Given a horizontal monoid M in a duoidal category F, we examine the relationship between bimonoid structures on M and monoidal structures on the category of right M-modules which lift the vertical monoidal structure of F. We obtain our result using a variant of the Tannaka adjunction. The approach taken utilizes hom-enriched categories rather than categories on which a monoidal category acts ("actegories"). The requirement of enrichment in F itself demands the existence of some internal homs, leading to the consideration of convolution for duoidal categories. Proving that certain hom-functors are monoidal, and so take monoids to monoids, unifies classical convolution in algebra and Day convolution for categories. Hopf bimonoids are defined leading to a lifting of closed structures. Warped monoidal structures permit the construction of new duoidal categories.
Deep Dive into Tannaka duality and convolution for duoidal categories.
Given a horizontal monoid M in a duoidal category F, we examine the relationship between bimonoid structures on M and monoidal structures on the category of right M-modules which lift the vertical monoidal structure of F. We obtain our result using a variant of the Tannaka adjunction. The approach taken utilizes hom-enriched categories rather than categories on which a monoidal category acts (“actegories”). The requirement of enrichment in F itself demands the existence of some internal homs, leading to the consideration of convolution for duoidal categories. Proving that certain hom-functors are monoidal, and so take monoids to monoids, unifies classical convolution in algebra and Day convolution for categories. Hopf bimonoids are defined leading to a lifting of closed structures. Warped monoidal structures permit the construction of new duoidal categories.
This paper initiates the development of a general theory of duoidal categories. In addition to providing the requisite definition of a duoidal V-category, various "classical" concepts are reinterpreted and new notions put forth, including: produoidal V-categories, convolution structures and duoidal cocompletion, enrichment in a duoidal V-category, Tannaka duality, lifting closed structures to a category of representations (Hopf opmonoidal monads), and discovering new duoidal categories by "warping" the monoidal structure of another. Duoidal categories, some examples, and applications, have appeared in the Aguiar-Mahajan book [1] (under the name "2-monoidal categories"), in the recently published work of Batanin-Markl [2] and in a series of lectures by the second author [23]. Taken together with this paper, the vast potential of duoidal category theory is only now becoming apparent.
An encapsulated definition is that a duoidal V-category F is a pseudomonoid in the 2-category Mon(V-Cat) of monoidal V-categories, monoidal V-functors and monoidal V-natural transformations. Since Mon(V-Cat) is equivalently the category of pseudomonoids in V-Cat we are motivated to call a pseudomonoid in a monoidal bicategory a monoidale (i.e. a monoidal object). Thus a duoidal Vcategory is an object of V-Cat equipped with two monoidal structures, one called horizontal and the other called vertical, such that one is monoidal with respect to the other. Calling such an object a duoidale encourages one to consider duoidales in other monoidal bicategories, in particular M = V-Mod. By giving a canonical monoidal structure on the V = M(I, I) valued-hom for any left unit closed monoidal bicategory M (see Section 2), we see that a duoidale in M = V-Mod is precisely the notion of promonoidal category lifted to the duoidal setting, that is, a produoidal V-category.
A study of duoidal cocompletion (in light of the produoidal V-category material) leads to Section 5 where we consider enrichment in a duoidal V-category base. We observe that if F is a duoidal V-category then the vertical monoidal structure • lifts to give a monoidal structure on F h -Cat. If F is then a horizontally left closed duoidal V-category then F is in fact a monoidale (F h , •, 1 ) in F h -Cat with multiplication • : F h • F h -→ F h defined using the evaluation of homs. That is, F h is an F h -category.
Section 6 revisits the Tannaka adjunction as it pertains to duoidal V-categories. We write F h -Cat ↓ ps F h for the 2-category F h -Cat ↓ F h restricted to having 1cells those triangles that commute up to an isomorphism. Post composition with the monoidale multiplication • yields a tensor product • on F h -Cat ↓ ps F h and we write F -Cat ↓ ps F for this monoidal 2-category. Let F * M be the F h -category of Eilenberg-Moore algebras for the monad - * M . There is a monoidal functor mod : (Mon F ) op -→ F -Cat ↓ ps F defined by taking a monoid M to the object U M : F * M -→ F h . Here Mon F is only being considered as a monoidal category, not a 2-category. Representable objects of F -Cat ↓ ps F are closed under the monoidal structure • which motivates restricting to F -Cat ↓ ps rep F . Since representable functors are “tractable” and the functor end : F -Cat ↓ ps rep F -→ Mon F is strong monoidal we have the biadjunction
giving the correspondence between bimonoid structures on M and isomorphism classes of monoidal structures on F * M such that the underlying functor is strong monoidal into the vertical structure on F . The non-duoidal version of this result is attributed to Bodo Pareigis (see [3], [4] and [5]).
The notion of a Hopf opmonoidal monad is found in the paper of Bruguières-Lack-Virelizier [6]. We adapt their work to the duoidal setting in order to lift closed structures on the monoidale (monoidal F h -category) (F , •, 1 ) to the F h -category of right modules F * M for a bimonoid M . In particular, Proposition 22 says that a monoidal F h -category (F , •, 1 ) is closed if and only if F v is a closed monoidal V-category and there exists Proposition 23 gives a refinement of this result which taken together with Proposition 22 yields two isomorphims
This result implies that in order to know • we only need to know * and J •or -• J. Similarly to know * we need only know • and 1 * -or - * 1. This extreme form of interpolation motivates the material of Section 8.
We would like a way to generate new duoidal categories. One possible method presented here is the notion of a warped monoidal structure. In its simplest presentation, a warping for a monoidal category A = (A, ⊗) is a purtabation of A’s tensor product by a “suitable” endo-functor T : A -→ A such that the new tensor product is defined by
We lift this definition to the level of a monoidale A in a monoidal bicategory M. Proposition 26 observes that a warping for a monoidale determines another monoidale structure on A. If F is a duoidal V-category satisfying the right-hand side of the second isomorphism
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