预立方集的定向退化映射
Directed degeneracy maps for precubical sets
AI总结：
本文通过引入q-实现函子和自然实现函子，证明了对称横截集上的定向同伦，并利用c-Reedy范畴性质证明二者同伦等价，将预立方集的结果推广至对称横截集。
摘要：
对称横截集的引入是为了使带同步的进程代数的并行乘积构造具有函子性。本文证明了可以在对称横截集上进行定向同伦，具体意义如下。利用流的q-余纤维替换函子，引入了从对称横截集到流的q-实现函子。通过对余横截映射进行拓扑化，构造了余横截拓扑立方体，它既可以视为余横截拓扑空间，也可以视为余横截Lawvere度量空间。利用Raussen的自然$d$-路径概念，并将其借助Lawvere度量空间结构推广到对称横截集，引入了从对称横截集到流的自然实现函子。利用定义对称横截集的小范畴在Shulman意义下是c-Reedy的这一事实，证明了这两个实现函子在余纤维对称横截集上是同伦等价的。这将先前针对预立方集获得的结果推广到了对称横截集。
英文摘要：
Symmetric transverse sets were introduced to make the construction of the parallel product with synchronization for process algebras functorial. It is proved that one can do directed homotopy on symmetric transverse sets in the following sense. A q-realization functor from symmetric transverse sets to flows is introduced using a q-cofibrant replacement functor of flows. By topologizing the cotransverse maps, the cotransverse topological cube is constructed. It can be regarded both as a cotransverse topological space and as a cotransverse Lawvere metric space. A natural realization functor from symmetric transverse sets to flows is introduced using Raussen's notion of natural $d$-path extended to symmetric transverse sets thanks to their structure of Lawvere metric space. It is proved that these two realization functors are homotopy equivalent on cofibrant symmetric transverse sets by using the fact that the small category defining symmetric transverse sets is c-Reedy in Shulman's sense. This generalizes to symmetric transverse sets results previously obtained for precubical sets.
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