Classification of dicoverings
In a previous paper [Homology Homotopy Appl. 5, No.~2, 1--17, electronic only (2003; Zbl 1030.68057)], the author proposed a construction of the universal dicovering \(\hat{X}\) of a well-pointed \(d\)-space \((X,x)\) (i.e. in which all points are the target of a \(d\)-path with source \(x\)). She provided examples where the universal map failed to be continuous and it was unclear whether the solution was to restrict the spaces \(X\) or the dicoverings allowed, or to change the topology on \(\hat{X}\). The author proves in this paper that the latter was the good one. The good space is obtained by taking the boxification of \(\hat{X}\) (the effect is to add opens). The boxification is the coreflector associated with the full coreflective subcategory of \(d\)-spaces generated by directed cubes. It is also proved in this paper that this subcategory is actually generated by the asynchronous segment \([0,1]\) and the directed segment. The author shows that the same proof applies for \(\Delta\)-generated spaces which are therefore generated by the segment \([0,1]\). In the last part of the paper, the dicoverings of a well pointed \(d\)-space are classified as quotients of the universal dicovering under congruence relations.
scientific article; zbMATH DE number 5791181
scientific article