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

Abstract

Categories of locally ordered spaces are especially well-adapted to the realization of most precubical sets, although their colimits are not so easy to determine (in comparison with colimits of d-spaces for example). We use the plural here, as the notion of a locally ordered space varies from an author to another, only differing according to seemingly anodyne technical details. However, these differences have dramatic consequences on colimits: most categories of locally ordered spaces are not cocomplete, which answers a question that was neglected so far. We proceed by identifying the image of a directed loop $\gamma$ on a locally ordered space ${\mathcal {X}}$ to a single point. If we worked in the category of d-spaces, such an identification would be likely to create a vortex, while locally ordered spaces have no vortices. In fact, the existence and the nature of the corresponding coequalizer strongly depends on the topology around the image of $\gamma$. Besides the pathologies, we provide an example of well-behaved colimit of locally ordered spaces related to the realization of a singular precubical set. Moreover, for a well-chosen notion of locally ordered spaces, the latter induce diagrams of ordered spaces whose colimits are expected to be well-behaved; the category of locally ordered spaces is the smallest extension of the category of ordered spaces satisfying this property. Our locally ordered spaces are compared to streams, d-spaces, and to the ‘original’ locally partially ordered spaces.