论文标题
部分可观测时空混沌系统的无模型预测
Generalized Composition via Nerves: Models and Algebra
论文作者
论文摘要
简单集的众所周知的条件是小类的神经，相对于两个参数：构成事物的尺寸n，以及事物的位置i，这是组成的结果。在小类的神经中，构成（地图）的事物的维度为n = 1。组成是2个简化（交换三角形），其中复合材料的位置是对面的1 simplex i = 1。这些条件概括为{0，...，n+1}中的所有n> 1和i。我们将如此简单的集合称为（n，i）-composer。本文探讨了作曲家的两个方面：模型和这种广义组成的代数。 模型：我们为作曲家开发了类似于普通类别的集合和功能模型的作曲家家族。获得基于集合的模型的关键（n，i） - composers是要观察到，在一系列集合和功能的神经中，1-Simplex是一个二进制关系，其两个面上的某些属性，即其代码域和域。在（n，i）composer的模型中，每个k-simplex是具有某些属性的面部的A（k+1） - Ary关系，并特别注意k = n（构成）和k = n+1（事物的组成）。这些模型的开发是通过要求的：（i）通常的设置和功能模型，即n = 1和i = 1时，属于家族； （ii）构成的事物具有类似于普通函数所具有的特性的属性。 代数：我们探讨了一些受普通构图代数启发的概念。其中包括“逗号组合商”概括逗号类别，功能复合物，代表性和通用映射的概括的作曲家结构。本文的这一部分在目前的形式上是临时的内容和组织。
The well-known conditions for a simplicial set to be the nerve of a small category generalize with respect to two parameters: the dimension n of the things which compose, and the position i of the thing which is the result of the composition. In the nerve of a small category, the dimension of the things which compose (maps) is n=1. Compositions are 2-simplices (commutative triangles) in which the position of the composite is the 1-simplex opposite vertex i=1. These conditions generalize to all n>1 and i in {0 , ... , n+1}. We call such a simplicial set an (n,i)-composer. This paper explores two aspects of composers: models and the algebra of such generalized composition. Models: we develop a family of set-based models for composers analogous to sets-and-functions models for ordinary categories. The key to obtaining set-based models of (n,i)-composers is to observe that in the nerve of a category of sets-and-functions, a 1-simplex is a binary relation with certain properties on its two faces, that is, its codomain and domain. In a model of an (n,i)-composer, each k-simplex is a (k+1)-ary relation on its faces having certain properties, with special attention to k=n (things which compose) and k=n+1 (a composition of things). The development of these models is guided by requiring: (i) the usual sets-and-functions model, i.e. when n=1 and i=1, belongs to the family; (ii) the things which compose have properties which resemble and generalize properties possessed by ordinary functions. Algebra: we explore some notions which are inspired by the algebra of ordinary composition. These include "comma-composers" generalizing comma categories, a composer structure for function complexes, representables, and a generalization of universal map. In its current form, this part of the paper is provisional in content and organization.
