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• CommentRowNumber1.
• CommentAuthorFosco
• CommentTimeDec 12th 2014

I’m on the fine-tuning phase of a short note about coend calculus. The discussion is still in a sketchy form, in particular

1. There are some subtleties I’d like to fix; not sure that all the constructions I make are possible, remaining in a single universe $\mathbf{U}$;
2. I feel I’m waving hands too vigorously in a couple of arguments (section 3.3 and the “introduction” to operads are extremely unsatisfying and messy);
3. Several references are still missing or can be extremely improved to clarify the discussion;
4. I’m still unable to retrieve the original paper by Yoneda On Ext and exact sequences. Jour. Fac. Sci. Univ. Tokyo 8 (1960), 507 - 576. which introduced the integral notation.

…plus several other mistakes which you will certainly notice if you begin reading! As always, any kind of advice, comment or criticism is welcome: the only reason why the note lacks an acknowledgements section is that the list will certainly grow bigger in the semi-public phase.

1. Tu devi fare le costruzioni che ti servono
e che sono matematicamente necessarie.
Il problema di sistemarle poi in una teoria matematica
degli insiemi, classi, classi di classi, ecc.
è una questione di ruotin da lasciare
agli ZF-specialisti.
Il problema interessante invece
è quello di inserire le nuove costruzioni categoriali
all'interno della categoria delle categorie di Lawvere,
eventualmente modificandola in parte.

Giancarlo Meloni
2. ERRATA
ruotin |--> routine
• CommentRowNumber4.
• CommentAuthorFosco
• CommentTimeDec 13th 2014

Apparently I summoned a troll… :)

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeDec 14th 2014

Fosco, I don’t know. You understand Italian and I don’t, but Giancarlo Meloni is the name of someone known in categorical circles and who has done reasonable work.

• CommentRowNumber6.
• CommentAuthorFosco
• CommentTimeDec 14th 2014

The author of the message is indeed the real Meloni. But I didn’t want to attack his authority or mathematical ability. Only the fact that his response could well have been written in English, but he didn’t: this looks rather strange. In any case, no offense implied. I met Giancarlo personally several times, and I know that he can handle this kind of jokes. :)

(any of you can feed Google translate with #2 to understand what Giancarlo wrote)

• CommentRowNumber7.
• CommentAuthorTodd_Trimble
• CommentTimeDec 14th 2014

Thanks for the reassurance, Fosco! I did use Google translate, but it’s not always the best. (There was a funny little thread at MO here where various users are commenting in languages other than English; when I tried Google translate on George Elencwajg’s comment in Russian – what is he, Belgian perhaps? – it started out “Expensive colleges, …” rather than “Dear colleagues, …” (-: ).

• CommentRowNumber8.
• CommentAuthorFosco
• CommentTimeDec 14th 2014
• (edited Dec 14th 2014)

I remember well that post on MO. :) Now that we exhausted the offtopics, and now that I remember, your note is acknowledged as an extremely good reference to get acquainted with “abstract operadic calculus”. I’ll be glad to know what do you think about the whole paper! Do you have any clue about where I can get the Yoneda paper “on Ext”?

• CommentRowNumber9.
• CommentAuthorTodd_Trimble
• CommentTimeDec 17th 2014
• (edited Dec 17th 2014)

Yes, I think it’s a nice note, although I think it might benefit by being trimmed down in places. A few quick reactions:

• For the purposes of describing end/coend calculus, I wouldn’t emphasize dinatural transformations so much as I would extranatural transformations. Most dinatural transformations that arise in the wild can be analyzed in terms of extranatural (extraordinary natural, in the old lingo) transformations. The old paper by Eilenberg and Kelly, A Generalization of the Functorial Calculus, ought to be mentioned. And can you draw some illustrative string diagrams to illustrate how extranatural transformations work? (Not commutative diagrams, but string diagrams.)

• To my mind, the very first example of a coend one should teach is that of general tensor product of two “modules” $F: C^{op} \to Set$, $G: C \to Set$, seeing the appropriate coend really as a tensor products

$\sum_{a, b} G(a) \times \hom(a, b) \times F(b) \stackrel{\to}{\to} \sum_c G(c) \times F(c) \to G \otimes_C F = \int^c G(c) \times F(c).$

For example, really see the geometric realization in this light. Connect all this up with extranatural transformations. (See for instance my MO post.) (I think you take too long to get to this type of example; it ought to come much earlier in my opinion.)

• As you know, all concepts in category theory are reducible to one another. For example, the coend $\int^c F(c, c)$ is a weighted colimit of $F$ along a hom-functor. Thus, hammering home the point that tensor products are probably the most illuminating source of intuition for coends, let me add that any coend $\int^c F$ for a functor $F: C^{op} \times C \to Set$ can be seen as a tensor product $F \otimes_{C^{op} \times C} \hom_{C^{op}}$. Somehow I think this gives another sense in which ends and coends are utterly natural. One should note that all this works in the enriched setting as well.

• My own taste would not emphasize the twisted category construction, subdivision, etc. These are “tricks” that do not port well to the enriched setting, and I don’t find them too illuminating personally.

• This isn’t important, but do you know the source of the phrase “Yoneda reduction”? As far as I know, I was the one who first coined that expression. :-) But I had it more in mind of a rhyme with “eta reduction”, thinking of a rewriting going in the direction $\int^c \hom(c, -) \cdot F(c) \rightsquigarrow F$ by analogy with $\lambda c. f(c) \rightsquigarrow f$. (Sometimes one ought to Yoneda-expand!) The original applications were to coherence theory (which I can explain if you like).

Sorry if these reactions come off as harsh.

• CommentRowNumber10.
• CommentAuthorFosco
• CommentTimeDec 20th 2014

Sorry if these reactions come off as harsh.

Not at all! Each point is extremely useful. I didn’t think that extranatural transformations played such an important role since the definition of co/end relies on the notion of co/wedge, which is an extremely particular example of dinatural transformation, which is a particular example of extranatural transformation. Please, help me understand your point, which is surely important.

Thanks for the references and for the clarification about Yoneda/eta-reduction!

• CommentRowNumber11.
• CommentAuthorTodd_Trimble
• CommentTimeDec 21st 2014

There might be some terminological difficulty here.

an extremely particular example of dinatural transformation, which is a particular example of extranatural transformation

As you say, there are wedges and cowedges, involving certain commutative squares, and this is the really relevant concept when discussing ends and coends. This generalizes in two directions: one, they are special examples of dinatural transformations, which involve certain commutative hexagons. Two, they are special cases of what I would call an extranatural (or extraordinary natural) transformation, which generally is a family of maps $F(a, a, b) \to G(b, c, c)$ which combines naturality in the argument $b$ with a cowedge condition on $a$ and a wedge condition on $c$. Or at least, that’s how I thought Eilenberg and Kelly use the term (it’s been a long time since I’ve looked at their article).

There is a kind of compositional calculus for extranatural transformations which is nicely illustrated by string diagrams. As you point out, dinatural transformations do not have a nice notion of composition.

But I didn’t think that dinatural transformations were a special case of extranatural transformation in this sense. (Although history has shown that I’ve misremembered or misapplied terminology in the past.) Could you clarify what you meant by the quote cited above? Particularly, how you understand the notion of extranatural transformation that includes dinatural transformation as a special case?

• CommentRowNumber12.
• CommentAuthorFosco
• CommentTimeDec 21st 2014

RIght after your comment I had a look to EK paper and I have the following interpretation:

• If in $F(a,a,b)\to G(b,c,c)$ the functor $F$ is constant in all variables, and $G(b,c,c)=\bar G(c,c)$ you get a wedge;
• dually you get a cowedge: $F(a,a,b)=\bar F(a,a)\to G(b,c,c)\equiv G$ for all $a,b,c$.

but maybe that’s my fault?

• CommentRowNumber13.
• CommentAuthorTodd_Trimble
• CommentTimeDec 21st 2014

I agree with your interpretation. (Is what your fault? (-:) But is that supposed to answer what I asked at the end of my previous comment?

• CommentRowNumber14.
• CommentAuthorFosco
• CommentTimeDec 21st 2014

Ah, sorry. A piece of conversation happened only in my mind.

I also believed that you can obtain a dinatural transformation out of $F(a,a,b)\to G(b,c,c)$, when the naturality piece is mute. But maybe this is where you disagree as I am wrong: I didn’t get that you have a wedge condition in $c$ and a cowedge condition in $a$, which is different from having a dinatural condition for $a=c$ in $F(a,a)\to G(a,a)$.

This is where I can ask: is that a fault? :)

• CommentRowNumber15.
• CommentAuthorTodd_Trimble
• CommentTimeDec 21st 2014
• (edited Dec 21st 2014)

Yes, I think I agree with

I also believed that you can obtain a dinatural transformation out of $F(a,a,b)\to G(b,c,c)$

in other words, looked through the right lens, you can see any extranatural transformation as a dinatural transformation (so dinatural transformations are more general than extranatural transformations, not the other around as you seemed to be saying in #10). But it takes a little effort to see this.

It’s extremely easy to screw up here, but roughly how it goes is this. Suppose you have functors $F: C^{op} \times C \times C \to Set$ and $G: C \times C \times C^{op} \to Set$. (The $Set$ is not important here.) Now put $D = C \times C^{op} \times C^{op}$, and form two new functors $F', G': D^{op} \times D \to Set$ by taking the composites

$\array{ F' & = & (C^{op} \times C \times C) \times (C \times C^{op} \times C^{op}) & \stackrel{proj}{\to} & C^{op} \times C \times C & \stackrel{F}{\to} & Set \\ & & (x', y', z'; x, y, z) & \mapsto & (x', x, y') & \stackrel{F}{\mapsto} & F(x', x, y') }$ $\,$ $\,$ $\array{ G' & = & (C^{op} \times C \times C) \times (C \times C^{op} \times C^{op}) & \stackrel{proj'}{\to} & C \times C \times C^{op} & \stackrel{G}{\to} & Set \\ & & (x', y', z'; x, y, z) & \mapsto & (y', z', z) & \stackrel{G}{\mapsto} & G(y', z', z) }$

Now let’s put $a' = (x', y', z')$ and $a = (x, y, z)$, considered as objects in $D$. An arrow $\phi: a' \to a$ in $D$ thus amounts to a triple of arrows $f: x' \to x$, $g: y \to y'$, $h: z \to z'$ all in $C$. Following the instructions above, we have $F'(a', a) = F(x', x, y')$ and $G(a', a) = G(y', z', z)$.

Now if we write down a dinaturality hexagon for $\alpha: F' \stackrel{\bullet}{\to} G'$, we get a diagram of shape

$\array{ F'(a, a') & \stackrel{F'(1, \phi)}{\to} & F'(a, a) & \stackrel{\alpha_a}{\to} & G'(a, a) \\ _\mathllap{F(\phi, 1)} \downarrow & & & & \downarrow_\mathrlap{G'(\phi, 1)} \\ F'(a', a') & \underset{\alpha_{a'}}{\to} & G'(a', a') & \underset{G'(1, \phi)}{\to} & G(a', a) }$

which translates to a hexagon of shape

$\array{ F(x, x', y) & \stackrel{F(1, f, 1)}{\to} & F(x, x, y) & \to & G(y, z, z) \\ _\mathllap{F(f, 1, g)} \downarrow & & & & \downarrow_\mathrlap{G(g, h, 1)} \\ F(x', x', y') & \to & G(y', z', z') & \underset{G(1, h, 1)}{\to} & G(y', z', z) }$

where the unlabeled arrows refer to the extranatural transformation. This contains two small miracles: (1) it seems to capture the general notion of extranatural transformation (all three forms at once: wedge, cowedge, and ordinary naturality), and (2) it seems that I didn’t screw up after all (or did I?).

This may seem like an advertisement for dinatural transformations, since it seems to generalize the general form of extranaturality. But my point is that, in retrospect, the notion of dinaturality may be too general, and in practice it’s the extranatural transformations which are more important. (We need to have a more convincing example of a dinatural that cannot be expressed in terms of extranaturality than currently exists at the nLab. I think Freyd once showed me one that involves equalizers in some tricky way, but I don’t remember it now.)

By the way, Fosco: have you been contemplating where to publish your note?

• CommentRowNumber16.
• CommentAuthorFosco
• CommentTimeDec 21st 2014

have you been contemplating where to publish your note?

I was, and still am, unsure about it, given that the note doesn’t contain anything “new”.

Once it has been polished, do you truly believe it is worth to be published somewhere? If yes, I’m open to suggestion!

(I’ll read carefully the rest of your answer in a while, thanks for the effort!!)

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeDec 21st 2014

An ideal place for such material would be the $n$Lab :-)

If you want an offician stamp-of-approval on it, then there are the Publications of the nLab .

• CommentRowNumber18.
• CommentAuthorTodd_Trimble
• CommentTimeDec 21st 2014
• (edited Dec 21st 2014)

If you want an offician stamp-of-approval on it, then there are the Publications of the nLab .

Urs, that’s exactly what I was thinking! But I didn’t want to suggest that before asking Fosco.

• CommentRowNumber19.
• CommentAuthorFosco
• CommentTimeDec 22nd 2014
• (edited Dec 22nd 2014)

Good to know, when the polishing phase is over I’ll surely think about it!

Now let’s go back to the discussion about extra VS dinatural. It seems to me we reached an important point. I feel slightly confused by the fact that dinaturals are particular examples of extranaturals, but also, as you point out (it doesn’t seem to me that you said something wrong), I can obtain a extranatural from a dinatural. This happens everywhere (limits are adjoints are terminal objects are representing objects), but now I am confused by what is the best choice, given this situation, to present ends and coends. Is it purely a matter of taste? I don’t believe.

For what concerns the diagrammatic calculus, I drew these diagrams. I also had a nice evening trying to draw them in LaTeX :-) I’ll put the result on the Lab page.

Finally, now I remember that I would like to add to the note the following example:

Let $\mathbf C$ be a rigid tensor category. The object

$\int^A A\otimes A^*$

(provided the coend exists in $\mathbf C$) is an internal Hopf algebra, i.e. some diagrams are commutative for maps $\mu,\eta, \Delta, \epsilon$ giving a bialgebra structure, and an antipode $S$.

This result is quoted in various sources, but I’m still unable to find a nifty proof, and (provided it exists) a proof relying on coend calculus. Any clue?

• CommentRowNumber20.
• CommentAuthorTodd_Trimble
• CommentTimeDec 22nd 2014

the fact that dinaturals are particular examples of extranaturals

Could you please explain how?

I’ll try to have a think later about your Hopf algebra question.

• CommentRowNumber21.
• CommentAuthorFosco
• CommentTimeDec 22nd 2014
• (edited Jan 27th 2015)

Uhm. I thought that the hexagon giving extranaturality, in the case where $F\colon \mathcal{A}\times \mathcal{B}^{op}\times\mathcal{B}\to\mathcal{D}$ is a functor $\bar F\colon\mathcal{B}^{op}\times\mathcal{B}\to\mathcal{D}$ and $G \colon \mathcal{A}\times \mathcal{C}^{op}\times\mathcal{C}\to\mathcal{D}$ is a functor $\bar G\colon\mathcal{B}^{op}\times\mathcal{B}\to\mathcal{D}$ gave the dinaturality condition for $\{\alpha_{A B C}\colon F(A,B,B)\to G(A,C,C)\}$ reduced to $\{ \alpha_{B B}\colon \bar F(B,B)\to \bar G(B,B)\}$. But this is not true! The two hexagons are slightly different.

Moral: always do the maths. :-)

• CommentRowNumber22.
• CommentAuthorTodd_Trimble
• CommentTimeJan 18th 2015

I just learned, courtesy of Ingo Blechschmidt commenting at the Café, of the appearance of Fosco’s article on the arXiv.

I haven’t tracked the changes (if there are any – none leapt to attention) between the version dated December 12, 2014 and the current arXiv version.

• CommentRowNumber23.
• CommentAuthorFosco
• CommentTimeJan 18th 2015

I am indeed revising the document after having spotted several typos and (not so) subtle errors, kindly pointed out by amg. Co/ends 0.2 will include an elaboration on the points you mention, too (especially extranatural transformations)!

• CommentRowNumber24.
• CommentAuthorFosco
• CommentTimeFeb 8th 2015

I completed a new subsection about extranaturality: it is available here. Let me know how it can be improved!

In a few days the version 2 will be uploaded on arXiv.

• CommentRowNumber25.
• CommentAuthorFosco
• CommentTimeJan 7th 2017

This is to inform you that the version 3 of the note is now on the arXiv (same address). I’d like to know 1. If such a paper has any hope to be published somewhere, given that it doesn’t contain much original work, and if yes, 2. Which journal minimizes the refereeing process being, at the same time, a nicely-ranked one.

• CommentRowNumber26.
• CommentAuthorDavidRoberts
• CommentTimeJan 7th 2017

I think Expositiones Mathematicae might fit (though I don’t like the publisher). There’s EMS surveys in mathematical sciences, or many others mentioned at this MO question.