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

Abstract

International audienceCategories 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 γ on a locally ordered space X to a single point. If we worked in the category of dspaces, 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 γ. 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