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1. • CommentRowNumber1.
   • CommentAuthorTim_Porter
   • CommentTimeApr 21st 2014
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   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
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   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
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   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.