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1. • CommentRowNumber1.
   • CommentAuthorporton
   • CommentTimeOct 23rd 2013
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   Author: porton
   Format: MarkdownItexXXX is a pair $(C;V)$ or a category $C$ and a functor $V:\mathbf{Rel}\rightarrow C$. How would you suggest to name XXX?

   XXX is a pair $(C;V)$ or a category $C$ and a functor $V:\mathbf{Rel}\rightarrow C$.

   How would you suggest to name XXX?

2. • CommentRowNumber2.
   • CommentAuthorporton
   • CommentTimeOct 23rd 2013
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   Author: porton
   Format: MarkdownItexProbably I will add more requirements, such as $V$ being full or faithful, I'm yet unsure.

   Probably I will add more requirements, such as $V$ being full or faithful, I’m yet unsure.

3. • CommentRowNumber3.
   • CommentAuthorporton
   • CommentTimeOct 23rd 2013
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   Author: porton
   Format: MarkdownItexWhat's about "category on top of $\mathbf{Rel}$"? Is this term taken?

   What’s about “category on top of $\mathbf{Rel}$”? Is this term taken?

4. • CommentRowNumber4.
   • CommentAuthorDavidRoberts
   • CommentTimeOct 23rd 2013
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   Author: DavidRoberts
   Format: MarkdownItex>category on top of Rel no, this is a category _under_ Rel. It's an object in Rel/Cat. But more important is the concept of morphism between these things, as I can imagine several contenders.

   category on top of Rel

   no, this is a category under Rel. It’s an object in Rel/Cat. But more important is the concept of morphism between these things, as I can imagine several contenders.

5. • CommentRowNumber5.
   • CommentAuthorporton
   • CommentTimeOct 23rd 2013
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   Author: porton
   Format: MarkdownItexAnd what's about to name the functor $V$ "remembrance functor"? (It is mapping a less structured category into a more structured category, just reverse to "forgetful" functors, so "remembrance".) If not "remembrance" then how to call it?

   And what’s about to name the functor $V$ “remembrance functor”? (It is mapping a less structured category into a more structured category, just reverse to “forgetful” functors, so “remembrance”.)

   If not “remembrance” then how to call it?

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeOct 23rd 2013
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   Author: Mike Shulman
   Format: MarkdownItex"Category under Rel" sounds fine to me.

   “Category under Rel” sounds fine to me.

7. • CommentRowNumber7.
   • CommentAuthorzskoda
   • CommentTimeOct 23rd 2013
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   Author: zskoda
   Format: MarkdownItexI second Mike's opinion.

   I second Mike’s opinion.