流范畴的交错理论
Theory of interleavings on categories with a flow
摘要：
交错距离最初是由Chazal等人在拓扑数据分析（TDA）领域中定义的，作为在实数线上参数化的持久性模块类的度量。Bubenik等人随后将定义扩展到偏序集上的函子范畴，这些范畴中的对象被视为“广义持久性模块”。这些度量通常依赖于偏序集的自同态的lax半群的选择。本文件的目的是使用“行为”理论为交错距离的概念建立一个更一般的框架。具体来说，我们扩展的概念交错距离的任意类别配备了一个流，即宽松monoidal行动的幺半群$[0,\infty)$。这样，在这样一个范畴中的对象类就获得了一个Lawvere度量空间的结构。为colax $[0,\infty)$ -等变的函子生成为$1$ -Lipschitz的映射。这导致简明的证明各种已知的稳定性结果TDA，通过考虑适当的共轴$[0,\infty)$ -等变函子。一路上，我们表明，几个常见的度量，包括Hausdorff距离和$L^{\infty}$规范，可以实现为交织距离在这个一般的角度来看。
英文摘要：
The interleaving distance was originally defined in the field of Topological Data Analysis (TDA) by Chazal et al. as a metric on the class of persistence modules parametrized over the real line. Bubenik et al. subsequently extended the definition to categories of functors on a poset, the objects in these categories being regarded as `generalized persistence modules'. These metrics typically depend on the choice of a lax semigroup of endomorphisms of the poset. The purpose of the present paper is to develop a more general framework for the notion of interleaving distance using the theory of `actegories'. Specifically, we extend the notion of interleaving distance to arbitrary categories equipped with a flow, i.e. a lax monoidal action by the monoid $[0,\infty)$. In this way, the class of objects in such a category acquires the structure of a Lawvere metric space. Functors that are colax $[0,\infty)$-equivariant yield maps that are $1$-Lipschitz. This leads to concise proofs of various known stability results from TDA, by considering appropriate colax $[0,\infty)$-equivariant functors. Along the way, we show that several common metrics, including the Hausdorff distance and the $L^{\infty}$-norm, can be realized as interleaving distances in this general perspective.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
