构造线性双范畴
Constructing linear bicategories
AI总结：
本文研究了线性双范畴的构造与性质，通过量词和量词oids定义了LD-量词及线性量词oids，并展示了多种标准构造在该框架下的应用，从而产生新的线性双范畴例子。
摘要：
线性分配范畴被引入以建模线性逻辑中的张量/平行片段，而无需使用否定。线性双范畴是线性分配范畴的双范畴版本。本质上，线性双范畴有两种形式的组合，每种组合确定一个双范畴的结构，而这两种组合通过线性分配相关联。本文的主要目标是证明存在许多线性双范畴的例子，这些例子通过考虑量词和量词oids来获得。在单子拓扑领域中，量词值关系的范畴是一个双范畴。在此，我们首先证明一个量词是Girard当且仅当对应的双范畴是一个Girard量词oids，这是线性双范畴的一个例子。热带和北极半环结构结合成一个Girard量词，因此此构造可能有多种应用。更一般地，我们定义了LD-量词，它们是带有两个量词结构的sup-格，这两个结构通过线性分配相关联，以及它们的双范畴类比，线性量词oids。我们证明Q-Rel是一个线性量词oids当且仅当Q是一个LD-量词。然后我们考虑了几种标准的富集双范畴理论构造，并展示了这些构造在线性量词oids设置中提升，并产生新的线性双范畴的例子。特别是，我们考虑了线性Q-范畴、Q中的矩阵和线性单子，其中Q是一个线性量词oids。我们还开发了非局部格例，Quant，量词、模块和模同态的双范畴，以及Qtld，量词oids、模块和模同态的双范畴。这些结果被发现是循环的* - 自律双范畴，本质上是线性双范畴的封闭版本。
英文摘要：
Linearly distributive categories were introduced to model the tensor/par fragment of linear logic, without resorting to the use of negation. Linear bicategories are the bicategorical version of linearly distributive categories. Essentially, a linear bicategory has two forms of composition, each determining the structure of a bicategory, and the two compositions are related by a linear distribution. The main goal of this paper is to demonstrate that there are many examples of linear bicategories, which are obtained by considering quantales and quantaloids. It is standard in the field of monoidal topology that the category of quantale-valued relations is a bicategory. Here we begin by showing that a quantale is Girard if and only if the corresponding bicategory is a Girard quantaloid, which is an example of linear bicategory. The tropical and arctic semiring structures fit together into a Girard quantale, so this construction is likely to have multiple applications. More generally, we define LD-quantales, which are sup-lattices with two quantale structures related by a linear distribution, and their bicategorical analogue, linear quantaloids. We show that Q-Rel is a linear quantaloid if and only if Q is an LD-quantale. We then consider several standard constructions from enriched bicategory theory, and show that these lift to the linear quantaloid setting and produce new examples of linear bicategories. In particular, we consider linear Q-categories, matrices in Q and linear monads in Q, where Q is a linear quantaloid. We develop non-locally posetal examples as well, Quant, the bicategory of quantales, modules and module homomorphisms, and Qtld, the bicategory of quantaloids, modules and module homomorphisms. These turn out to be cyclic *-autonomous bicategories, which are in essence a closed version of linear bicategories.
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