From higher-order rewriting systems to higher-order categorial algebras and higher-order Curry-Howard isomorphisms

This ongoing project aims to define and investigate, from the standpoint of category theory, order theory and universal algebra, the notions of higher-order many-sorted rewriting system and of higher-order many-sorted categorial algebra and their relationships, via the higher-order Curry-Howard isomorphisms. The ultimate goal, to be developed in future versions of this work, is to define and investigate the category of towers, whose objects will consist of families, indexed by $\mathbb{N}$, of higher-order many-sorted rewriting systems and of higher-order many-sorted categorial algebras, including higher-order Curry-Howard type results for the latter, together with an additional structure that intertwines such $\mathbb{N}$-families; and whose morphism from a tower to another will be families, indexed by $\mathbb{N}$, of morphisms between its higher-order many-sorted rewriting systems and of higher-order many-sorted categorial algebras compatible with their structures. All feedback is appreciated.
Comment: 814 pages, three parts and an appendix