Duoidal categories, measuring comonoids and enrichment

Map monoidales and duoidal $\infty$-categories

In this paper we give an example of duoidal $\infty$-categories. We introduce map $\mathcal{O}$-monoidales in an $\mathcal{O}$-monoidal $(\infty,2)$-category for an $\infty$-operad $\mathcal{O}^{\otimes}$. We show that the endomorphism mapping $\infty$-category of a map $\mathcal{O}$-monoidale is a coCartesian $(Δ^{\rm op},\mathcal{O})$-duoidal $\infty$-category. After that, we introduce a convolution product on the mapping $\infty$-category from an $\mathcal{O}$-comonoidale to an $\mathcal{O}$-monoidale. We show that the $\mathcal{O}$-monoidal structure on the duoidal endomorphism mapping $\i…

BiHom Hopf algebras viewed as Hopf monoids

We introduce monoidal categories whose monoidal products of any positive number of factors are lax coherent and whose nullary products are oplax coherent. We call them $\mathsf{Lax}^+\mathsf{Oplax}^0$-monoidal. Dually, we consider $\mathsf{Lax}_0\mathsf{Oplax}_+$-monoidal categories which are oplax coherent for positive numbers of factors and lax coherent for nullary monoidal products. We define $\mathsf{Lax}^+_0\mathsf{Oplax}^0_+$-duoidal categories with compatible $\mathsf{Lax}^+\mathsf{Oplax}^0$- and $\mathsf{Lax}_0\mathsf{Oplax}_+$-monoidal structures. We introduce comonoids in $\mathsf{La…

论双幺半$\infty$-范畴

On duoidal $\infty$-categories

本文在$\infty$-范畴框架下引入双幺半$\infty$-范畴，定义三类函子及相应的三种范畴表述，并据此定义双幺半群等结构。

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 …

Hopf measuring comonoids and enrichment

We study the existence of universal measuring comonoids $P(A,B)$ for a pair of monoids $A$, $B$ in a braided monoidal closed category, and the associated enrichment of a category of monoids over the monoidal category of comonoids. In symmetric categories, we show that if $A$ is a bimonoid and $B$ is a commutative monoid, then $P(A,B)$ is a bimonoid; in addition, if $A$ is a cocommutative Hopf monoid then $P(A,B)$ always is Hopf. If $A$ is a Hopf monoid, not necessarily cocommutative, then $P(A,B)$ is Hopf if the fundamental theorem of comodules holds; to prove this we give an alternative descr…

Monads on higher monoidal categories

We study the action of monads on categories equipped with several monoidal structures. We identify the structure and conditions that guarantee that the higher monoidal structure is inherited by the category of algebras over the monad. Monoidal monads and comonoidal monads appear as the base cases in this hierarchy. Monads acting on duoidal categories constitute the next case. We cover the general case of $n$-monoidal categories and discuss several naturally occurring examples in which $n\leq 3$.