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    • CommentRowNumber1.
    • CommentAuthorporton
    • CommentTimeOct 23rd 2013

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

    How would you suggest to name XXX?

    • CommentRowNumber2.
    • CommentAuthorporton
    • CommentTimeOct 23rd 2013

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

    • CommentRowNumber3.
    • CommentAuthorporton
    • CommentTimeOct 23rd 2013

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

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 23rd 2013

    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.

    • CommentRowNumber5.
    • CommentAuthorporton
    • CommentTimeOct 23rd 2013

    And what’s about to name the functor VV “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?

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeOct 23rd 2013

    “Category under Rel” sounds fine to me.

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeOct 23rd 2013

    I second Mike’s opinion.