In this work we develop some aspects of the theory of Hopf algebras to thecontext of autonomous map pseudomonoids. We concentrate in the Hopf modules andthe Centre or Drinfel'd double. If $A$ is a map pseudomonoid in a monoidalbicategory \M, the analogue of the category of Hopf modules for $A$ is anEilenberg-Moore construction for a certain monad in$\mathbf{Hom}(\M^{\mathrm{op}},\mathbf{Cat})$. We study the existence of theinternalisation of this notion, called the Hopf module construction, byextending the completion under Eilenberg-Moore objects of a 2-category to aendo-homomorphism of tricategories on $\mathbf{Bicat}$. Our main result is the equivalence between the existence of a leftdualization for $A$ ({\em i.e.}, $A$ is left autonomous) and the validity of ananalogue of the structure theorem of Hopf modules. In this case the Hopf moduleconstruction for $A$ always exists. We use these results to study the lax centre of a left autonomous mappseudomonoid. We show that the lax centre is the Eilenberg-Moore constructionfor a certain monad on $A$ (one existing if the other does). If $A$ is alsoright autonomous, then the lax centre equals the centre. We look at theexamples of the bicategories of \V-modules and of comodules in \V, and obtainthe Drinfel'd double of a coquasi-Hopf algebra $H$ as the centre of $H$.
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机译：在这项工作中，我们将Hopf代数理论的某些方面扩展到了自主图假单态的背景下。我们专注于Hopf模块和Center或Drinfel'd double。如果$ A $是\ monoidalbicategory \ M中的映射拟单态，则$ A $的Hopf模块类别的类似物是$ \ mathbf {Hom}（\ M ^ {\ mathrm { op}}，\ mathbf {Cat}）$。我们通过将在$ \ mathbf {Bicat} $上的三类的2类扩展为三类的同胚同态来研究这种概念的内在化（称为Hopf模块构造）的存在。我们的主要结果是存在$ A $的对偶化（{\ em i.e.}，$ A $保持自治）与Hopf模块的结构定理类似的有效性之间的等价关系。在这种情况下，始终存在$ A $的Hopf模块构造。我们使用这些结果来研究左自主mappseudomonoid的松弛中心。我们显示松弛中心是在$ A $上某个单子的Eilenberg-Moore构造（如果存在则一个存在）。如果$ A $同样是自治的，则松弛中心等于中心。我们看一下\ V-modules和\ V中的co-modules两类的例子，并获得了Co-Hopf代数$ H $的Drinfel'd两倍为$ H $的中心。
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