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1. • CommentRowNumber1.
   • CommentAuthorHarry Gindi
   • CommentTimeJun 6th 2010
   • (edited Jun 6th 2010)
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   Author: Harry Gindi
   Format: MarkdownItexThe last non-yoneda step of the proof over at [[presheaf]] is that $\lim_{Y(V)\to F\in C_F}B(V)\cong Hom_{[C^{op},Set]}(F,B)$, which it says follows by inspection. I feel like an idiot, but how exactly does that follow? I'm worse off not knowing than looking like a fool =(. Hmm... I think I've got it. By the definition of a limit, we've got $\lim_{Y(V)\to F\in C_F}B(V)=Hom_{[C_F^{op}, Set]}(pt,B)$, then for each element of $Hom_{[C_F^{op}, Set]}(pt,B)$, for each object of $Y(V)\to F\in C_F$, we pick out an element of $B(V)$, that is, by Yoneda, for each element of $F(V)$, we pick an element of $B(V)$, that is to say, a function $F(V)\to B(V)$. The naturality of this assignment is guaranteed by the naturality of the map $pt\to B$. But there must be a much easier way, no? If not, I'll add the paragraph above to the proof, since that's where the actual work seems to be.

   The last non-yoneda step of the proof over at presheaf is that $\lim_{Y(V)\to F\in C_F}B(V)\cong Hom_{[C^{op},Set]}(F,B)$, which it says follows by inspection. I feel like an idiot, but how exactly does that follow?

   I’m worse off not knowing than looking like a fool =(.

   Hmm… I think I’ve got it. By the definition of a limit, we’ve got $\lim_{Y(V)\to F\in C_F}B(V)=Hom_{[C_F^{op}, Set]}(pt,B)$, then for each element of $Hom_{[C_F^{op}, Set]}(pt,B)$, for each object of $Y(V)\to F\in C_F$, we pick out an element of $B(V)$, that is, by Yoneda, for each element of $F(V)$, we pick an element of $B(V)$, that is to say, a function $F(V)\to B(V)$. The naturality of this assignment is guaranteed by the naturality of the map $pt\to B$.

   But there must be a much easier way, no? If not, I’ll add the paragraph above to the proof, since that’s where the actual work seems to be.

2. • CommentRowNumber2.
   • CommentAuthorFinnLawler
   • CommentTimeJun 6th 2010
   • (edited Jun 6th 2010)
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   Author: FinnLawler
   Format: MarkdownItex[Edit: crossed with your edit.] There may be an easier way than this, but: The category $C_F = Y/F$ is the [[category of elements]] of $F$, with objects $(V \in C, x \in F V)$ and the obvious arrows. You can write the limit $\lim_{(V,x)} B A$ as a subset of a product $\Pi_{V,x \in F V} B V$, and this is the set of maps $V \mapsto x \mapsto \alpha_V(x) \in B V$, so it's the set of *non*-natural transformations $\alpha$ from $F$ to $B$. Then the limit is the equalizer of the two maps $$ \Pi_{x \in F V} \rightrightarrows \Pi_{f \colon (U,y) \to (V,x)} B U $$ where the top arrow takes $\alpha$ to $\alpha_U(y)$ and the bottom $\alpha$ to $B f(\alpha_V(x))$. But because the right-hand product is over all $f$ such that $F f(x) = y$, the equalizer contains precisely those $\alpha$ such that $\alpha_U(F f(x)) = B f(\alpha_V(x))$, i.e. the set $\mathrm{Nat}(F, B)$. I have this proof in some notes of mine, though I'm pretty sure I found something like it somewhere on nLab (maybe the page you're talking about). > I'm worse off not knowing than looking like a fool =(. Asking intelligent questions never makes anyone look like a fool. :)

   [Edit: crossed with your edit.]

   There may be an easier way than this, but:

   The category $C_F = Y/F$ is the category of elements of $F$, with objects $(V \in C, x \in F V)$ and the obvious arrows. You can write the limit $\lim_{(V,x)} B A$ as a subset of a product $\Pi_{V,x \in F V} B V$, and this is the set of maps $V \mapsto x \mapsto \alpha_V(x) \in B V$, so it’s the set of non-natural transformations $\alpha$ from $F$ to $B$. Then the limit is the equalizer of the two maps

   $$\Pi_{x \in F V} \rightrightarrows \Pi_{f \colon (U,y) \to (V,x)} B U$$

   where the top arrow takes $\alpha$ to $\alpha_U(y)$ and the bottom $\alpha$ to $B f(\alpha_V(x))$. But because the right-hand product is over all $f$ such that $F f(x) = y$, the equalizer contains precisely those $\alpha$ such that $\alpha_U(F f(x)) = B f(\alpha_V(x))$, i.e. the set $\mathrm{Nat}(F, B)$.

   I have this proof in some notes of mine, though I’m pretty sure I found something like it somewhere on nLab (maybe the page you’re talking about).

   I’m worse off not knowing than looking like a fool =(.

   Asking intelligent questions never makes anyone look like a fool. :)

3. • CommentRowNumber3.
   • CommentAuthorHarry Gindi
   • CommentTimeJun 6th 2010
   • (edited Jun 6th 2010)
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   Author: Harry Gindi
   Format: MarkdownItexYes, my proof is easier than the one you gave since it uses the slick definition of the limit =). Usually when a step in the proof is something like "the proof follows by inspection", one doesn't expect the majority of the argument to be covered by that (hence my despair!). Perhaps it would be more honest to say, "the proof follows by _introspection_"! Not to be too much of a complainer, but there's nothing (apart from confusing typos) that is more annoying than having the main body of the argument left with something like "immediate" or "obvious" or "by inspection". You might as well leave the whole thing as an exercise =(. I mean, that last part isn't hard, but it's harder than the stupid part of the proof where you just flip the colimit out and apply Yoneda.

   Yes, my proof is easier than the one you gave since it uses the slick definition of the limit =).

   Usually when a step in the proof is something like “the proof follows by inspection”, one doesn’t expect the majority of the argument to be covered by that (hence my despair!).

   Perhaps it would be more honest to say, “the proof follows by introspection”! Not to be too much of a complainer, but there’s nothing (apart from confusing typos) that is more annoying than having the main body of the argument left with something like “immediate” or “obvious” or “by inspection”. You might as well leave the whole thing as an exercise =(.

   I mean, that last part isn’t hard, but it’s harder than the stupid part of the proof where you just flip the colimit out and apply Yoneda.

4. • CommentRowNumber4.
   • CommentAuthorFinnLawler
   • CommentTimeJun 6th 2010
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   Author: FinnLawler
   Format: MarkdownItexRight: you should add your proof to the page. Maybe an even slicker proof is to note that we're calculating the [[weighted limit]] $lim^F B$, so the whole lot follows by enriched nonsense as at ibid. (though I think the coends there should be ends).

   Right: you should add your proof to the page.

   Maybe an even slicker proof is to note that we’re calculating the weighted limit $lim^F B$, so the whole lot follows by enriched nonsense as at ibid. (though I think the coends there should be ends).

5. • CommentRowNumber5.
   • CommentAuthorHarry Gindi
   • CommentTimeJun 6th 2010
   • (edited Jun 6th 2010)
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   Author: Harry Gindi
   Format: MarkdownItexAlright, added it. By the way, to prove that it's the weighted limit would be exactly the same, since we're switching from taking limits over the comma category vs taking limits with a weight.

   Alright, added it. By the way, to prove that it’s the weighted limit would be exactly the same, since we’re switching from taking limits over the comma category vs taking limits with a weight.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJun 6th 2010
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   Author: Urs
   Format: MarkdownItexThanks, Harry. I edited it slightly, for instance added formal proposition/proof environments and a link to [[co-Yoneda lemma]]. Also added a remark that the proof is a special case for the formula of left [[Kan extension]]: the co-Yoneda lemma is the formula for the left Kan extension along the identity functor.

   Thanks, Harry.

   I edited it slightly, for instance added formal proposition/proof environments and a link to co-Yoneda lemma.

   Also added a remark that the proof is a special case for the formula of left Kan extension: the co-Yoneda lemma is the formula for the left Kan extension along the identity functor.

7. • CommentRowNumber7.
   • CommentAuthorHarry Gindi
   • CommentTimeJun 6th 2010
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   Author: Harry Gindi
   Format: MarkdownItexThanks!

   Thanks!

8. • CommentRowNumber8.
   • CommentAuthorTodd_Trimble
   • CommentTimeJun 6th 2010
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   Author: Todd_Trimble
   Format: MarkdownItexHere's the way I view this: to show that a presheaf $F: C^{op} \to Set$ is canonically presented as a colimit of representables, we exhibit a natural isomorphism $$\int^{c: C} F(c) \times \hom_C(-, c) \cong F$$ By the definition of coend, maps $\int^c F(c) \times \hom_C(-, c) \to G(-)$ are in natural bijection with families of maps $F(c) \times \hom_C(d, c) \to G(d)$ extranatural in $c$ and natural in $d$. Those are in natural bijection with families of maps $F(c) \to \hom(\hom_C(d, c), G(d))$ natural in $c$ and extranatural in $d$. These are in natural bijection with families of maps $F(c) \to Nat(\hom_C(-, c), G) \cong G(c)$ (natural in $c$), by Yoneda. Thus we have exhibited a natural isomorphism $$Nat(\int^c F(c) \times \hom_C(-, c), G) \cong Nat(F, G)$$ (natural in $G$). By Yoneda again, this gives $\int^c F(c) \times \hom_C(-, c) \cong F$. The more conceptual way of viewing this is in terms of bimodules (aka profunctors), but I won't go into this here.

   Here’s the way I view this: to show that a presheaf $F: C^{op} \to Set$ is canonically presented as a colimit of representables, we exhibit a natural isomorphism

   $$\int^{c: C} F(c) \times \hom_C(-, c) \cong F$$

   By the definition of coend, maps $\int^c F(c) \times \hom_C(-, c) \to G(-)$ are in natural bijection with families of maps $F(c) \times \hom_C(d, c) \to G(d)$ extranatural in $c$ and natural in $d$. Those are in natural bijection with families of maps $F(c) \to \hom(\hom_C(d, c), G(d))$ natural in $c$ and extranatural in $d$. These are in natural bijection with families of maps $F(c) \to Nat(\hom_C(-, c), G) \cong G(c)$ (natural in $c$), by Yoneda. Thus we have exhibited a natural isomorphism

   $$Nat(\int^c F(c) \times \hom_C(-, c), G) \cong Nat(F, G)$$

   (natural in $G$). By Yoneda again, this gives $\int^c F(c) \times \hom_C(-, c) \cong F$.

   The more conceptual way of viewing this is in terms of bimodules (aka profunctors), but I won’t go into this here.

9. • CommentRowNumber9.
   • CommentAuthorHarry Gindi
   • CommentTimeJun 7th 2010
   • (edited Jun 7th 2010)
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   Author: Harry Gindi
   Format: MarkdownItexI still know basically nothing about ends and coends since Mac Lane's treatment is really dry (as is all of [[Categories Work]]. I really dislike that book.), so whenever I see them, I immediately run in the opposite direction. Do you know of a better source to learn about them?

   I still know basically nothing about ends and coends since Mac Lane’s treatment is really dry (as is all of Categories Work. I really dislike that book.), so whenever I see them, I immediately run in the opposite direction.

   Do you know of a better source to learn about them?

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJun 7th 2010
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    Author: Urs
    Format: MarkdownItex> Do you know of a better source to learn about them? As we recently dicussed elsewhere, a standard reference is Kelly's "Basic concepts of enriched category theory". (use Google.)

    Do you know of a better source to learn about them?

    As we recently dicussed elsewhere, a standard reference is Kelly’s “Basic concepts of enriched category theory”. (use Google.)

11. • CommentRowNumber11.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 7th 2010
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    Author: Todd_Trimble
    Format: MarkdownItexThat's too bad. Have you had better luck looking around nLab pages? Some people seem to like the discussion which I think is at [[extranatural transformation]]. If that's under your belt, then coends/ends are just universal examples of extranatural wedges. I like Cats Work (grew up on it, really), but I don't think the section on ends/coends is one of the stronger efforts in that book. I remember struggling with them myself, wondering what they were really about. Anyway, staring down your fears over ends/coends is well worth the effort (I think Urs would agree with that!).

    That’s too bad. Have you had better luck looking around nLab pages? Some people seem to like the discussion which I think is at extranatural transformation. If that’s under your belt, then coends/ends are just universal examples of extranatural wedges.

    I like Cats Work (grew up on it, really), but I don’t think the section on ends/coends is one of the stronger efforts in that book. I remember struggling with them myself, wondering what they were really about. Anyway, staring down your fears over ends/coends is well worth the effort (I think Urs would agree with that!).

12. • CommentRowNumber12.
    • CommentAuthorHarry Gindi
    • CommentTimeJun 7th 2010
    • (edited Jun 7th 2010)
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    Author: Harry Gindi
    Format: MarkdownItex[redacted]

    [redacted]