论文标题
双线性地图的概括 - 技术报告
Generalizations of Bilinear Maps -- Technical Report
论文作者
论文摘要
双线性图及其分类张量产物在线性代数理论中是众所周知的，并且它们对交换单元代数的概括是Monad理论的经典结果。由有限模型理论的分类方法中所需的结构所激发，我们将两态性的概念进一步推广。为了说明这些图是数学上自然概念，我们表明类别理论中的许多常见公理可以用某些形态为双态性。我们还表明，许多既定的双态理论都具有更大的一般性。我们的结果仔细地确定了该理论不同组成部分所需的假设，包括何时在全球范围内拥有良好的特性，或者至少可以在本地建立。我们在双态公理中包括一个简短的弦图示意，并通过恢复经典定理的简单证明来结束，并强调双态观点的功效。
Bilinear maps and their classifying tensor products are well-known in the theory of linear algebra, and their generalization to algebras of commutative monads is a classical result of monad theory. Motivated by constructions needed in categorical approaches to finite model theory, we generalize the notion of bimorphism much further. To illustrate these maps are mathematically natural notions, we show that many common axioms in category theory can be phrased as certain morphisms being bimorphisms. We also show that much of the established theory of bimorphisms goes through in much greater generality. Our results carefully identify which assumptions are needed for the different components of the theory, including when good properties hold globally, or can at least be established locally. We include a brief string diagrammatic account of the bimorphism axiom, and conclude by recovering a simple proof of a classical theorem, emphasizing the efficacy of the bimorphism perspective.
