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1. • CommentRowNumber1.
   • CommentAuthorChris SchommerPries
   • CommentTimeMar 12th 2014
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   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 $Cyl(C) \to C \times C$, 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.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeMar 12th 2014
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   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 $Cocyl(C)$ 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.