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I moved the definition of promonoidal categories from Day convolution to promonoidal category, and expanded on it a bit.
I noticed that the definition of promonoidal category given here is op’d from the one in Day’s original paper (and the one with Panchadcharam and Street), in the sense that a Day-promonoidal structure on $C$ corresponds to a nlab-promonoidal structure on $C^{op}$. I imagine this reversal has to do with the convention for the direction of profunctors. Should we care? I don’t think it matters for the promonoidal structure generated by a monoidal category (since a monoidal structure on $C$ transports to a monoidal structure on $C^op$), but on the other hand, another example Street gives is of the promonoidal structure generated by a biclosed category, with $P(A,B,C) = Hom(B, A\setminus C) = Hom(A, C/B)$. That wouldn’t work with the current definition in the article, since the existence of a biclosed structure is not preserved under op’ing.
Good catch, Noam; that’s a little annoying. I think we should care, and I’d say we should change over to the standard Day definition, or at the very least leave a remark.
What about the equivalence between promonoidal structures on $C$ and biclosed monoidal structures on $V^{C^{op}}$?
Looking at (Day 1974) as well as the (Day & LaPlaza 1978) paper cited at closed category, it seems the idea is that a (Day-)promonoidal structure on the closed category $C$ is used to construct a biclosed monoidal structure on $V^C$, and then $C$ is embedded into contravariant presheaves over a small subcategory $A \subset V^C$. So perhaps it is just worth adding a note that a closed structure on $C$ induces a (nlab-)promonoidal structure on $C^op$, and that this is op’d from Day’s original convention?
(and that actually clarifies for me why you have to “take presheaves twice” in the construction of a biclosed monoidal category over a closed category – when one suffices for getting a biclosed monoidal category over a monoidal category – so I see the advantage of the current convention used in the nlab article.)
More generally, any multicategory structure on $C$ induces an nlab-promonoidal structure on $C^{op}$.
I find it very pleasing when using the convention where $Prof$ is equivalent to the 2-category of presheaf categories and cocontinuous functors, which makes it utterly obvious that a promonoidal structure on $C$ is the same as a nice monoidal structure on $V^{C^{op}}$.
So I went ahead and added a little note about the relationship to Day’s definition. (Also note that he had actually first called these “premonoidal categories”, but sometime between 1970 and 1974 he switched to “promonoidal” and reformulated the definition in terms of profunctor composition.)
Thanks!
I have fixed the numbered lists with the diagrams. The trick is to have all content of a numbered item be indented by exactly 3 whitespace.
(Single lines of running text wrapping the edit pane width just look like they are not fully indented, but as long as they don’t contain an explicit “carriage return” they are.)
For a more compact way of presenting the pentagon and triangle identities for a promonoidal category, see diagrams (11.47) and (11.51), and the paragraph preceding them, in Ross Street’s Skew-closed categories.
I have added hyperlinks to a bunch of further keywords.
(Noticed that we really ought to make triangle identity a disambiguation page…)
Thanks, Urs! I indeed didn’t notice the indentation in the other lists.
(Speaking of triangle identities; it does feel weird to call these diagrams the triangle and pentagon identities; but I guess the “octagon and tridecagon identities” would be quite weirder :P)
@Guest: This is a point I was/am a bit confused about; don’t we need to include the isomorphisms $(P\diamond Q)\diamond R\cong P\diamond(Q\diamond R)$ in the diagrams too?
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