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See Day convolution
I started writing up the actual theorem from Day’s paper “On closed categories of functors”, regarding an extension of the “usual” Day convolution. He identifies an equivalence of categories between biclosed monoidal structures on the presheaf category $V^{A^{op}}$ and what are called pro-monoidal structures on A (with appropriate notions of morphisms between them) (“pro-monoidal” structures were originally called “pre-monoidal”, but in the second paper in the series, he changed the name to “pro-monoidal” (probably because they are equivalent to monoidal structures on the category of “pro-objects”, that is to say, presheaves)).
This is quite a bit stronger than the version that was up on the lab, and it is very powerful. For instance, it allows us to seamlessly extend the Crans-Gray tensor product from strict ω-categories to cellular sets (such that the reflector and Θ-nerve functors are strong monoidal). This is the key ingredient to defining lax constructions for ω-quasicategories, and in particular, it’s an important step towards the higher Grothendieck construction, which makes use of lax cones constructed using the Crans-Gray tensor product.
That’s quite true; promonoidal structures are important. Thanks for doing this, Harry.
The name ’promonoidal’ was of course ill-chosen (conflict with ’pro-object’, profinite, etc.).
Yeah, I spent at least a few hours trying to figure out how to extend the GC Tensor product from ω-categories to cellular sets by proving that it was dense or something stupid, but then I went back and bam, it’s the first major theorem.
Hereby I am moving the following ancient discussion from the enty Day convolution to here:
Eric says: When I see “convolution”, I think “Fourier transform”. Is Day convolution somehow related to a categorified version of Fourier transforms?
Todd says: Yes, something like that. I talk a little about this in the article on operads, in the detailed theoretical section.
The usual Fourier transform (for periodic functions) passes between Fourier coefficients $a_n$ and functions $\sum_n a_n z^n$ on $S^1$. One way of categorifying this is to pass from the category of functors $a: \mathbb{N} \to Set$ (considered as a monoidal category with respect to Day convolution) to their so-called “analytic functors” $\hat{a}: Set \to Set$, mapping a set $x$ to $\hat{a}(x) = \sum_n a_n \cdot x^n$. The “categorified Fourier transform” $a \mapsto \hat{a}$ takes Day convolution products to (pointwise) cartesian products.
If the “Fourier transform” is properly formulated (using enriched tensor products), then the same holds for any monoidal category in place of the discrete monoidal category $\mathbb{N}$.
AnonymousCoward says: The passage to analytic functors seems more like a z-transform or Laplace transform. In the particular case of species, it is the Laplace transform formula that applies to the analytic functor of a derivative of a species, not the Fourier transform one involving multiplication by the imaginary unit.
The use of hom above is reminiscent of the Dirac delta. Is there a connection?
John Baez says: It’s true that the passage from a sequence $a_n$ to a power series $\sum_n a_n z^n$ is precisely the $z$-transform. If we set $z = exp(i \theta)$, we get the Fourier transform — but as you note this makes use of the imaginary unit $i$, which plays no evident role in Day convolution. So, the analogies Todd is discussing become most precise if we work with the $z$-transform. But the Fourier transform is closely related.
On the other hand, I’ve discovered that many ’pure mathematicians’ don’t know about the $z$-transform — at least, not under that name. I think it’s ’engineers’ who talk most about the $z$-transform. So, if you’re trying to explain Day convolution to pure mathematicians, it’s pedagogically best to start talking about the Fourier transform, and then later mention the $z$-transform.
In general $hom$ is a categorified version of an inner product. I’m too lazy to figure out how this is related to the Dirac delta, but I would not be surprised if there were a connection.
Urs: maybe all that “anonymous coward” is looking for is this statement:
if $C$ is a discrete category (i.e. just a set regarded as a category with only identity morphisms) then a functor $C \o Set$ is like a $\mathb{Z}$-valued function on the set $C$ and then for every object $c$ in $C$ the functor $Hom_C(c,-) = \delta_c$ is the Kronecker delta on $C$ at $c$, in that
$Hom_C(c,d) = \left\{ \array{ * & if c = d \\ \emptyset & if c \neq d } \right.$I have added this remark now explicitly to the entry above.
Just as well to remove that passage – the exchange strikes me as one where people are talking past one another. I’m happy to say this is the first time I’ve seen most of it.
I have added to Day convolution statement (here) of the examples of symmetric smash products of spectra.
(While MMSS00 make the smash product on orthogonal and symmetric spectra, among others, explicit as a Day convolution product, the analogous statement in Lydakis 98 is less explicit, not the least since Lydakis chooses to speak in $Cat^{\Delta^{op}}$ as opposed to in $sSet Cat$. But otherwise the model in Lydakis 98 is particularly nice, I think. It would be good to make its Day convolution structure more explicit in the entry (in any entry).)
I have added at Day convolution in the section Definition – For monoidal categories the statement that Day convolution of a tensor product is equivalently the left Kan extension of the “exterenal tensor product” along the original tensor product.
I am very surprised that hadn’t been there already.
I think this article is due for some cleaning up, now that I have a look.
That would be great if you would look into it again. Please just give me five minutes to finish something…
Okay, I am done editing that entry for the moment.
I have now added in the subsection Properties – Monoids discussion that monoids with respect to Day convolution are equivalently lax monoidal functors out of the base monoidal category (“FSPs”).
I have added to the proposition in that subsection Properties – Monoids also the corresponding statement for modules: under the equivalence of monoids with respect to Day convolution with monoidal functors, module objects with respect to Day convolution correspond to what is sometimes called “modules over monoidal functors”. And so for completeness I created a brief page for the latter concept.
Maybe we could jointly think about how to streamline further arguments about Day convolution from the article “Model categories of diagram spectra”.
For instance theorem 2.2 there states that with $R$ a monoid object in $([\mathcal{D},V], \otimes_{Day})$ (for some monoidal $V$-category $\mathcal{D}$) then (paraphrasing the formulation in the article) we have an equivalence of categories
$R Mod \simeq [ R FreeMod^{op}, V] \,.$On p. 63 of the article a somewhat laborious proof is given, but I suppose there ought to be a quick abstract proof of this?
Hmm, that sounds like some kind of Kleisli / Eilenberg-Moore thing?
With Day convolution cropping up on the focusing thread, I took again at Day convolution.
Presumably in definition 1, the $F$s should be $X$ and $Y$. And why is there $[c_1 \otimes c_2, c]$, with tensor product as domain, when above in the definition of $F \star G$, the tensor product is the codomain, $Hom_C(-, c \otimes d)$?
Presumably in definition 1, the $F$s should be $X$ and $Y$.
Yes, thanks. I have fixed it now.
And why is there $[c_1 \otimes c_2, c]$, with tensor product as domain, when above in the definition of $F \star G$, the tensor product is the codomain, $Hom_C(-, c \otimes d)$?
Because the text switches from considering functors on $C^{op}$ to considering functors on $\mathcal{C}$.
Of course this ought to be dealt with better in the presentation. I had been meaning to edit further, but didn’t find the time yet.
Because the text switches…
Oh, yes.
Of course this ought to be dealt with better in the presentation.
Yes, it is a bit suboptimal. Later it’s proved twice that the yoneda embedding of the unit is the unit for Day convolution.
We needs some more recruits with time.
I’ll get to Day convolution in a few weeks. Then I’ll be streamlining the entry a bit more.
(But of course if anyone else feels like looking into it right now, please do.)
I have filled in all the remaining proofs in Properties – Closed monoidal structure.
Also I tried to harmonize notation and organization of the entry a little more.
Coming back to #12:
I have tried to fill in more details of the proof of the proposition (here) that for $R$ a monoid with respect to Day convolution over some monoidal $V$-enriched category $\mathcal{C}$, then right $R$-modules are a $V$-presheaf category over the free $R$-modules free on objects of $\mathcal{C}$:
$Mod_R \simeq [Free_{\mathcal{C}}Mod^{op},V] \,.$This is one of these cases where the basic idea is really simple, but making the details fully explicit is really tedious. Or maybe somebody knows a shortcut.
The most tedious bit is omitted in the proof offered in MMSS 00. It’s the check that not only are both sides of the above equivalence given by functors with the same structure, but that also the compatibility condition on this data is the same on both sides.
Verifying this essentially comes down to checking that under identifying the hom-objects in $R Free_{\mathcal{C}}Mod$ like so
$\begin{aligned} R Free_{\mathcal{C}}Mod(c_2,c_1) & = R Mod( y(c_2) \otimes_{Day} R , y(c_1) \otimes_{Day} R) \\ & \simeq [\mathcal{C},V](y(c_2), y(c_1) \otimes_{Day} R) \\ & \simeq (y(c_1) \otimes_{Day} R)(c_2) \\ & \simeq \overset{c_3,c_4}{\int} \mathcal{C}(c_3 \otimes c_4,c_2) \otimes_V \mathcal{C}(c_1, c_3) \otimes_V R(c_4) \\ & \simeq \overset{c_4}{\int} \mathcal{C}(c_1 \otimes c_4,c_2) \otimes_V R(c_4) \end{aligned}$that then composition in $R Free_{\mathcal{C}}Mod$ is given by
$\begin{aligned} R FreeMod(c_2, c_1) \otimes_V R FreeMod(c_3, c_2) & = \left( \overset{c_4}{\int} \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_4, c_2) \otimes_V R(c_4) \right) \otimes_V \left( \overset{c_5}{\int} \mathcal{C}(c_2 \otimes_{\mathcal{C}} c_5, c_3 ) \otimes_V R(c_5) \right) \\ & \simeq \overset{c_4, c_5}{\int} \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_4 , c_2 ) \otimes_V \mathcal{C}(c_2 \otimes_{\mathcal{C}} c_5, c_3) \otimes_V R(c_4) \otimes_V R(c_5) \\ & \longrightarrow \overset{c_4,c_5}{\int} \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_4 \otimes_{\mathcal{C}} c_5 , c_2 \otimes_{\mathcal{C}} c_5 ) \otimes_V \mathcal{C}(c_2 \otimes c_5, c_3) \otimes_V R(c_4 \otimes_{\mathcal{C}} c_5 ) \\ & \longrightarrow \overset{c_4, c_5}{\int} \mathcal{C}(c_1\otimes_{\mathcal{C}} c_4 \otimes_{\mathcal{C}} c_5 , c_3) \otimes_V R(c_4 \otimes_{\mathcal{C}} c_5 ) \\ & \longrightarrow \overset{c_4}{\int} \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_4 , c_3) \otimes_V R(c_4 ) \end{aligned} \,,$where
the first morphism is, in the integrand, the tensor product of
forming the tensor product of hom-objects of $\mathcal{C}$ with the identity of $c_5$
$\mathcal{C}(c_1 \otimes_{\mathcal{C}} c_4, c_2 ) \simeq \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_4, c_2 ) \otimes_V 1_V \overset{}{\longrightarrow} \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_4, c_2 ) \otimes \mathcal{C}(c_5,c_5) \longrightarrow \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_4 \otimes_{\mathcal{C}} c_5, c_2 \otimes_{\mathcal{C}} c_5)$the monoidal functor incarnation $R(c_4) \otimes_V R(c_5)\longrightarrow R(c_4 \otimes_{\mathcal{C}} c_5 )$ of the monoid structure on $R$;
the second morphism is, in the integrand, given by composition in $\mathcal{C}$;
the last morphism is the morphism induced on coends by regarding extranaturality in $c_4$ and $c_5$ separately as a special case of extranaturality in $c_6 \coloneqq c_4 \otimes c_5$ (and then renaming).
I think that’s clear, but writing out a full proof will be as tedious as boring. There must be some abstract argument that gives all this.
Yes, there must…
Can we give $Free_{\mathcal{C}}Mod^{op}$ a universal property? It feels like it should be some kind of lax colimit or collage.
That would be nice. I don’t know, but I’ll be thinking about it. If you come up with anything, please drop a note.
Does it simplify things any to do it the following way? Let $C_R$ be the $V$-category whose hom-objects and composition are defined by the coend given above. Show that diagrams on $C_R$ are equivalent to $R$-modules. Now use the fact that for any $V$-category $D$, the representables $D(d,-)$ are the free $D$-diagrams on the objects of $D$ (by the Yoneda lemma) and the hom-objects of diagrams between representables are just the hom-objects of $D$ (by the Yoneda embedding); thus $D$ is the category of free $D$-diagrams on objects of $D$.
I see. Thanks! That sounds good.
What is your preferred way to write down a fully explicit proof of the coherence laws for the Day convolution product over some monoidal $V$-category $\mathcal{C}$?
I find that explicitly exhibiting the pentagon identity for the Day-associator is easy, because we may fix canonical isomorphisms going horizontally in the following diagram
$\array{ ((X \otimes_{Day} Y) \otimes_{Day} Z)(c) &\simeq& \overset{c_1,c_2, c_3}{\int} \mathcal{C}((c_1 \otimes c_2) \otimes c_3) \otimes_V ((X(c_1) \otimes_V Y(c_2)) \otimes_V Z(c_3)) \\ {}^{\mathllap{ \alpha^{Day}_{X,Y,Z}(c) }}\downarrow && \downarrow^{\mathrlap{ \overset{c_1,c_2,c_3}{\int} \mathcal{C}( \alpha^{\mathcal{C}}_{c_1,c_2,c_3} , c ) \otimes_V \alpha^{V}_{X(c_1), X(c_2), X(c_3)} }} \\ (X \otimes_{Day} (Y \otimes_{Day} Z) )(c) &\simeq& \overset{c_1, c_2, c_3}{\int} \mathcal{C}(c_1\otimes ( c_2 \otimes c_3), c ) \otimes_V (X(c_1) \otimes_V (Y(c_2) \otimes_V Z(c_3))) }$and this way the pentagon for $\otimes_{Day}$ follows immediately from that of $\mathcal{C}$ and $V$.
Similarly for the braiding, in case $\mathcal{C}$ is braided.
But for the triangle identities the situation is messier. The right Day unitor may be exhibited as
$\array{ (X \otimes_{Day} y(1))(c) &\simeq& \overset{c_1}{\int} \mathcal{C}(c_1 \otimes 1, c) \otimes_V X(c_1) \\ {}^{\mathllap{r^{Day}_{X}(c) }}\downarrow && \downarrow^{\mathrlap{ \overset{c_1}{\int} \mathcal{C}(r^{\mathcal{C}}_{c_1},c) \otimes_V X(c_1) }} \\ X(c) &\simeq& \overset{c_1}{\int} \mathcal{C}(c_1,c) \otimes_V X(c_1) }$and similarly for the left unitor, and it is clear that the proof of the triangle identity for the Day-unitors similarly wants to come, under the integral sign, from those of $\mathcal{C}$. But writing this down fully rigorously now seems to get a little more awkward, since in $(X \otimes_{Day} (y(1) \otimes_{Day} Y) )$ the associator, as above, is defined after an application of the co-Yoneda lemma which produces the coend over three variables, while the unitors apply before passing through that co-Yoneda lemma, while there are still two coends, each over two variables. So to appeal to proof via the triangle identity in $\mathcal{C}$, some extra words need to be said.
I know that it’s all just by universality and coherence. But what would be a concise and yet complete way to articulate the proof of the triangle identity for Day convolution?
I like to think of it as the pseudofunctor $Cat\to Prof$ taking the monoidal category $C$ to a pseudomonoid in $Prof$ (a promonoidal category), and then doing everything in terms of composites of profunctors without ever needing to write them as coends. Does that help?
I see. You are saying that Day convolution is just the image of $\otimes \colon \mathcal{C} \times \mathcal{C} \longrightarrow \mathcal{C}$ under the embedding $Cat \longrightarrow Prof$?
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 $P C$.
Might you have a reference?
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.
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?
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.
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?
I would think not much work, since composition of profunctors is a coend by definition.
Oh I see. So in particular it is immediate that the profunctor description matches the coend description.
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