论文标题
开放系统：双重分类视角
Open Systems: A Double Categorical Perspective
论文作者
论文摘要
Fong开发了“装饰的Cospan”来建模各种开放系统：即具有输入和输出的系统。在此框架中，开放系统被视为类别的形态，可以这样组成，从而使较大的开放系统可以从较小的系统中构建。在这个方向上已经做了很多工作，但是存在一个问题：装饰的Cospans之间的同构概念通常太限制了。在这里，我们介绍并比较了有关此问题的两种方式：结构化的Cospans和新版本的装饰Cospans。结构化的COSPAN非常简单：给定一个函子$ l \ colon \ Mathsf {a} \ to \ Mathsf {x} $，一个“结构化的Cospan”是$ \ Mathsf {x} $的图表，$ \ Mathsf {x} $ $ l（a）\ rightarrow x \ rightarrow x \ leftarrow l（a）如果$ \ mathsf {a} $和$ \ mathsf {x} $具有有限的colimits，而$ l $是左伴，则有一个对称的单体类别，其对象是$ \ mathsf {a a} $的对象，其形态是结构性cospans的形态类别。但是，此类别源于更基本的结构：对称的单体双重类别。在某些条件下，这种对称的单体双类别相当于使用我们新版本的装饰Cospans构建的单体类别。我们将这些想法应用于开放电路，开放马尔可夫工艺和开放培养皿网的对称单体双重类别。
Fong developed `decorated cospans' to model various kinds of open systems: that is, systems with inputs and outputs. In this framework, open systems are seen as the morphisms of a category and can be composed as such, allowing larger open systems to be built up from smaller ones. Much work has already been done in this direction, but there is a problem: the notion of isomorphism between decorated cospans is often too restrictive. Here we introduce and compare two ways around this problem: structured cospans, and a new version of decorated cospans. Structured cospans are very simple: given a functor $L \colon \mathsf{A} \to \mathsf{X}$, a `structured cospan' is a diagram in $\mathsf{X}$ of the form $L(a) \rightarrow x \leftarrow L(b)$. If $\mathsf{A}$ and $\mathsf{X}$ have finite colimits and $L$ is a left adjoint, there is a symmetric monoidal category whose objects are those of $\mathsf{A}$ and whose morphisms are isomorphism classes of structured cospans. However, this category arises from a more fundamental structure: a symmetric monoidal double category. Under certain conditions this symmetric monoidal double category is equivalent to one built using our new version of decorated cospans. We apply these ideas to symmetric monoidal double categories of open electrical circuits, open Markov processes and open Petri nets.
