nForum - Discussion Feed (Day convolution) 2022-05-26T12:33:13-04:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher Mike Shulman comments on "Day convolution" (86044) https://nforum.ncatlab.org/discussion/3165/?Focus=86044#Comment_86044 2020-07-29T21:06:44-04:00 2022-05-26T12:33:12-04:00 Mike Shulman https://nforum.ncatlab.org/account/3/ Mentioned the simpler formula for the internal-hom. diff, v65, current

Mentioned the simpler formula for the internal-hom.

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Sam Staton comments on "Day convolution" (85794) https://nforum.ncatlab.org/discussion/3165/?Focus=85794#Comment_85794 2020-07-13T14:58:48-04:00 2022-05-26T12:33:12-04:00 Sam Staton https://nforum.ncatlab.org/account/1642/ I think this is quite widely used, at least in CS, for example Pym-O’Hearn-Yang, end of Section 2. Would still be interested to hear a canonical reference.

I think this is quite widely used, at least in CS, for example Pym-O’Hearn-Yang, end of Section 2. Would still be interested to hear a canonical reference.

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Mike Shulman comments on "Day convolution" (85781) https://nforum.ncatlab.org/discussion/3165/?Focus=85781#Comment_85781 2020-07-12T10:42:07-04:00 2022-05-26T12:33:12-04:00 Mike Shulman https://nforum.ncatlab.org/account/3/ The formula on the page for the internal-hom: [X,Y] Day(c)=&Integral;c 1,c 2V(&Cscr;(c&otimes; &Cscr;c 1,c 2),V(X(c 1),Y(c 2))). [X,Y]_{Day}(c) = \underset{c_1,c_2}{\int} ...

The formula on the page for the internal-hom:

$[X,Y]_{Day}(c) = \underset{c_1,c_2}{\int} V\left( \mathcal{C}(c \otimes_{\mathcal{C}} c_1,c_2), V(X(c_1), Y(c_2)) \right) \,.$

seems like it can be simplified in the monoidal (rather than promonoidal) case (which is the case that it’s currently written for): by the $V$-enriched Yoneda lemma isn’t this isomorphic to

$\int_{c_1} V(X(c_1), Y(c\otimes_{\mathcal{C}} c_1)) \quad?$

Is there a reference for this?

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Mike Shulman comments on "Day convolution" (85780) https://nforum.ncatlab.org/discussion/3165/?Focus=85780#Comment_85780 2020-07-12T10:37:45-04:00 2022-05-26T12:33:12-04:00 Mike Shulman https://nforum.ncatlab.org/account/3/ Fix variance in the promonoidal case. diff, v64, current

Fix variance in the promonoidal case.

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Théo de Oliveira S. comments on "Day convolution" (84775) https://nforum.ncatlab.org/discussion/3165/?Focus=84775#Comment_84775 2020-05-21T16:35:24-04:00 2022-05-26T12:33:12-04:00 Théo de Oliveira S. https://nforum.ncatlab.org/account/2018/ Hi Richard, thanks for merging the threads! Happy to know mentioning the edit was appropriate :)

Hi Richard, thanks for merging the threads! Happy to know mentioning the edit was appropriate :)

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Richard Williamson comments on "Day convolution" (84762) https://nforum.ncatlab.org/discussion/3165/?Focus=84762#Comment_84762 2020-05-21T05:36:09-04:00 2022-05-26T12:33:12-04:00 Richard Williamson https://nforum.ncatlab.org/account/822/ Now done. Announcements of future edits to the page should now appear here.

Now done. Announcements of future edits to the page should now appear here.

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Richard Williamson comments on "Day convolution" (84761) https://nforum.ncatlab.org/discussion/3165/?Focus=84761#Comment_84761 2020-05-21T04:37:19-04:00 2022-05-26T12:33:12-04:00 Richard Williamson https://nforum.ncatlab.org/account/822/ Hi Théo, I’ll merge the threads later when I’m at my computer. Yes, it is appropriate to announce this edit :-).

Hi Théo, I’ll merge the threads later when I’m at my computer. Yes, it is appropriate to announce this edit :-).

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Théo de Oliveira S. comments on "Day convolution" (84753) https://nforum.ncatlab.org/discussion/3165/?Focus=84753#Comment_84753 2020-05-20T21:23:16-04:00 2022-05-26T12:33:12-04:00 Théo de Oliveira S. https://nforum.ncatlab.org/account/2018/ Oh, this should be on the other page on Day convolution. I think writing the above directly from the nLab edit page prompt created an extra page. Is it possible to move this to the other discussion ...

Oh, this should be on the other page on Day convolution. I think writing the above directly from the nLab edit page prompt created an extra page. Is it possible to move this to the other discussion page?

(Also, am I correct in assuming this is an edit that should be mentioned on the discussion page (well, the correct page, that is)?)

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Theo comments on "Day convolution" (84752) https://nforum.ncatlab.org/discussion/3165/?Focus=84752#Comment_84752 2020-05-20T21:19:45-04:00 2022-05-26T12:33:12-04:00 Theo https://nforum.ncatlab.org/account/1016/ Started adding more material on Day convolution for promonoidal categories. (I’ll add more later) diff, v62, current

Started adding more material on Day convolution for promonoidal categories. (I’ll add more later)

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Théo de Oliveira S. comments on "Day convolution" (84731) https://nforum.ncatlab.org/discussion/3165/?Focus=84731#Comment_84731 2020-05-20T01:43:08-04:00 2022-05-26T12:33:12-04:00 Théo de Oliveira S. https://nforum.ncatlab.org/account/2018/ @Mike Shulman: Thanks! That makes much more sense.

@Mike Shulman: Thanks! That makes much more sense.

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Mike Shulman comments on "Day convolution" (84723) https://nforum.ncatlab.org/discussion/3165/?Focus=84723#Comment_84723 2020-05-19T20:14:00-04:00 2022-05-26T12:33:12-04:00 Mike Shulman https://nforum.ncatlab.org/account/3/ Theo: I would presume the categories in question have as objects (pro)monoidal structures on CC and as morphisms (pro)monoidal functors that are the identity on objects (which explains why the ...

Theo: I would presume the categories in question have as objects (pro)monoidal structures on $C$ and as morphisms (pro)monoidal functors that are the identity on objects (which explains why the dimension stops there), and that what Day meant by “correspond bijectively to within isomorphism” is that there is a bijection between the set of isomorphism classes of these categories (a weaker version of saying there is an equivalence of categories between them).

I’ve never heard the name “functor category theorem”, nor seen a proof of it written out.

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Mike Shulman comments on "Day convolution" (84722) https://nforum.ncatlab.org/discussion/3165/?Focus=84722#Comment_84722 2020-05-19T20:09:38-04:00 2022-05-26T12:33:12-04:00 Mike Shulman https://nforum.ncatlab.org/account/3/ Max: I think Sam was talking about the universal property in Corollary 2.4 on the page, which expresses the binary form of that same universal property (though without the language of ...

Max: I think Sam was talking about the universal property in Corollary 2.4 on the page, which expresses the binary form of that same universal property (though without the language of multicategories).

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Théo de Oliveira S. comments on "Day convolution" (84679) https://nforum.ncatlab.org/discussion/3165/?Focus=84679#Comment_84679 2020-05-18T18:26:51-04:00 2022-05-26T12:33:13-04:00 Théo de Oliveira S. https://nforum.ncatlab.org/account/2018/ Some questions regarding Proposition 2.5 (sorry if they are too basic; I’m still learning the basics): 1) Is an equivalence of categories the correct way to formulate this result? I ask this ...

Some questions regarding Proposition 2.5 (sorry if they are too basic; I’m still learning the basics):

1) Is an equivalence of categories the correct way to formulate this result?

I ask this because Day speaks of a bijection both in his report [link, beginning of page 5]:

As one would expect, biclosed structures on $[\mathcal{A},\mathcal{V}]$ correspond bijectively to premonoldal structures on $\mathcal{A}$ to within “isomorphism”.

and in his thesis [link, page 68 (corresponding to page 74 of the PDF), Theorem 3.1.2]:

[…] Then there exists a bijection between promonoidal completions of the data $(P,J)$ on $\mathcal{A}$ and biclosed completions of $(\overline{\otimes},J)$ on $F$. [Here $F$ is the $\mathcal{V}$-functor category $[\mathcal{A},\mathcal{V}]$.]

That is, there are three levels of category-theoretical “highness” we might use here: (stated below for the promonoidal side only)

1. The set of promonoidal structures on $\mathcal{C}$.

2. The full subcategory $\mathcal{X}$ (for lack of better notation) with $\mathrm{Obj}(\mathcal{X})=\{\mathcal{C}\}$ of the $1$-category $\mathsf{ProMonCats}_{\mathcal{V}}$ of

• promonoidal $\mathcal{V}$-categories and
• promonoidal $\mathcal{V}$-functors between them.
3. The full sub-bicategory $\mathcal{X}$ with $\mathrm{Obj}(\mathcal{X})=\{\mathcal{C}\}$ of the bicategory $\mathsf{ProMonCats}_{\mathcal{V}}^\mathsf{bi}$ of

• promonoidal $\mathcal{V}$-categories,
• promonoidal $\mathcal{V}$-functors between them, and
• promonoidal $\mathcal{V}$-natural transformation between these.

Which of these should we use to state Day’s theorem?

2) On terminology: Day calls this bijection (equivalence?) between promonoidal structures on $\mathcal{C}$ and biclosed monoidal structures on $[\mathcal{C},\mathcal{V}]$ the “functor category theorem” in his PhD thesis. Is this name used today as well?

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maxsnew comments on "Day convolution" (84674) https://nforum.ncatlab.org/discussion/3165/?Focus=84674#Comment_84674 2020-05-18T15:46:05-04:00 2022-05-26T12:33:13-04:00 maxsnew https://nforum.ncatlab.org/account/1534/ Could you spell it out a bit for me? I don’t see the connection

Could you spell it out a bit for me? I don’t see the connection

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Sam Staton comments on "Day convolution" (84653) https://nforum.ncatlab.org/discussion/3165/?Focus=84653#Comment_84653 2020-05-17T08:29:08-04:00 2022-05-26T12:33:13-04:00 Sam Staton https://nforum.ncatlab.org/account/1642/ Hi Max, There is a bit about this kind of thing under Day convolution#Monoids and Day convolution#Modules, but not this exactly. I don’t have a reference.

Hi Max, There is a bit about this kind of thing under Day convolution#Monoids and Day convolution#Modules, but not this exactly. I don’t have a reference.

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maxsnew comments on "Day convolution" (84632) https://nforum.ncatlab.org/discussion/3165/?Focus=84632#Comment_84632 2020-05-16T15:31:52-04:00 2022-05-26T12:33:13-04:00 maxsnew https://nforum.ncatlab.org/account/1534/ In Mike Shulman’s new paper https://arxiv.org/abs/2004.08487 he gives a universal property of a Day convolution-like product on modules of polycategories as the tensor product representing a ...

In Mike Shulman’s new paper https://arxiv.org/abs/2004.08487 he gives a universal property of a Day convolution-like product on modules of polycategories as the tensor product representing a symmetric multicategory of modules.

It seems pretty clear to me that the standard Day convolution could be similarly described as a tensor product for a presheaves on a monoidal category (and probably presheaves on a multicategory) where the multi-arrows

$f : P_1,P_2\ldots \to Q$

are given by maps

$f : P_1(m_1) * P_2(m_2) \cdots \to Q(m_1 * m_2 * \cdots)$

with a naturality condition. Does anyone have a reference for this universal property? It looks fairly obvious in retrospect. If not, I’ll just cite Shulman.

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Urs comments on "Day convolution" (57979) https://nforum.ncatlab.org/discussion/3165/?Focus=57979#Comment_57979 2016-06-23T04:44:44-04:00 2022-05-26T12:33:13-04:00 Urs https://nforum.ncatlab.org/account/4/ Oh I see. So in particular it is immediate that the profunctor description matches the coend description.

Oh I see. So in particular it is immediate that the profunctor description matches the coend description.

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Mike Shulman comments on "Day convolution" (57977) https://nforum.ncatlab.org/discussion/3165/?Focus=57977#Comment_57977 2016-06-23T04:32:55-04:00 2022-05-26T12:33:13-04:00 Mike Shulman https://nforum.ncatlab.org/account/3/ I would think not much work, since composition of profunctors is a coend by definition.

I would think not much work, since composition of profunctors is a coend by definition.

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Urs comments on "Day convolution" (57969) https://nforum.ncatlab.org/discussion/3165/?Focus=57969#Comment_57969 2016-06-23T02:45:38-04:00 2022-05-26T12:33:13-04:00 Urs https://nforum.ncatlab.org/account/4/ Mike, how about this: one reason for keeping the coend version is that it gives a fairly explicit description of what the structure maps actually are. For instance for the application to the ...

one reason for keeping the coend version is that it gives a fairly explicit description of what the structure maps actually are. For instance for the application to the symmetric smash product of spectra, one wants an explicit formula for what the braiding $\tau^{Day}$ in the Day convolution structure is. The coend formula provides this (it’s $\tau_{X,Y}^{Day}(c) \simeq \overset{c_1,c_2}{\int} \mathcal{C}(\tau_{c_1,c_2}^{\mathcal{C}},c) \otimes_V \tau_{X(c_1), X(c_2)}^V$) and this may be directly evaluated to yield the explicit operation on symmetric and orthogonal spectra known from the literature.

How much work is it to extract such explicit expressions from the profunctorial description?

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Urs comments on "Day convolution" (57962) https://nforum.ncatlab.org/discussion/3165/?Focus=57962#Comment_57962 2016-06-22T13:32:08-04:00 2022-05-26T12:33:13-04:00 Urs https://nforum.ncatlab.org/account/4/ Thanks! Do you think these abstract arguments should replace the explicit coendy ones, or live alongside them? For the usability of the article it would probably be good to have both arguments ...

Thanks!

Do you think these abstract arguments should replace the explicit coendy ones, or live alongside them?

For the usability of the article it would probably be good to have both arguments stated in the entry. But for the time being I removed the one in terms of coends, since it was very incomplete. I have more details written out in the respective section at model structure on orthogonal spectra, but there I chose to specify $(V, \otimes_V)$ to $(Top^{\ast/}_{cg}, \wedge)$ and so when copying that over to Day convolution I would have to change notation throughout, for which I don’t have the energy now.

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Mike Shulman comments on "Day convolution" (57960) https://nforum.ncatlab.org/discussion/3165/?Focus=57960#Comment_57960 2016-06-22T12:53:43-04:00 2022-05-26T12:33:13-04:00 Mike Shulman https://nforum.ncatlab.org/account/3/ I added a brief description of the profunctory approach to the page Day convolution. It would be nice to spell out the details of the proof of the monoidal structure this way, and also to include ...

I added a brief description of the profunctory approach to the page Day convolution. It would be nice to spell out the details of the proof of the monoidal structure this way, and also to include the argument I suggested in #23 for the other statement. Do you think these abstract arguments should replace the explicit coendy ones, or live alongside them?

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Mike Shulman comments on "Day convolution" (57957) https://nforum.ncatlab.org/discussion/3165/?Focus=57957#Comment_57957 2016-06-22T12:13:05-04:00 2022-05-26T12:33:13-04:00 Mike Shulman https://nforum.ncatlab.org/account/3/ Not a very explicit one, unfortunately. In Theorem 11.22 of enriched indexed categories I sketched how to do this in a more general case, but I didn’t give a lot of details.

Not a very explicit one, unfortunately. In Theorem 11.22 of enriched indexed categories I sketched how to do this in a more general case, but I didn’t give a lot of details.

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Urs comments on "Day convolution" (57956) https://nforum.ncatlab.org/discussion/3165/?Focus=57956#Comment_57956 2016-06-22T12:08:35-04:00 2022-05-26T12:33:13-04:00 Urs https://nforum.ncatlab.org/account/4/ Might you have a reference?

Might you have a reference?

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Mike Shulman comments on "Day convolution" (57955) https://nforum.ncatlab.org/discussion/3165/?Focus=57955#Comment_57955 2016-06-22T12:04:00-04:00 2022-05-26T12:33:13-04:00 Mike Shulman https://nforum.ncatlab.org/account/3/ Well, that together with something Yoneda-y in Prof. The image of &otimes;\otimes in ProfProf is the induced promonoidal structure on CC, which is still a step away from the Day convolution on ...

Well, that together with something Yoneda-y in Prof. The image of $\otimes$ in $Prof$ is the induced promonoidal structure on $C$, which is still a step away from the Day convolution on $P C$.

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