由幺半范畴中对象的分解所产生的偏序集
Posets arising from decompositions of objects in a monoidal category
AI总结：
本文在对称幺半范畴中统一研究由对象分解产生的偏序集及相关复形，通过显式同伦公式传递 Cohen–Macaulay 性，并涵盖自由群、向量空间和拟阵等经典情形。
摘要：
给定一个以 $\sqcup$ 为积、且该积的中性元为初始对象的对称幺半范畴 $C$，我们考察给定对象 $X$ 的 $\sqcup$-补子对象所构成的偏序集。当该偏序集具有有限高度时，我们定义与 $\sqcup$ 相容的 $X$ 的分解和部分分解，并按加细关系对其排序。基于这些偏序集，我们定义标架复形与部分基复形、增广 Bergman 复形以及相关的有序版本。我们提出一种统一方法来研究它们的组合性质与同伦类型，建立它们之间的多种性质和关系。借助显式同伦公式，我们将能够传递 Cohen–Macaulay 性等结构性质。在已得到充分研究的情形中，$\sqcup$-补子对象偏序集可特化为自由群的自由因子偏序集、向量空间的子空间偏序集、具有非退化形式的向量空间中非退化子空间的偏序集，以及拟阵的平集格。分解与部分分解偏序集、标架复形和部分基复形连同有序版本，或者与熟知结构重合，或者推广这些结构，或者产生新的有趣对象。在这些具体情形中，我们给出了新的结果以及开放问题和猜想。
英文摘要：
Given a symmetric monoidal category $C$ with product $\sqcup$, where the neutral element for the product is an initial object, we consider the poset of $\sqcup$-complemented subobjects of a given object $X$. When this poset has finite height, we define decompositions and partial decompositions of $X$ which are coherent with $\sqcup$, and order them by refinement. From these posets, we define complexes of frames and partial bases, augmented Bergman complexes and related ordered versions. We propose a unified approach to the study of their combinatorics and homotopy type, establishing various properties and relations between them. Via explicit homotopy formulas, we will be able to transfer structural properties, such as Cohen-Macaulayness. In well-studied scenarios, the poset of $\sqcup$-complemented subobjects specializes to the poset of free factors of a free group, the subspace poset of a vector space, the poset of non-degenerate subspaces of a vector space with a non-degenerate form, and the lattice of flats of a matroid. The decomposition and partial decomposition posets, the complex of frames and partial bases together with the ordered versions, either coincide with well-known structures, generalize them, or yield new interesting objects. In these particular cases, we provide new results along with open questions and conjectures.
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。
