Mentioned the simpler formula for the internal-hom.

]]>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.

]]>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?

]]>Fix variance in the promonoidal case.

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

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

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

]]>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)?)

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

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

]]>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.

]]>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).

]]>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)

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

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.

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?

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

]]>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.

]]>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.

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

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

]]>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 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?

]]>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.

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?

]]>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.

]]>Might you have a reference?

]]>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 $P C$.

]]>