度量空间的因果序复形与幅度同伦型
Causal order complex and magnitude homotopy type of metric spaces
AI总结：
本文通过时空因果偏序构造度量空间的幅度同伦型，建立从度量空间范畴到带基点拓扑空间范畴的协变函子，并利用离散 Morse 理论重新证明 Mayer-Vietoris 型定理及有限度量空间在 sycamore twist 下的幅度不变性。
摘要：
在本文中，我们为给定的度量空间 $X$ 和实参数 $\ell \geq 0$ 构造了一个带基点的 CW 复形，称为幅度同伦型。该空间大致由所有长度为 $\ell$ 的路径组成，并且其约化同调群与 $X$ 的幅度同调群同构。为了构造幅度同伦型，我们考虑时空 $X\times\mathbb{R}$ 上由因果（类时或类光）关系定义的偏序集结构。幅度同伦型被定义为 $X\times\mathbb{R}$ 上区间的序复形对一个特定子复形的商。幅度同伦型给出了从带 $1$-Lipschitz 映射的度量空间范畴到带基点拓扑空间范畴的一个协变函子。对于某些度量空间，幅度同伦型还具有类似“路径积分”的表达式。通过对幅度同伦型应用离散 Morse 理论，我们得到了 Mayer-Vietoris 型定理的一个新证明，以及若干新结果，包括有限度量空间在 sycamore twist 下幅度的不变性。
英文摘要：
In this paper, we construct a pointed CW complex called the magnitude homotopy type for a given metric space $X$ and a real parameter $\ell \geq 0$. This space is roughly consisting of all paths of length $\ell$ and has the reduced homology group that is isomorphic to the magnitude homology group of $X$. To construct the magnitude homotopy type, we consider the poset structure on the spacetime $X\times\mathbb{R}$ defined by causal (time- or light-like) relations. The magnitude homotopy type is defined as the quotient of the order complex of an intervals on $X\times\mathbb{R}$ by a certain subcomplex. The magnitude homotopy type gives a covariant functor from the category of metric spaces with $1$-Lipschitz maps to the category of pointed topological spaces. The magnitude homotopy type also has a ``path integral'' like expression for certain metric spaces. By applying discrete Morse theory to the magnitude homotopy type, we obtain a new proof of the Mayer-Vietoris type theorem and several new results including the invariance of the magnitude under sycamore twist of finite metric spaces.
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。
