论文标题
Bihom Hopf代数被视为Hopf Monoids
BiHom Hopf algebras viewed as Hopf monoids
论文作者
论文摘要
我们介绍了单素类别的任何阳性因素的单体产物是宽松的连贯性，并且其无效产物是oplax相干的。我们称它们为$ \ mathsf {lax}^+\ Mathsf {oplax}^0 $ -Monoidal。双重，我们考虑$ \ mathsf {lax} _0 \ mathsf {oplax} _+$ - 单型类别，这些类别对于阳性因子的阳性数量和无量子单相结合而言是oplax相干的。我们定义$ \ MATHSF {LAX}^+_ 0 \ MATHSF {oplax}^0 _+$ - upuoidal类别，具有兼容$ \ MathSf {lax}^+\ Mathsf {oplax}^0 $ - 和$ \ Mathsf {laxsf {lax}我们在$ \ mathsf {lax}^+\ mathsf {oplax}^0 $ -Monoidal类别中介绍comonoids，$ \ Mathsf {lax} _0 \ Mathsf {oplax} _++$ - +$ - 单体类别和bimonoids中的单体类别$ \ mathsf {lax}^+_ 0 \ Mathsf {oplax}^0 _+$ - upuoidal类别。 这些概念的动机来自于caenepeel和goyvaerts引起的建筑的概括。这将分配$ \ MATHSF {LAX}^+_ 0 \ MATHSF {oplax}^0 _+$ - duoidal类别$ \ MATHSF D $ d $ UNITAL $ \ MATHSF {BIHOM} $ - 单型，$ \ Mathsf {BiHom} $ - comonoids，以及Unital and Counital $ \ Mathsf {BiHom} $ - $ \ Mathsf V $中的bimonoids-与MONOIDS，COMONOIDS和BIMONOIDS和BIMONOIDS和BIMONOIDS和BIMONOIDS和BIMONOIDS和BIMONOIDS和BIMONOIDS和$ c.
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 $\mathsf{Lax}^+\mathsf{Oplax}^0$-monoidal categories, monoids in $\mathsf{Lax}_0\mathsf{Oplax}_+$-monoidal categories and bimonoids in $\mathsf{Lax}^+_0\mathsf{Oplax}^0_+$- duoidal categories. Motivation for these notions comes from a generalization of a construction due to Caenepeel and Goyvaerts. This assigns a $\mathsf{Lax}^+_0\mathsf{Oplax}^0_+$-duoidal category $\mathsf D$ to any symmetric monoidal category $\mathsf V$. The unital $\mathsf{BiHom}$-monoids, counital $\mathsf{BiHom}$-comonoids, and unital and counital $\mathsf{BiHom}$-bimonoids in $\mathsf V$ are identified with the monoids, comonoids and bimonoids in $\mathsf D$, respectively.
