  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.
