Applied Categorical Structures ( IF 0.6 ) Pub Date : 2021-03-26 , DOI: 10.1007/s10485-021-09629-x J. R. B. Cockett , G. S. H. Cruttwell , J. -S. P. Lemay
This paper describes how to define and work with differential equations in the abstract setting of tangent categories. The key notion is that of a curve object which is, for differential geometry, the structural analogue of a natural number object. A curve object is a preinitial object for dynamical systems; dynamical systems may, in turn, be viewed as determining systems of differential equations. The unique map from the curve object to a dynamical system is a solution of the system, and a dynamical system is said to be complete when for all initial conditions there is a solution. A subtle issue concerns the question of when a dynamical system is complete, and the paper provides abstract conditions for this. This abstract formulation also allows new perspectives on topics such as commutative vector fields and flows. In addition, the stronger notion of a differential curve object, which is the centrepiece of the last section of the paper, has exponential maps and forms a differential exponential rig. This rig then, somewhat surprisingly, has an action on every differential object and bundle in the setting. In this manner, in a very strong sense, such a curve object plays the role of the real numbers in standard differential geometry.
中文翻译:
切线类别I中的微分方程:完整的矢量场,流和指数
本文介绍了如何在切线类别的抽象设置中定义和使用微分方程。关键概念是曲线对象对于微分几何,它是自然数对象的结构类似物。曲线对象是动力学系统的初始对象。动力学系统又可以看作是微分方程的确定系统。从曲线对象到动力学系统的唯一映射是该系统的一种解决方案,并且在所有初始条件都存在一个解决方案的情况下,称动力学系统是完整的。一个微妙的问题涉及动力系统何时完成的问题,本文为此提供了抽象条件。这种抽象的表达方式也使人们对诸如可交换矢量场和流动等主题有了新的认识。另外,微分曲线对象的概念更强,是本文最后一部分的重点,它具有指数映射并形成差分指数绑定。然后,这台钻机对设置中的每个差分对象和捆束都有作用,这有些出乎意料。这样,在非常强烈的意义上,这样的曲线对象在标准微分几何中扮演了实数的角色。