Towards a Homotopy Domain Theory

An appropriate framework is put forward for the construction of $\lambda$-models with $\mathrm{\infty }$-groupoid structure, which we call $\mathit{\text{homotopic}}\lambda \mathit{\text{-models}}$, through the use of an $\mathrm{\infty }$-category 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 $\mathit{\text{higher}}\mathit{\text{proof}}$ (homotopy).