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1. • CommentRowNumber1.
   • CommentAuthorEric
   • CommentTimeMay 21st 2010
   • PermaLink
   Author: Eric
   Format: MarkdownItexOn the page [[hom-functor]], it says >There is also a **contravariant hom-functor** $$ hom(-,c) : C^{op} \to Set, $$ where $C^{op}$ is the [[opposite category]] to $C$, which sends any object $x \in C^{op}$ to the hom-set $hom(x,c)$. If you write it like this, should you really call it "contravariant"? When you write $C^{op}$, I thought you should call it just "functor" or "covariant". By saying it is contravariant AND writing $C^{op}$, 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.

   On the page hom-functor, it says

   There is also a contravariant hom-functor

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

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

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

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 21st 2010
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexOne 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: $Hom(-,c),Hom(c,-)$ or $Hom(-,-)$. But I don't really know the rationale.

   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: $Hom(-,c),Hom(c,-)$ or $Hom(-,-)$. But I don’t really know the rationale.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 21st 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks for spotting this. I completely rewrote the page, and expanded it. Also created a stub for [[product category]].

   Thanks for spotting this.

   I completely rewrote the page, and expanded it.

   Also created a stub for product category.

4. • CommentRowNumber4.
   • CommentAuthorEric
   • CommentTimeMay 21st 2010
   • (edited May 21st 2010)
   • PermaLink
   Author: Eric
   Format: MarkdownItexNice. 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 $C^{op}\times C\to Set$ is also a [[profunctor]] $C ⇸ C$.

   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 $C^{op}\times C\to Set$ is also a profunctor $C ⇸ C$.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMay 21st 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexThat's right Eric: The hom-functor $C^{op} \times C \to Set$ is the _identity_ profunctor $Id : C ⇸ C$ !

   That’s right Eric:

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

6. • CommentRowNumber6.
   • CommentAuthorEric
   • CommentTimeMay 21st 2010
   • PermaLink
   Author: Eric
   Format: MarkdownItexWhoa :) Ok. Changed it to >### Relation to profunctors >A hom-functor $C^{op}\times C\to Set$ is also the identity [[profunctor]] $1_C: C ⇸ C$. :)

   Whoa :)

   Ok. Changed it to

   Relation to profunctors

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

   :)

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeMay 21st 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItex> 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.

   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.

8. • CommentRowNumber8.
   • CommentAuthorTodd_Trimble
   • CommentTimeMay 21st 2010
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexEric, this statement (that $hom$ is a unit profunctor) is also essentially the key fact of [[Yoneda reduction]] (which I really need to get into better shape).

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

9. • CommentRowNumber9.
   • CommentAuthorEric
   • CommentTimeMay 21st 2010
   • PermaLink
   Author: Eric
   Format: MarkdownItexThanks. 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.

   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.

10. • CommentRowNumber10.
    • CommentAuthorAndrew Stacey
    • CommentTimeMay 21st 2010
    • PermaLink
    Author: Andrew Stacey
    Format: MarkdownItex(Just testing to see if I can post here)

    (Just testing to see if I can post here)

11. • CommentRowNumber11.
    • CommentAuthorEric
    • CommentTimeMay 21st 2010
    • PermaLink
    Author: Eric
    Format: MarkdownItexTest

    Test

12. • CommentRowNumber12.
    • CommentAuthornLab edit announcer
    • CommentTimeJun 7th 2022
    • PermaLink
    Author: nLab edit announcer
    Format: MarkdownItexadding text from HoTT wiki Anonymous <a href="https://ncatlab.org/nlab/revision/diff/hom-functor/23">diff</a>, <a href="https://ncatlab.org/nlab/revision/hom-functor/23">v23</a>, <a href="https://ncatlab.org/nlab/show/hom-functor">current</a>

    adding text from HoTT wiki

    Anonymous

    diff, v23, current

13. • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJun 7th 2022
    • PermaLink
    Author: Urs
    Format: MarkdownItexWait, 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.

    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.