I read about this? Benabou has something like this in the end of his paper on Bicategories, but he doesn't show these properties and I haven't had time to tease out whether these properties hold. This should be well known, I would think. I just don't know where to look.

   Let B and C be bicategories. Does there exist a “cylinder object”, a bicategory Cyl(C) with a strict homomorphism , with the following properties: (1) homomorphisms are the same as pseudo-natural transformations between homomorphisms and (2) strict homomorphisms are the same as strict natural transformations between strict homomorphisms?

   If so where could I read about this? Benabou has something like this in the end of his paper on Bicategories, but he doesn’t show these properties and I haven’t had time to tease out whether these properties hold. This should be well known, I would think. I just don’t know where to look.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeMar 12th 2014
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI think what you describe would be a cocylinder or path object, not a cylinder object. But I think the answer is no to what you suggest, because the objects of $Cocyl(C)$ must be morphisms in $C$, and strictness of a natural transformation represented as a functor into $Cocyl(C)$ is going to have to do with whether or not the morphisms in $Cocyl(C)$ are strictly or pseudo-ly commutative squares, not whether the functor into $Cocycl(C)$ is itself strict.

   I think what you describe would be a cocylinder or path object, not a cylinder object. But I think the answer is no to what you suggest, because the objects of must be morphisms in , and strictness of a natural transformation represented as a functor into is going to have to do with whether or not the morphisms in are strictly or pseudo-ly commutative squares, not whether the functor into is itself strict.