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The Internal Operads of Combinatory Algebras
We argue that operads provide a general framework for dealing with
polynomials and combinatory completeness of combinatory algebras, including the
classical -algebras, linear -algebras, planar
-algebras as well as the braided -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
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