Clones, closed categories, and combinatory logic

Abstract

We explain how to recast the semantics of the simply-typed $$\uplambda $$ λ -calculus, and its linear and ordered variants, using multi-ary structures. We define universal properties for multicategories, and use these to derive familiar rules for products, tensors, and exponentials. Finally we outline how to recover both the category-theoretic syntactic model and its semantic interpretation from the multi-ary framework. We then use these ideas to study the semantic interpretation of combinatory logic and the simply-typed $$\uplambda $$ λ -calculus without products. We introduce extensional SK-clones and show these are sound and complete for both combinatory logic with extensional weak equality and the simply-typed $$\uplambda $$ λ -calculus without products. We then show such SK-clones are equivalent to a variant of closed categories called SK-categories, so the simply-typed $$\uplambda $$ λ -calculus without products is the internal language of SK-categories.