The $K_{\mathrm{\infty }}$ Homotopy $\lambda$-Model

We extend the complete ordered set Dana Scott’s $D_{\mathrm{\infty }}$ to a complete weakly ordered Kan complex $K_{\mathrm{\infty }}$, with properties that guarantee the non-equivalence of the interpretation of some higher conversions of $\beta \eta$-conversions of $\lambda$-terms.