We introduce constructible directed complexes, a combinatorial presentation of higher categories inspired by constructible complexes in poset topology. Constructible directed complexes with a greatest element, called atoms, encompass common classes of higher-categorical cell shapes, including globes, cubes, oriented simplices, and a large sub-class of opetopes, and are closed under lax Gray products and joins. We define constructible polygraphs to be presheaves on a category of atoms and inclusions, and extend the monoidal structures. We show that constructible directed complexes are a well-behaved subclass of Steiner’s directed complexes, which we use to define a realisation functor from constructible polygraphs to $$\omega $$ ω -categories. We prove that the realisation of a constructible polygraph is a polygraph in restricted cases, and in all cases conditionally to a conjecture. Finally, we define the geometric realisation of a constructible polygraph, and prove that it is a CW complex with one cell for each of its elements.

中文翻译：

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测谎的组合拓扑形状类别

我们引入了可构造的有向复合体，这是一种受偏序拓扑中的可构造复合体启发的更高类别的组合表示。具有最大元素（称为原子）的可构造定向复合体包含常见类别的更高分类单元形状，包括球体、立方体、定向单形和一大类 opetopes，并在松散的格雷乘积和连接下封闭。我们将可构造的测谎仪定义为一类原子和内含物的预层，并扩展了幺半群结构。我们证明可构造有向复合物是 Steiner 有向复合物的一个良好的子类，我们用它来定义从可构造测谎到 $$\omega $$ ω -categories 的实现函子。我们证明了可构造测谎仪的实现是受限情况下的测谎仪，并且在所有情况下都有条件地进行猜想。最后，我们定义了可构造测谎仪的几何实现，并证明它是一个 CW 复合体，每个元素都有一个单元格。