The Internal Operads of Combinatory Algebras

Abstract

We argue that operads provide a general framework for dealing with polynomials and combinatory completeness of combinatory algebras, including the classical $\mathbf{SK}$-algebras, linear $\mathbf{BCI}$-algebras, planar $\mathbf{BI}(\_{)}^{\bullet }$-algebras as well as the braided $\mathbf{B}{\mathbf{C}}^{\pm }\mathbf{I}$-algebras. We show that every extensional combinatory algebra gives rise to a canonical closed operad, which we shall call the internal operad of the combinatory algebra. The internal operad construction gives a left adjoint to the forgetful functor from closed operads to extensional combinatory algebras. As a by-product, we derive extensionality axioms for the classes of combinatory algebras mentioned above.

References in corpus (1)

• A Braided Lambda Calculus

Cited by in corpus (1)

• Braids, twists, trace and duality in combinatory algebras