added a bit about when the functor category between two promonoidal categories has a convolution monoidal structure and what the algebras are w/r/t it

Jonathan Beardsley

]]>trying to add some details about promonoidal versions of these statements

Jonathan Beardsley

]]>Add reference to monoidally cocomplete category.

]]>Performed the above suggestion, adding a comment before the proposition about the covariant vs contravariant distinction and dualising the proof.

]]>I would probably just fix it by putting the op on the domain, since as you say the whole page is written for the covariant version.

]]>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.

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]]>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 :-).

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

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]]>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).

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]]>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.

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