Relation between finite topological spaces and finitely presentable groups

Directed algebraic topology and higher dimensional transition system

Cattani-Sassone's notion of higher dimensional transition system is interpreted as a small-orthogonality class of a locally finitely presentable topological category of weak higher dimensional transition systems. In particular, the higher dimensional transition system associated with the labelled n-cube turns out to be the free higher dimensional transition system generated by one n-dimensional transition. As a first application of this construction, it is proved that a localization of the category of higher dimensional transition systems is equivalent to a locally finitely presentable reflect…

Derivations of Group Algebras

In the paper, a method of describing the outer derivations of the group algebra of a finitely presentable group is given. The description of derivations is given in terms of characters of the groupoid of the adjoint action of the group.

Some Properties of Finitely Presented Groups with Topological Viewpoints

In this paper, using some properties of fundamental groups and covering spaces of connected polyhedra and CW-complexes, we present topological proof for some famous theorems about finitely presented groups.

Ambidexterity and the universality of finite spans

Pursuing the notions of ambidexterity and higher semiadditivity as developed by Hopkins and Lurie, we prove that the span $\infty$-category of $m$-finite spaces is the free $m$-semiadditive $\infty$-category generated by a single object. Passing to presentable $\infty$-categories we obtain a description of the free presentable $m$-semiadditive $\infty$-category in terms of a new notion of $m$-commutative monoids, which can be described as spaces in which families of points parameterized by $m$-finite spaces can be coherently summed. Such an abstract summation procedure can be used to give a fo…

刚性化单纯复形与实现群作用

Rigidifying simplicial complexes and realizing group actions

本文证明有限群在有限展示群及有限生成阿贝尔群上的作用均可实现为某空间自同伦等价群在同伦群上的作用，并证明任意有限连通单纯复形可在保持同伦型的前提下刚性化，使其自同构群约化为任意给定子群。

The homotopy-invariance of constructible sheaves

The purpose of this paper is to explain why the functor that sends a stratified topological space $S$ to the $\infty$-category of constructible (hyper)sheaves on $S$ with coefficients in a large class of presentable $\infty$categories is homotopy-invariant. To do this, we first establish a number of results in the unstratified setting, i.e., the setting of locally constant (hyper)sheaves. For example, if $X$ is a locally weakly contractible topological space and $\mathcal{E}$ is a presentable $\infty$-category, then we give a concrete formula for the constant hypersheaf functor $\mathcal{E}\to…