论文标题
刚性单体类别
Pre-rigid Monoidal Categories
论文作者
论文摘要
在\ cite {gv-inthediality}的意义上，编织的单体类别之间的伴随函数的可升起对提供了双gebras相关类别之间的自动辅助。通过找到有趣的示例，我们研究了一般刚性的单体类别。粗略地说，这些是单一类别的类别，其中每个对象$ x $，$ x^{\ ast} $，以及从$ x^{\ ast} \ otimes x $从单位对象的$ x^{\ ast} \ otimes x $的良好行为评估映射。一个典型的示例是一个字段上向量空间的类别，其中$ x^{\ ast} $如果$ x $不是有限维度，则不是一个分类二重要。我们探索与相关概念（例如正确的封闭性）的联系，并提出有意义的例子。我们还研究了Turaev的Hopf Group-（CO）代数的分类框架，鉴于依据和封闭性，填补了沿途文献的一些空白。最后，我们表明，在LOC的意义上，编织的刚性单体类别确实为提升性提供了适当的设置。引用而且，我们提出了一个应用程序，以矢量空间为主题有所不同，以表明 - 在有利的情况下 - 依赖性的概念允许在不可用的类别的正确闭合性时构造可伴随函数的可升起对。
Liftable pairs of adjoint functors between braided monoidal categories in the sense of \cite{GV-OnTheDuality} provide auto-adjunctions between the associated categories of bialgebras. Motivated by finding interesting examples of such pairs, we study general pre-rigid monoidal categories. Roughly speaking, these are monoidal categories in which for every object $X$, an object $X^{\ast}$ and a nicely behaving evaluation map from $X^{\ast}\otimes X$ to the unit object exist. A prototypical example is the category of vector spaces over a field, where $X^{\ast}$ is not a categorical dual if $X$ is not finite-dimensional. We explore the connection with related notions such as right closedness, and present meaningful examples. We also study the categorical frameworks for Turaev's Hopf group-(co)algebras in the light of pre-rigidity and closedness, filling some gaps in literature along the way. Finally, we show that braided pre-rigid monoidal categories indeed provide an appropriate setting for liftability in the sense of loc. cit. and we present an application, varying on the theme of vector spaces, showing how -- in favorable cases -- the notion of pre-rigidity allows to construct liftable pairs of adjoint functors when right closedness of the category is not available.
