Gabriella Böhm,

*(Hopf) bimonoids in duoidal categories*, Chapter in*Hopf algebras and their generalizations from a category theoretical point of view*, Springer Lecture Notes in Mathematics __2226_ (2018) (chapter doi)Gabriella Böhm, Y. Chen, L. Zhang,

*On Hopf monoids in duoidal categories*, J. Algebra**394**(2013) 139–172 (2013) MR3092715 doiY. Chen, G. Böhm,

*Weak bimonoids in duoidal categories*, J. Pure Appl. Math.**218**:12 (2014) 2240–2273 doi

added arXiv:math/0701325 and doi:10.1007/s10485-010-9228-x and author link to *Kosta Došen*

Added a related paper of Došen and Petrić.

]]>Added a reference to *Duoidal categories, measuring comonoids and enrichment*.

States the definition of produoidals from “Tannaka Duality and Convolution for Duoidal Categories (Booker, Street, 2013)”. I plan to start the article on produoidals next.

Mario Román

]]>I was looking at this entry recently and had a question about it which has a possibly obvious answer: it is mentioned that if A and B are monoidal then Fun(A,B) is equipped with a duoidal structure with one tensor product being the pointwise one, and the other being Day convolution. Later in the entry, it states that the category of monoids with respect to one monoidal structure, in a duoidal category, is monoidal with respect to the other monoidal structure. My question is the following: suppose we take B to be a monoidal category and A=1, the unit category (with respect to the cartesian product of categories). Then algebras in Fun(1,B), with respect to the pointwise monoidal structure are exactly algebras in B, and algebras in Fun(1,B) with respect to the Day convolution are also algebras in B. Then it seems that for Alg(Fun(1,B)), with respect to either monoidal structure, has a monoidal structure given by the underlying monoidal structure in B. However, at least in the usual cases, this cannot happen in a monoidal category, as far as I know.

So, for this to make sense, do we need to require B to be (at least) braided monoidal?

]]>At duoidal category I have:

- corrected the example of endofunctors (they are not normal in general) and added an example of profunctors.
- generalized the notion of “commutative monoid” to the case of a non-normal duoidal category, in which case the object first has to be assumed to be “strong”.
- mentioned the notion of “virtual duoidal category”, to which it seems nearly all the definitions in a duoidal category can be generalized.

Added another example, and a reference.

]]>I have added some more hyperlinks and cross-linked with *monoidal category*, *bimonoidal category*, etc.