A directed space is a topological space $X$ together with a subspace $\vec {P}(X)\subset X^I$ of directed paths on $X$. A symmetry of a directed space should therefore respect both the topology of the underlying space and the topology of the associated spaces $\vec {P}(X)_-^+$ of directed paths between a source ($-$) and a target ($+$)—up to homotopy. If it is, moreover, homotopic to the identity map—in a directed sense—such a symmetry will be called an inessential d-map, and the paper explores the algebra and topology of inessential d-maps. Comparing two d-spaces $X$ and $Y$ ‘up to symmetry’ yields the notion of a directed homotopy equivalence between them. Under appropriate conditions, all directed homotopy equivalences are shown to satisfy a 2-out-of-3 property. Our notion of directed homotopy equivalence does not agree completely with the one defined in Goubault (2017, arxiv:1709:05702v2) and Goubault, Farber and Sagnier (2020, J. Appl. Comput. Topol. 4, 11–27); the deviation is motivated by examples. Nevertheless, directed topological complexity, introduced in Goubault, Farber and Sagnier (2020) is shown to be invariant under our notion of directed homotopy equivalence. Finally, we show that directed homotopy equivalences result in isomorphisms on the pair component categories of directed spaces introduced in Goubault, Farber and Sagnier (2020).

中文翻译：

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非必要的有向映射和有向同伦等价

有向空间是拓扑空间$X$连同一个子空间$\vec {P}(X)\子集 X^I$的定向路径$X$. 因此，有向空间的对称性应该同时尊重底层空间的拓扑结构和相关空间的拓扑结构$\vec {P}(X)_-^+$源之间的有向路径（$-$) 和目标 ($+$)——直到同伦。此外，如果它与恒等映射同伦——在有向的意义上——这种对称性将被称为非实体 d-map，本文探讨了非实体 d-map 的代数和拓扑。比较两个 d 空间$X$和$Y$“直到对称”产生了它们之间有向同伦等价的概念。在适当的条件下，所有有向同伦等价都被证明满足 2-out-of-3 属性。我们的定向同伦等价概念与 Goubault (2017, arxiv:1709:05702v2) 和 Goubault, Farber 和 Sagnier (2020, J. Appl. Comput. Topol. 4, 11-27) 中定义的概念并不完全一致；这种偏差是由例子引起的。然而，Goubault、Farber 和 Sagnier (2020) 中引入的有向拓扑复杂性在我们的有向同伦等价概念下被证明是不变的。最后，我们展示了有向同伦等价导致 Goubault、Farber 和 Sagnier (2020) 中引入的有向空间的对分量类别的同构。