On strict extensional reflexivity in compact closed categories

Abstract

This article studies the categorical setting of Abramsky, Haghverdi, and Scott's untyped linear combinatory algebras, and relates this to more recent work of Abramsky and Heunen on Frobenius algebras in the infinitary setting. The key to this is extensional reflexivity (the property of an object being isomorphic to its own internal hom. $N\cong [N\rightarrow N]$). We characterise extensional reflexivity in compact closed categories, and consider how this may be `strictified' to give monoidally equivalent categories where the isomorphisms exhibiting reflexivity are identity arrows. This results in two-object compact closed categories consisting of a unit object, and a non-unit extensionally reflexive object. We study the endomorphism monoids of such `strictly extensionally reflexive' objects from an algebraic viewpoint. They necessarily contain an interesting monoid that may be thought of as Richard Thompson's iconic group F together with the equally iconic bicyclic monoid of semigroup theory, with non-trivial interactions between the two derived from the Frobenius algebra identity. We claim this as a significant example of the (unitless) Frobenius algebras of Abramsky and Heunen. We then give concrete examples, based on the algebra and category theory behind the Geometry of Interaction program. These are based on the traced monoidal category of partial injections, and reflexive objects in the compact closed category that results from applying the Int or GoI construction. We give compact closed categories, monoidally equivalent to compact closed subcategories of Int(pInj), where this reflexivity is exhibited by identity arrows, and show how the above algebraic structures (Thompson's F, the bicyclic monoid, and Frobenius algebras) arise in a fundamental manner.