Components of the Fundamental Category

Invariants of Directed Spaces

“…On the other hand, it is not always clear, how natural this extra requirement is. In particular, in the presence of loops, this method (and also those of the previous [7]) does not give satisfying results.…”

“…[9,10]) or as elements of subspaces of spaces of paths in an underlying topological space (with suitable concatenations). We will apply the latter approach, compare also [14,7].…”

“…It was the aim of [7], to divide the underlying space into "components" in a systematic way, so that one only needs to investigate the (dihomotopy classes of) directed paths between components. The outcome was the component category of the fundamental category of a space, with certain generalizations to component categories of other categories.…”

“…The outcome was the component category of the fundamental category of a space, with certain generalizations to component categories of other categories. The approach chosen in [7] had the drawback, that it…”

“…Morphisms in the indexing category are then regarded as (weakly) invertible if they are mapped into isomorphisms by the functor. As a result, one can apply the localization method leading to components suggested in [7] and modified in [11] to the indexing category.…”