论文标题
弗罗贝尼乌斯（Frobenius）的结构
Frobenius structures in star-autonomous categories
论文作者
论文摘要
众所周知，在且仅当晶格完全分布时，从完整的晶格中提供SUP保留地图的量化才是Frobenius的量化。由于完全分布的晶格是完整晶格和保留图的自主类别中的核对象，因此我们在分类环境中研究上述陈述。我们在任意自主类别中介绍了Frobenius结构的概念，从而推广了Frobenius Quantale的概念。我们证明，核物体的内态的单态具有弗罗贝尼乌斯的结构。如果环境类别是恒星自主的，并且具有Epi-Mono因素化，则该定理的一种变体允许开发抽象的阶段语义，并概括了先前的陈述。相反，我们认为，如果对象的内态性具有frobenius结构，而单位单位将单个单位嵌入该对象后，则单位单位是双重对象，而单位的单态性是缩回的，则该对象是核的。
It is known that the quantale of sup-preserving maps from a complete lattice to itself is a Frobenius quantale if and only if the lattice is completely distributive. Since completely distributive lattices are the nuclear objects in the autonomous category of complete lattices and sup-preserving maps, we study the above statement in a categorical setting. We introduce the notion of Frobenius structure in an arbitrary autonomous category, generalizing that of Frobenius quantale. We prove that the monoid of endomorphisms of a nuclear object has a Frobenius structure. If the environment category is star-autonomous and has epi-mono factorizations, a variant of this theorem allows to develop an abstract phase semantics and to generalise the previous statement. Conversely, we argue that, in a star-autonomous category where the monoidal unit is a dualizing object, if the monoid of endomorphisms of an object has a Frobenius structure and the monoidal unit embeds into this object as a retract, then the object is nuclear.
