The lambda calculus is a universal programming language. It can represent the computable functions, and such offers a formal counterpart to the point of view of functions as rules. Terms represent functions and this allows for the application of a term/function to any other term/function, including itself. The calculus can be seen as a formal theory with certain pre-established axioms and inference rules, which can be interpreted by models. Dana Scott proposed the first non-trivial model of the extensional lambda calculus, known as $ D_{\infty }$, to represent the $\lambda $-terms as the typical functions of set theory, where it is not allowed to apply a function to itself. Here we propose a construction of an $\infty $-groupoid from any lambda model endowed with a topology. We apply this construction for the particular case $D_{\infty }$, and we see that the Scott topology does not provide enough information about the relationship between higher homotopies. This motivates a new line of research focused on the exploration of $\lambda $-models with the structure of a non-trivial $\infty $-groupoid to generalize the proofs of term conversion (e.g., $\beta $-equality, $\eta $-equality) to higher-proofs in $\lambda $-calculus.

中文翻译：

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由任意拓扑 λ 模型生成的 ∞-Groupoid

lambda 演算是一种通用编程语言。它可以表示可计算的函数，这提供了作为规则的函数观点的正式对应物。术语代表功能，这允许将术语/功能应用于任何其他术语/功能，包括其自身。微积分可以看作是具有某些预先建立的公理和推理规则的形式理论，可以通过模型来解释。Dana Scott 提出了外延 lambda 演算的第一个非平凡模型，称为 $ D_{\infty }$，将 $\lambda $-项表示为集合论的典型函数，其中不允许应用对自己起作用。在这里，我们建议从任何具有拓扑结构的 lambda 模型构造一个 $\infty $-groupoid。我们将此构造应用于特定情况 $D_{\infty }$, 我们看到 Scott 拓扑没有提供关于更高同伦之间关系的足够信息。这激发了一个新的研究方向，专注于探索具有非平凡 $\infty $-groupoid 结构的 $\lambda $-模型，以推广术语转换的证明（例如，$\beta $-equality，$ \eta $-equality) 到 $\lambda $-calculus 中的更高证明。