Galois theory of simplicial complexes
type: Item
instance of: scholarly article
title: Galois theory of simplicial complexes
zbMATH Open document ID: 1029.18007
DOI: 10.1016/S0166-8641(03)00036-1
author: Marco Grandis
author: George Janelidze
published in: Topology and its Applications
publication date: 2003-09-04T00:00:00Z
review text: By the ``categorical Galois theory'' of covering spaces the authors mean the classification of covering spaces via the fundamental groupoid and its actions, presented from a categorical point of view by \textit{P. Gabriel} and \textit{M. Zisman} in [Calculus of fractions and homotopy theory. Berlin: Springer (1967; Zbl 0186.56802)]. The authors apply categorical Galois theory to the adjunction \[ {\mathcal S}m{\mathcal C}{\overset \Pi{_0} {\underset {D}\rightleftarrows}}{\mathcal S}et(\Pi_0\dashv D) \] where \({\mathcal S}m {\mathcal C}\) is the category of simplicial complexes, \(\Pi_0(A)\) the set of connected components of a simplicial complex \(A\) and \(D\) embeds \({\mathcal S}et\) into \({\mathcal S}m {\mathcal C}\) regarding sets as discrete simplicial complexes (i.e., simplicial complexes with no \(n\)-simplex for \(n\neq 1)\). The category \({\mathcal S}m{\mathcal C}\) is equivalent to the category \(\text{Fam}({\mathcal A})\) of families on the category \({\mathcal A}\) of connected simplicial complexes. By this equivalence, the previous adjunction appears to be a special case of the basic adjunction of the Galois theory of abstract families \[ \text{Fam}({\mathcal A}){\overset {I}{ \underset {H}\rightleftarrows}} \text{Fam(\textbf{1}})= {\mathcal S}et(I\dashv H) \] where \({\mathcal A}\) is an arbitrary category with terminal object and \(\text{Fam} ({\mathcal A})\) has pullbacks and \(I\) sends a family to its set of indices, or, equivalently, to the set of its connected components. A morphism \(\alpha:A \to B\) in the category \({\mathcal C}=\text{Fam}({\mathcal A})\) is called a covering if there exists a pullback diagram \[ \begin{tikzcd} E\times_BA \ar[r,"pr_2"]\ar[d,"pr_1" '] & A\ar[d,"\alpha"]\\ E \ar[r,"p" '] & B\end{tikzcd} \] in which \(p\) is an effective descent morphism, and \(pr_1:E\times_B A\to E\) is a trivial covering. The authors prove properties of these coverings. For example, it is proved that \(\alpha\) is a covering iff this morphi…
Mathematics Subject Classification ID: 18G30
Mathematics Subject Classification ID: 57Q05
Mathematics Subject Classification ID: 57M10
Mathematics Subject Classification ID: 18B30
Mathematics Subject Classification ID: 55U35
Mathematics Subject Classification ID: 18A40
zbMATH DE Number: 1974779
zbMATH Keywords: categorical Galois theory
zbMATH Keywords: covering spaces
zbMATH Keywords: fundamental groupoid
zbMATH Keywords: simplicial complexes
zbMATH Keywords: effective descent
zbMATH Keywords: projective subobject lifting property
MaRDI profile type: Publication
cites work: Etale homotopy
cites work: Q3926769
cites work: Galois theories
cites work: Q4040708
cites work: Galois theory of second order covering maps of simplicial sets
cites work: On localization and stabilization for factorization systems
cites work: Q3691902
cites work: Q5580926
cites work: Higher fundamental functors for simplicial sets
cites work: An intrinsic homotopy theory for simplicial complexes, with applications to image analysis
cites work: Q3978974
cites work: Q4342440
cites work: Facets of descent. I
cites work: What is the fundamental group?
cites work: Homotopical algebra
cites work: Effective descent morphisms in some quasivarieties of algebraic, relational, and more general structures
cites work: Q5522742
full work available at URL: s0166-8641(03)00036-1
OpenAlex ID: W1992958487
description: scientific article; zbMATH DE number 1974779