Many definitions of weak and strict $\infty$-categories have been proposed. In this paper we present a definition for $\infty$-categories with strict associators, but which is otherwise fully weak. Our approach is based on the existing type theory Catt, whose models are known to correspond to weak $\infty$-categories. We add a definitional equality relation to this theory which identifies terms with the same associativity structure, yielding a new type theory Catt$_{sa}$, for strictly associative $\infty$-categories. We also provide a reduction relation which generates definitional equality, and show it is confluent and terminating, giving an algorithm for deciding equality of terms, and making typechecking decidable. Our key contribution, on which our reduction is based, is an operation on terms which we call insertion. This has a direct geometrical interpretation, allowing a subterm to be inserted into the head of the term, flatting its syntactic structure. We describe this operation combinatorially in terms of pasting diagrams, and also show can be characterized as a pushout of contexts. This allows reasoning about insertion using just its universal property.

翻译：提出了许多薄弱和严格的美元类别的定义。 在本文中,我们提出了一个与严格关联的美元类别的定义,但该定义则完全薄弱。我们的方法基于现有的类型理论Catt,其模型已知与疲软的美元类别相对应。我们对这一理论添加了一个定义平等关系,该理论将术语与相同的联合结构确定为同一术语,产生一种新的类型理论Catt$@sa},用于严格关联的美元类别。我们还提供了产生定义平等的削减关系,并显示它具有互通性和终止性,为决定条件平等提供算法,并进行排序校正。我们削减所依据的是我们要求插入的术语。我们有一个直接的几何等解释,允许在术语标题中插入一个子术语,固定其组合结构。我们用粘贴图的术语来描述这一操作,并显示它具有互通性和终止性,同时能够以推动性推动性推动性推动性推动性推动性推动性推动性推动性推动性推动性推动性。