Algebraic fundamental group and simplicial complexes

{\tt Macaulay2} 中的抽象单纯复形

Abstract simplicial complexes in {\tt Macaulay2}

该研究为计算机代数系统 {\tt Macaulay2} 开发了 {\tt AbstractSimplicialComplexes.m2} 包，以哈希表形式的分次列表实现抽象单纯复形，支持链复形直接实现及随机复形构造，补充现有框架。

Simplicial Structure on Complexes

While chain complexes are equipped with a differential $d$ satisfying $d^2 = 0$, their generalizations called $N$-complexes have a differential $d$ satisfying $d^N = 0$. In this paper we show that the lax nerve of the category of chain complexes is pointwise adjoint equivalent to the décalage of the simplicial category of $N$-complexes. This reveals additional simplicial structure on the lax nerve of the category of chain complexes which provides a categorfication of the triangulated homotopy category of chain complexes. We study this phenomena in general and present evidence that the axioms o…