论文标题
富含Oplax Monoidal类别的类别
Categories enriched over oplax monoidal categories
论文作者
论文摘要
我们定义了一个富含Oplax单类别类别$ V $的类别的概念，从而扩展了富含单类类别的类别的通常定义。即使Oplax Monoidal结构涉及无限的许多函子$ v^n \ to V $，但定义富含$ V $的类别仅需要较低的Arity Maps $（N \ LEQ 3）$，类似于单型案例。在Oplax情况下，富集理论的焦点从$ v $ - 类别的概念（由对象和对象的集合以及构图和单位图给出）转移到富含$ v $的类别之一（配备了其他结构的真实类别）。超过$ v $的类别概念的优点之一是，定义丰富的函子和自然转换变得很简单。此外，我们表明的是，所得的2类$ \ MATHSF {CAT} _v $可以将其与2类模块以上的2类模块相当于$ v $。我们在作战框架中举例说明了这样一个丰富的类别：每一个共同的对称单体类别$ c $都在$ c $中的序列类别中富集，这些序列赋予了源自oplax himoidal结构，这些结构是由通常的运营组成产品所构成的，它们的单型物体仍然是运营的。 As an application of the study of the 2-functor $V\mapsto\mathsf{Cat}_V$, we show that when $V$ is also endowed with a compatible lax monoidal structure - thus forming a lax-oplax duoidal category - the 2-category $\mathsf{Cat}_V$ inherits a lax 2-monoidal structure, thereby generalising the corresponding result when the富集基础是一个编织的单体类别。我们通过讨论$（r^\ mathrm {e}，r^\ mathrm {e}）$ bimodules的类别上的Lax-Oplax结构来说明这一结果，其双子体是双子型。我们还评论与其他富集理论的关系（Monoidal，Mutivergories，Skew and Lax）。
We define a notion of category enriched over an oplax monoidal category $V$, extending the usual definition of category enriched over a monoidal category. Even though oplax monoidal structures involve infinitely many functors $V^n\to V$, defining categories enriched over $V$ only requires the lower arity maps $(n \leq 3)$, similarly to the monoidal case. The focal point of the enrichment theory shifts, in the oplax case, from the notion of $V$-category (given by collections of objects and hom-objects together with composition and unit maps) to the one of categories enriched over $V$ (genuine categories equipped with additional structures). One of the merits of the notion of categories enriched over $V$ is that it becomes straightforward to define enriched functors and natural transformations. We show moreover that the resulting 2-category $\mathsf{Cat}_V$ can be put in correspondence (via the theory of distributors) with the 2-category of modules over $V$. We give an example of such an enriched category in the framework of operads: every cocomplete symmetric monoidal category $C$ is enriched over the category of sequences in $C$ endowed with an oplax monoidal structure stemming from the usual operadic composition product, whose monoids are still the operads. As an application of the study of the 2-functor $V\mapsto\mathsf{Cat}_V$, we show that when $V$ is also endowed with a compatible lax monoidal structure - thus forming a lax-oplax duoidal category - the 2-category $\mathsf{Cat}_V$ inherits a lax 2-monoidal structure, thereby generalising the corresponding result when the enrichment base is a braided monoidal category. We illustrate this result by discussing the lax-oplax structure on the category of $(R^\mathrm{e}, R^\mathrm{e})$-bimodules, whose bimonoids are the bialgebroids. We also comment on the relations with other enrichment theories (monoidal, multicategories, skew and lax).
