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nLab > Latest Changes: representable functor theorem

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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 22nd 2023

    Added:

    Specifically, a continuous functor C→SetC\to Set is a right adjoint functor if and only if it is representable, in which case the left adjoint functor Set→CSet\to C sends the singleton set to the representing object

    Related concepts

    diff, v3, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 22nd 2023

    I see now that, originating with the creation of the entry back in 2011, it has a line starting with the words

    As representable functors are ubiquitous,…

    This does not make sense to me, neither the claim itself nor the suggestion that it implies the statement that follows. I wonder what was really meant here. But it looks like just deleting these words would not take anything away from the paragraphs that follow. (?)

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 22nd 2023

    I changed the opening of the first sentence under “Related facts” a little so that it ties in better with the phrase that follows (the earlier version, about the ubiquity of representable functors, seemingly echoes similar statements about the ubiquity of (the concept of) adjoint functors – see the quotations of Mac Lane given in Categories for the Working Mathematician).

    diff, v5, current

    • CommentRowNumber4.
    • CommentAuthorncfavier
    • CommentTimeOct 31st 2023

    Added some details and corrections. Please double-check! I don’t have a reference, sadly, I’m just going off what I could convince Agda of.

    diff, v6, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeOct 31st 2023

    I have added two references (here)

    You changed “complete” to “cocomplete” in the first sentence. But it looks to me like the original version was correct.

    diff, v7, current

    • CommentRowNumber6.
    • CommentAuthorvarkor
    • CommentTimeOct 31st 2023

    Reformulated the statement of the representable functor theorem to make explicit the distinction between representability and corepresentability, which was leading to confusion.

    diff, v8, current

    • CommentRowNumber7.
    • CommentAuthorncfavier
    • CommentTimeNov 1st 2023
    • (edited Nov 1st 2023)

    Okay, I think my confusion came from the fact that you need CC to be cocomplete in order to be able to say that F:C→SetF : C \to Set has a left adjoint iff it is representable, and I assumed that the representable functor theorem followed from the adjoint functor theorem and this equivalence; but the representable functor theorem actually stands on its own (without the assumption that CC is cocomplete), and then in order to deduce the adjoint functor theorem from that you only need D(x,Fy)D(x, Fy) to be representable, so there’s no copowering nonsense.

    I think the “specifically” paragraph should be rewritten now, but I am not sure how.

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 3rd 2023

    I don’t think there’s any “nonsense”.

1 to 8 of 8

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