Towards a Homotopy Domain Theory

An appropriate framework is put forward for the construction of $\lambda$-models with $\infty$-groupoid structure, which we call \textit{homotopic $\lambda$-models} through the use of an $\infty$-bicategory with cartesian closure and enough points. With this, we establish the start of a project of generalization of Domain Theory and $\lambda$-calculus, in the sense that the concept of proof (path) of equality of $\lambda$-terms is raised to \textit{higher proof} (homotopy)

The Theory of an Arbitrary Higher λ\lambda-Model

One takes advantage of some basic properties of every homotopic $\lambda$-model (e.g. extensional Kan complex) to explore the higher $\beta\eta$-conversions, which would correspond to proofs of equality between terms of a theory of equality of any extensional Kan complex. Besides, Identity types based on computational paths are adapted to a type-free theory with higher $\lambda$-terms, whose equality rules would be contained in the theory of any $\lambda$-homotopic model