We prove that $$ \cat X\le \frac{\cd(π_1(X))+\dim X}{2}$$ for every CW complex $X$ where $\cd(π_1(X))$ denotes the cohomological dimension of the fundamental group of $X$. We obtain this as a corollary of the inequality $$ \cat X\le\frac{\cat (u_X)+\dim X}{2} $$ where $u_X:X\to Bπ_1(X)$ is a classifying map for the universal covering of $X$.
An Upper Bound for the Depth of Some Classes of Polyhedra
K. Borsuk在70年代介绍了能力和深度的概念，连同一些相关的问题。在本文中，首先，我们引入了任意范畴中对象的（强）容量和（强）深度的概念。然后在群的群范畴中，计算了一些著名群的（强）容量和（强）深度。最后，我们得到了几类有限多面体深度的一个上界，推广了D. Kolodziejczyk在这个问题上。
For a finite group $G$, we define the $G$-cobordism category in dimension two. We show there is a one-to-one correspondence between the connected components of its classifying space and the abelianization of $G$. Also, we find an isomorphism of its fundamental group onto the direct sum $\mathbb{Z}\oplus Ω_2^{SO}(BG)$, where $Ω_2^{SO}(BG)$ is the free oriented $G$-bordism group in dimension two, and we study the classifying space of some important subcategories. We obtain the classifying space has the homotopy type of the product $G/[G,G]\times S^1\times X^G$, where $π_1(X^G)=Ω_2^{SO}(BG)$. Fin…
We develop parametrized generalizations of a number of fundamental concepts in the theory of $\infty$-categories, including factorization systems, free fibrations, exponentiable fibrations, relative colimits and relative Kan extensions, filtered and sifted diagrams, and the universal constructions Ind and $P^Σ$.
We introduce the category of finite étale covers of an arbitrary schematic finite space $X$ and show that, equipped with an appropriate natural fiber functor, it is a Galois Category. This allows us to define the étale fundamental group of schematic spaces. If $X$ is a finite model of a scheme $S$, we show that the resulting Galois theory on $X$ coincides with the classical theory of finite étale covers on $S$ and therefore we recover the classical étale fundamental group introduced by Grothendieck. In order to prove these results it is crucial to find a suitable geometric notion of connectedn…
Given a Noetherian formal scheme $\hat X$ over ${\rm Spf}(R)$, where $R$ is a complete DVR, we first prove a theorem of meromorphic descent along a possibly infinite cover of $\hat{X}$. Using this we construct a specialization functor from the category of continuous representations of the pro-étale fundamental group of the special fiber to the category of $F$-divided sheaves on the generic fiber. This specialization functor partially recovers the specialization functor of the étale fundamental groups. We also express the pro-étale fundamental group of a connected scheme $X$ of finite type over…
Epitopological and pseudotopological fundamental group functors
在这些注记中，我们引入了外延和伪拓扑基本群函子。这些函子分别从指向的外延空间和伪拓扑空间的范畴到它们各自的群对象的范畴。它们对拓扑空间满子范畴的限制是[Daniel K. Biss，The topological基本群and generalized covering spaces，Topology and its Applications，124（3）：355-371，2002]，从而保留其信息。同时，它们表现出更大的规律性，继承自EpiTop和PsTop的方便属性。此外，使用这些方便的范畴，原则上允许应用来自丰富同伦理论的一般技术。我们的方法应该与[Jeremy Brazas，The基本群as a拓扑群，Topology and its Applications，160（1）：170-188，2013]中在拓扑设置内开发的Biss函子的替代改进进行比较。几个开放的问题，包括那些旨在了解确切的关系，以布拉萨斯的方法，分散在整个文本。
