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I have added some more hyperlinks and cross-linked with monoidal category, bimonoidal category, etc.
Added another example, and a reference.
At duoidal category I have:
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?
added arXiv:math/0701325 and doi:10.1007/s10485-010-9228-x and author link to Kosta Došen
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 doi
Y. Chen, G. Böhm, Weak bimonoids in duoidal categories, J. Pure Appl. Math. 218:12 (2014) 2240–2273 doi
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