We discuss two categorical characterizations of the class of acyclic maps between spaces. The first one is in terms of the higher categorical notion of an epimorphism. The second one employs the notion of a balanced map, that is, a map whose homotopy pullbacks along \(\pi _0\)-surjective maps define also homotopy pushouts. We also identify the modality in the homotopy theory of spaces that is defined by the class of acyclic maps, and discuss the content of the generalized Blakers–Massey theorem for this modality.

中文翻译：

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非循环图的一些特征

我们讨论了空间之间无环图类的两种分类表征。第一个是关于上位型的更高分类概念。第二种使用平衡图的概念，即，沿着\（\ pi _0 \）-射影图的同位性回撤也定义了同位推出的图。我们还确定了由非循环图类定义的空间同伦理论中的模态，并讨论了该模态的广义Blakers-Massey定理的内容。