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    • CommentRowNumber1.
    • CommentAuthorHarry Gindi
    • CommentTimeSep 22nd 2011

    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 opV^{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.

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 22nd 2011

    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.).

    • CommentRowNumber3.
    • CommentAuthorHarry Gindi
    • CommentTimeSep 22nd 2011

    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.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeAug 8th 2013

    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 na_n and functions na nz n\sum_n a_n z^n on S 1S^1. One way of categorifying this is to pass from the category of functors a:Seta: \mathbb{N} \to Set (considered as a monoidal category with respect to Day convolution) to their so-called “analytic functors” a^:SetSet\hat{a}: Set \to Set, mapping a set xx to a^(x)= na nx n\hat{a}(x) = \sum_n a_n \cdot x^n. The “categorified Fourier transform” aa^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 na_n to a power series na nz n\sum_n a_n z^n is precisely the zz-transform. If we set z=exp(iθ)z = exp(i \theta), we get the Fourier transform — but as you note this makes use of the imaginary unit ii, which plays no evident role in Day convolution. So, the analogies Todd is discussing become most precise if we work with the zz-transform. But the Fourier transform is closely related.

    On the other hand, I’ve discovered that many ’pure mathematicians’ don’t know about the zz-transform — at least, not under that name. I think it’s ’engineers’ who talk most about the zz-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 zz-transform.

    In general homhom 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 CC is a discrete category (i.e. just a set regarded as a category with only identity morphisms) then a functor CoSetC \o Set is like a mathbZ\mathb{Z}-valued function on the set CC and then for every object cc in CC the functor Hom C(c,)=δ cHom_C(c,-) = \delta_c is the Kronecker delta on CC at cc, in that

    Hom C(c,d)={* ifc=d ifcd 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.

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 8th 2013

    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.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeFeb 16th 2016
    • (edited Feb 16th 2016)

    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 Δ opCat^{\Delta^{op}} as opposed to in sSetCatsSet 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).)

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMar 3rd 2016
    • (edited Mar 3rd 2016)

    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.

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 3rd 2016

    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.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMar 3rd 2016

    That would be great if you would look into it again. Please just give me five minutes to finish something…

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMar 3rd 2016
    • (edited Mar 3rd 2016)

    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”).

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMar 22nd 2016
    • (edited Mar 22nd 2016)

    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.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMar 24th 2016
    • (edited Mar 24th 2016)

    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 RR a monoid object in ([𝒟,V], Day)([\mathcal{D},V], \otimes_{Day}) (for some monoidal VV-category 𝒟\mathcal{D}) then (paraphrasing the formulation in the article) we have an equivalence of categories

    RMod[RFreeMod op,V]. 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?

    • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeMar 24th 2016

    Hmm, that sounds like some kind of Kleisli / Eilenberg-Moore thing?

    • CommentRowNumber14.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 14th 2016

    With Day convolution cropping up on the focusing thread, I took again at Day convolution.

    Presumably in definition 1, the FFs should be XX and YY. And why is there [c 1c 2,c][c_1 \otimes c_2, c], with tensor product as domain, when above in the definition of FGF \star G, the tensor product is the codomain, Hom C(,cd)Hom_C(-, c \otimes d)?

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeApr 14th 2016
    • (edited Apr 14th 2016)

    Presumably in definition 1, the FFs should be XX and YY.

    Yes, thanks. I have fixed it now.

    And why is there [c 1c 2,c][c_1 \otimes c_2, c], with tensor product as domain, when above in the definition of FGF \star G, the tensor product is the codomain, Hom C(,cd)Hom_C(-, c \otimes d)?

    Because the text switches from considering functors on C opC^{op} to considering functors on 𝒞\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.

    • CommentRowNumber16.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 14th 2016

    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.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeApr 14th 2016

    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.)

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeJun 2nd 2016

    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.

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeJun 14th 2016

    Coming back to #12:

    I have tried to fill in more details of the proof of the proposition (here) that for RR a monoid with respect to Day convolution over some monoidal VV-enriched category 𝒞\mathcal{C}, then right RR-modules are a VV-presheaf category over the free RR-modules free on objects of 𝒞\mathcal{C}:

    Mod R[Free 𝒞Mod op,V]. 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 RFree 𝒞ModR Free_{\mathcal{C}}Mod like so

    RFree 𝒞Mod(c 2,c 1) =RMod(y(c 2) DayR,y(c 1) DayR) [𝒞,V](y(c 2),y(c 1) DayR) (y(c 1) DayR)(c 2) c 3,c 4𝒞(c 3c 4,c 2) V𝒞(c 1,c 3) VR(c 4) c 4𝒞(c 1c 4,c 2) VR(c 4) \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 RFree 𝒞ModR Free_{\mathcal{C}}Mod is given by

    RFreeMod(c 2,c 1) VRFreeMod(c 3,c 2) =(c 4𝒞(c 1 𝒞c 4,c 2) VR(c 4)) V(c 5𝒞(c 2 𝒞c 5,c 3) VR(c 5)) c 4,c 5𝒞(c 1 𝒞c 4,c 2) V𝒞(c 2 𝒞c 5,c 3) VR(c 4) VR(c 5) c 4,c 5𝒞(c 1 𝒞c 4 𝒞c 5,c 2 𝒞c 5) V𝒞(c 2c 5,c 3) VR(c 4 𝒞c 5) c 4,c 5𝒞(c 1 𝒞c 4 𝒞c 5,c 3) VR(c 4 𝒞c 5) c 4𝒞(c 1 𝒞c 4,c 3) VR(c 4), \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} \,,


    1. the first morphism is, in the integrand, the tensor product of

      1. forming the tensor product of hom-objects of 𝒞\mathcal{C} with the identity of c 5c_5

        𝒞(c 1 𝒞c 4,c 2)𝒞(c 1 𝒞c 4,c 2) V1 V𝒞(c 1 𝒞c 4,c 2)𝒞(c 5,c 5)𝒞(c 1 𝒞c 4 𝒞c 5,c 2 𝒞c 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)
      2. the monoidal functor incarnation R(c 4) VR(c 5)R(c 4 𝒞c 5)R(c_4) \otimes_V R(c_5)\longrightarrow R(c_4 \otimes_{\mathcal{C}} c_5 ) of the monoid structure on RR;

    2. the second morphism is, in the integrand, given by composition in 𝒞\mathcal{C};

    3. the last morphism is the morphism induced on coends by regarding extranaturality in c 4c_4 and c 5c_5 separately as a special case of extranaturality in c 6c 4c 5c_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.

    • CommentRowNumber20.
    • CommentAuthorMike Shulman
    • CommentTimeJun 14th 2016

    Yes, there must…

    • CommentRowNumber21.
    • CommentAuthorMike Shulman
    • CommentTimeJun 14th 2016

    Can we give Free 𝒞Mod opFree_{\mathcal{C}}Mod^{op} a universal property? It feels like it should be some kind of lax colimit or collage.

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeJun 15th 2016

    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.

    • CommentRowNumber23.
    • CommentAuthorMike Shulman
    • CommentTimeJun 15th 2016

    Does it simplify things any to do it the following way? Let C RC_R be the VV-category whose hom-objects and composition are defined by the coend given above. Show that diagrams on C RC_R are equivalent to RR-modules. Now use the fact that for any VV-category DD, the representables D(d,)D(d,-) are the free DD-diagrams on the objects of DD (by the Yoneda lemma) and the hom-objects of diagrams between representables are just the hom-objects of DD (by the Yoneda embedding); thus DD is the category of free DD-diagrams on objects of DD.

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeJun 15th 2016

    I see. Thanks! That sounds good.

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeJun 21st 2016
    • (edited Jun 21st 2016)

    What is your preferred way to write down a fully explicit proof of the coherence laws for the Day convolution product over some monoidal VV-category 𝒞\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

    ((X DayY) DayZ)(c) c 1,c 2,c 3𝒞((c 1c 2)c 3) V((X(c 1) VY(c 2)) VZ(c 3)) α X,Y,Z Day(c) c 1,c 2,c 3𝒞(α c 1,c 2,c 3 𝒞,c) Vα X(c 1),X(c 2),X(c 3) V (X Day(Y DayZ))(c) c 1,c 2,c 3𝒞(c 1(c 2c 3),c) V(X(c 1) V(Y(c 2) VZ(c 3))) \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 Day\otimes_{Day} follows immediately from that of 𝒞\mathcal{C} and VV.

    Similarly for the braiding, in case 𝒞\mathcal{C} is braided.

    But for the triangle identities the situation is messier. The right Day unitor may be exhibited as

    (X Dayy(1))(c) c 1𝒞(c 11,c) VX(c 1) r X Day(c) c 1𝒞(r c 1 𝒞,c) VX(c 1) X(c) c 1𝒞(c 1,c) VX(c 1) \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 𝒞\mathcal{C}. But writing this down fully rigorously now seems to get a little more awkward, since in (X Day(y(1) DayY))(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 𝒞\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?

    • CommentRowNumber26.
    • CommentAuthorMike Shulman
    • CommentTimeJun 22nd 2016

    I like to think of it as the pseudofunctor CatProfCat\to Prof taking the monoidal category CC to a pseudomonoid in ProfProf (a promonoidal category), and then doing everything in terms of composites of profunctors without ever needing to write them as coends. Does that help?

    • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeJun 22nd 2016
    • (edited Jun 22nd 2016)

    I see. You are saying that Day convolution is just the image of :𝒞×𝒞𝒞\otimes \colon \mathcal{C} \times \mathcal{C} \longrightarrow \mathcal{C} under the embedding CatProfCat \longrightarrow Prof?

    • CommentRowNumber28.
    • CommentAuthorMike Shulman
    • CommentTimeJun 22nd 2016

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

    • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTimeJun 22nd 2016

    Might you have a reference?

    • CommentRowNumber30.
    • CommentAuthorMike Shulman
    • CommentTimeJun 22nd 2016

    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.

    • CommentRowNumber31.
    • CommentAuthorMike Shulman
    • CommentTimeJun 22nd 2016

    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?

    • CommentRowNumber32.
    • CommentAuthorUrs
    • CommentTimeJun 22nd 2016


    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, V)(V, \otimes_V) to (Top cg */,)(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.

    • CommentRowNumber33.
    • CommentAuthorUrs
    • CommentTimeJun 23rd 2016
    • (edited Jun 23rd 2016)

    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 τ Day\tau^{Day} in the Day convolution structure is. The coend formula provides this (it’s τ X,Y Day(c)c 1,c 2𝒞(τ c 1,c 2 𝒞,c) Vτ X(c 1),X(c 2) V\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?

    • CommentRowNumber34.
    • CommentAuthorMike Shulman
    • CommentTimeJun 23rd 2016

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

    • CommentRowNumber35.
    • CommentAuthorUrs
    • CommentTimeJun 23rd 2016

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

    • CommentRowNumber36.
    • CommentAuthormaxsnew
    • CommentTimeMay 16th 2020

    In Mike Shulman’s new paper 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 2Qf : P_1,P_2\ldots \to Q

    are given by maps

    f:P 1(m 1)*P 2(m 2)Q(m 1*m 2*)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.

    • CommentRowNumber37.
    • CommentAuthorSam Staton
    • CommentTimeMay 17th 2020

    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.

    • CommentRowNumber38.
    • CommentAuthormaxsnew
    • CommentTimeMay 18th 2020

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

  1. 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 [𝒜,𝒱][\mathcal{A},\mathcal{V}] correspond bijectively to premonoldal structures on 𝒜\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)(P,J) on 𝒜\mathcal{A} and biclosed completions of (¯,J)(\overline{\otimes},J) on FF. [Here FF is the 𝒱\mathcal{V}-functor category [𝒜,𝒱][\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 𝒞\mathcal{C}.

    2. The full subcategory 𝒳\mathcal{X} (for lack of better notation) with Obj(𝒳)={𝒞}\mathrm{Obj}(\mathcal{X})=\{\mathcal{C}\} of the 11-category ProMonCats 𝒱\mathsf{ProMonCats}_{\mathcal{V}} of

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

      • promonoidal 𝒱\mathcal{V}-categories,
      • promonoidal 𝒱\mathcal{V}-functors between them, and
      • promonoidal 𝒱\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 𝒞\mathcal{C} and biclosed monoidal structures on [𝒞,𝒱][\mathcal{C},\mathcal{V}] the “functor category theorem” in his PhD thesis. Is this name used today as well?

    • CommentRowNumber40.
    • CommentAuthorMike Shulman
    • CommentTimeMay 19th 2020

    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).

    • CommentRowNumber41.
    • CommentAuthorMike Shulman
    • CommentTimeMay 19th 2020

    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 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.

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

    • CommentRowNumber43.
    • CommentAuthorTheo
    • CommentTimeMay 20th 2020

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

    diff, v62, current

    • CommentRowNumber44.
    • CommentAuthorThéo de Oliveira S.
    • CommentTimeMay 20th 2020
    • (edited May 20th 2020)

    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)?)

    • CommentRowNumber45.
    • CommentAuthorRichard Williamson
    • CommentTimeMay 21st 2020
    • (edited May 21st 2020)

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

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

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

    • CommentRowNumber48.
    • CommentAuthorMike Shulman
    • CommentTimeJul 12th 2020

    Fix variance in the promonoidal case.

    diff, v64, current

    • CommentRowNumber49.
    • CommentAuthorMike Shulman
    • CommentTimeJul 12th 2020

    The formula on the page for the internal-hom:

    [X,Y] Day(c)=c 1,c 2V(𝒞(c 𝒞c 1,c 2),V(X(c 1),Y(c 2))). [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 VV-enriched Yoneda lemma isn’t this isomorphic to

    c 1V(X(c 1),Y(c 𝒞c 1))? \int_{c_1} V(X(c_1), Y(c\otimes_{\mathcal{C}} c_1)) \quad?

    Is there a reference for this?

    • CommentRowNumber50.
    • CommentAuthorSam Staton
    • CommentTimeJul 13th 2020

    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.

    • CommentRowNumber51.
    • CommentAuthorMike Shulman
    • CommentTimeJul 29th 2020

    Mentioned the simpler formula for the internal-hom.

    diff, v65, current

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