标题： $(\infty, n)$-范畴的半严格化 显示英文标题

标题： Semi-strictification of $(\infty, n)$-categories

摘要： 我们证明了弱非代数模型与半严格代数模型之间的第一个等价关系，该模型是$(\infty, n)$-范畴。 这表现为一种自然的半严格化，其中弱$(\infty, n)$-范畴通过一个循环上纤维被嵌入到一个半严格模型中，使得弱函子可以提升为半严格函子；这构成了弱模型范畴之间 Quillen 等价的导出单位，其纤维对象分别为弱$(\infty, n)$-范畴和（在循环纤维之上）半严格模型。 半严格模型具有代数单位和圆粘贴图的复合，满足如 Henry 的 Simpson 弱单位猜想的正规版本中所描述的严格结合性和交换性形式；半严格函子严格保持圆复合，但仅弱保持单位。 全局复合操作是从单位和圆复合的组合中得到的。 由于这些模型在情况$n = 0$下满足同伦假设，因此该结果也展示了第一个具有代数单位和复合的经典的同伦类型的半严格模型。 这些构造基于正则定向复形的组合学，并且完全显式和组合，符合 Mac Lane 的双范畴严格化的精神。 显示英文摘要

摘要： We prove the first equivalence between a weak non-algebraic model and a semi-strict algebraic model of $(\infty, n)$-categories. This takes the form of a natural semi-strictification, whereby a weak $(\infty, n)$-category is embedded into a semi-strict one through an acyclic cofibration, in such a way that weak functors lift to semi-strict functors; this constitutes the derived unit of a Quillen equivalence between weak model categories whose fibrant objects are, respectively, the weak $(\infty, n)$-categories and (up to an acyclic fibration) the semi-strict ones. The semi-strict model has algebraic units and composition of round pasting diagrams, satisfying a strict form of associativity and interchange as in Henry's regular version of Simpson's weak units conjecture; semi-strict functors strictly preserve round composition, but only weakly preserve units. Globular composition operations are obtained from a combination of units and round composition. Since the models satisfy the homotopy hypothesis in the case $n = 0$, this result also exhibits the first semi-strict model of the classical homotopy types that has algebraic units and composition. The constructions are based on the combinatorics of regular directed complexes and are entirely explicit and combinatorial, in the spirit of Mac Lane's strictification of bicategories.