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.
翻译:本地定购空间的类别特别适合于实现大多数预设空间,尽管它们的共限并不那么容易确定(比如d-空间类别中的共限 ) 。 我们在这里使用复数, 因为本地定购空间的概念从作者到另一个作者各不相同, 与表面上的反常技术细节不同。 正如我们在本篇文章中解释的那样, 这些差异对共同限具有巨大的影响。 特别是, 我们显示大多数本地定购空间的类别不完全, 从而回答一个迄今为止被忽视的问题。 战略如下: 给本地定购空间X 的直线循环 {gamma} 带来直接的循环 {gamma}, 我们试图用一个点来识别 {gamma} 的图像。 如果在 d- 空间类别中采用, 这样的识别可能会造成一个涡旋, 而本地定购定的空间空间没有软性。 具体地说, 本地定购定的反正对称性空间的度比重, 而不是仅仅在上方位的。 然而, 这个现象的趋性效果在某种意义上, 它的直等性 。