Non-existing and ill-behaved coequalizers of locally ordered spaces

TL;DR

This paper investigates the limitations of colimits in categories of locally ordered spaces, showing that many such categories are not cocomplete due to the complex behavior of coequalizers related to directed loops.

Contribution

It demonstrates that most categories of locally ordered spaces lack cocompleteness, highlighting how technical differences affect colimit existence and providing insights into their topological and order-theoretic properties.

Findings

Most categories of locally ordered spaces are not cocomplete.

The existence of coequalizers depends on local topological structure around directed loops.

Connected components allow for coequalizers matching topological quotients.

Abstract

Categories of locally ordered spaces are especially well-adapted to the realization of most precubical sets, though their colimits are not so easy to determine (in comparison with colimits in the category of d-spaces for example). We use the plural here, as the notion of a locally ordered space vary from an author to another, only differing according to seemingly anodyne technical details. As we explain in this article, these differences have dramatic consequences on colimits. In particular, we show that most categories of locally ordered spaces are not cocomplete, thus answering a question that was neglected so far. The strategy is the following: given a directed loop {\gamma} on a locally ordered space X, we try to identify the image of {\gamma} with a single point. If it were taken in the category of d-spaces, such an identification would be likely to create a vortex, while locally ordered spaces have no vortices. Concretely, the antisymmetry of local orders gets more points to be identified than in a mere topological quotient. However, the effect of this phenomenon is in some sense limited to the neighbourhood of (the image of) {\gamma}. So the existence and the nature of the corresponding coequalizer strongly depends on the topology around the image of {\gamma}. As an extreme example, if the latter forms a connected component, the coequalizer exists and its underlying space matches with the topological coequalizer.