Galois theory of simplicial complexes (Q1403827)

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 morphism has the unique connected projective subobject lifting property. Moreover, the following equivalence of categories is proved: \(\text{Cov}(B)\simeq {\mathcal S}et^{\Pi_1(B)}\), where \(\text{Cov}(B)\) is the category of coverings of \(B\) and \(\Pi_1(B)\) is the fundamental groupoid of the simplicial complex \(B\).