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    • CommentRowNumber1.
    • CommentAuthorEric
    • CommentTimeMay 21st 2010

    On the page hom-functor, it says

    There is also a contravariant hom-functor

    hom(,c):C opSet, hom(-,c) : C^{op} \to Set,

    where C opC^{op} is the opposite category to CC, which sends any object xC opx \in C^{op} to the hom-set hom(x,c)hom(x,c).

    If you write it like this, should you really call it “contravariant”? When you write C opC^{op}, I thought you should call it just “functor” or “covariant”. By saying it is contravariant AND writing C opC^{op}, it seems like double counting.

    I hope to add some illustrations to these pages. It is a shame there are not more illustrations on the nLab since nStuff is so amenable to nice pictures.

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 21st 2010

    One could argue that the functor is just called the ”contravariant hom-functor, to distinguish it, as saying ’hom-functor’ could mean one of three things: Hom(,c),Hom(c,)Hom(-,c),Hom(c,-) or Hom(,)Hom(-,-). But I don’t really know the rationale.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 21st 2010

    Thanks for spotting this.

    I completely rewrote the page, and expanded it.

    Also created a stub for product category.

    • CommentRowNumber4.
    • CommentAuthorEric
    • CommentTimeMay 21st 2010
    • (edited May 21st 2010)

    Nice. Thanks :)

    Edit: For a second, I changed functor to profunctor, but realized it wasn’t quite right so changed it back. It would be interesting to relate hom-functor and profunctor somehow though (I just did it wrong).

    Edit^2: Added a blurb:

    Relation to profunctors

    A hom-functor C op×CSetC^{op}\times C\to Set is also a profunctor CCC ⇸ C.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 21st 2010

    That’s right Eric:

    The hom-functor C op×CSetC^{op} \times C \to Set is the identity profunctor Id:CCId : C ⇸ C !

    • CommentRowNumber6.
    • CommentAuthorEric
    • CommentTimeMay 21st 2010

    Whoa :)

    Ok. Changed it to

    Relation to profunctors

    A hom-functor C op×CSetC^{op}\times C\to Set is also the identity profunctor 1 C:CC1_C: C ⇸ C.


    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMay 21st 2010

    Whoa :)

    Yeah, speaking of striking category theory. ;-)

    I added to hom-functor some remarks on why this is the case. And also to the beginning of profunctor.

    A good warm-up excercise if you want to play around with the Yoneda embedding.

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 21st 2010

    Eric, this statement (that homhom is a unit profunctor) is also essentially the key fact of Yoneda reduction (which I really need to get into better shape).

    • CommentRowNumber9.
    • CommentAuthorEric
    • CommentTimeMay 21st 2010

    Thanks. The reason I’m reading this page on hom-functor is that I’m working on Yoneda lemma from Vistoli’s paper. His explanation of Yoneda lemma is very nice. As usual, I will not understand until I’ve drawn some pictures so was hoping to add some illustrations to the nLab.

    • CommentRowNumber10.
    • CommentAuthorAndrew Stacey
    • CommentTimeMay 21st 2010

    (Just testing to see if I can post here)

    • CommentRowNumber11.
    • CommentAuthorEric
    • CommentTimeMay 21st 2010


  1. adding text from HoTT wiki


    diff, v23, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJun 7th 2022

    Wait, there is nothing specific to homotopy type theory in what you just added. You are essentially just re-stating the usual definition.

    It’s great that you are copying material from the HoTT-wiki, but let’s do it a little less mechanically, more with an eye towards retaining logic and cosistency of existing entries.