Arboreal范畴：一种资源的公理化理论
Arboreal Categories: An Axiomatic Theory of Resources
AI总结：
本文提出arboreal categories的公理化框架，通过内在过程结构和arboreal covers捕获余代数范畴的本质属性，统一并恢复了有限模型论与描述复杂性中的基本概念。
摘要：
Game comonads提供了一种面向有限模型论的无语法范畴方法，其Eilenberg-Moore余代数通常编码结构的重要组合参数。在本文中，我们开发了一个框架，以纯公理化的方式捕获这些余代数范畴的本质属性。为此，我们引入了arboreal categories，它具有内在的过程结构，允许定义诸如互模拟和往返博弈等动态概念，以及诸如博弈轮数等资源概念。这些通过arboreal covers与外延的或“静态”的结构相关联，后者是资源索引的余伴随。这些思想在一般的公理化环境中发展，并应用于关系结构，其中恢复了Abramsky等人最近引入的用于pebbling、Ehrenfeucht-Fraïssé和模态互模拟博弈的余代数构造，表明有限模型论和描述复杂性的许多基本概念源于arboreal covers的实例。
英文摘要：
Game comonads provide a categorical syntax-free approach to finite model theory, and their Eilenberg-Moore coalgebras typically encode important combinatorial parameters of structures. In this paper, we develop a framework whereby the essential properties of these categories of coalgebras are captured in a purely axiomatic fashion. To this end, we introduce arboreal categories, which have an intrinsic process structure, allowing dynamic notions such as bisimulation and back-and-forth games, and resource notions such as number of rounds of a game, to be defined. These are related to extensional or "static" structures via arboreal covers, which are resource-indexed comonadic adjunctions. These ideas are developed in a general, axiomatic setting, and applied to relational structures, where the comonadic constructions for pebbling, Ehrenfeucht-Fraïssé and modal bisimulation games recently introduced by Abramsky et al. are recovered, showing that many of the fundamental notions of finite model theory and descriptive complexity arise from instances of arboreal covers.
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