By extending type theory with a universe of definitionally associative and unital polynomial monads, we show how to arrive at a definition of opetopic type which is able to encode a number of fully coherent algebraic structures. In particular, our approach leads to a definition of $\infty$-groupoid internal to type theory and we prove that the type of such $\infty$-groupoids is equivalent to the universe of types. That is, every type admits the structure of an $\infty$-groupoid internally, and this structure is unique.

中文翻译：

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类型是内部$ \ infty $ -Groupoids

通过将类型理论扩展到定义性关联和单位多项式单子态的宇宙中，我们展示了如何得出对位类型的定义，该对位类型能够对许多完全相干的代数结构进行编码。特别地，我们的方法导致了类型理论内部的$ \ infty $ -groupoid的定义，并且证明了此类$ \ infty $ -groupoids的类型等同于类型范围。也就是说，每种类型都在内部接受$ \ infty $ -groupoid的结构，并且这种结构是唯一的。