Clarify the characterisation of promonoidal categories in terms of multicategories.

]]>Added a reference to *Lax monoids, pseudo-operads, and convolution*.

Remove boolean algebra comment from idea section

]]>Checking the page history, this statement as added by “Anonymous” in revision 3, back in 2013.

I suppose we should just delete it.

]]>The idea section says that a promonoidal category is a categorification of a boolean algebra. Does anyone know what this is supposed to mean? This is a categorification of many things, and doesn’t seem particularly similar to boolean algebras, or any more similar to boolean algebras than just semi-lattices.

]]>[deleted]

]]>I have added hyperlinks to a bunch of further keywords.

(Noticed that we really ought to make *triangle identity* a disambiguation page…)

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 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.)

]]>Added the diagrams corresponding to the triangle and pentagon identities for promonoidal categories. (First time making a non-trivial contribution to the nLab; I hope everything is okay.)

]]>Thanks!

]]>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.)

]]>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 $V^{C^{op}}$.

]]>(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.)

]]>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?

]]>What about the equivalence between promonoidal structures on $C$ and biclosed monoidal structures on $V^{C^{op}}$?

]]>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.

]]>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.

]]>I moved the definition of promonoidal categories from Day convolution to promonoidal category, and expanded on it a bit.

]]>