Elgot Theories: A New Perspective of Iteration Theories (Extended Abstract)

The concept of iteration theory of Bloom and Ésik summarizes all equational properties that iteration has in usual applications, e.g., in Domain Theory where to every system of recursive equations the least solution is assigned. However, this assignment in Domain Theory is also functorial. Yet, functoriality is not included in the definition of iteration theory. Pitty: functorial iteration theories have a particularly simple axiomatization, and most of examples of iteration theories are functorial. The reason for excluding functoriality was the view that this property cannot be called equational. This is true from the perspective of the category Sgn of signatures as the base category: whereas iteration theories are monadic (thus, equationally presentable) over Sgn, functorial iteration theories are not. In the present paper we propose to change the perspective and work, in lieu of Sgn, in the category of sets in context (the presheaf category of finite sets and functions). We prove that Elgot theories, which is our name for functorial iteration theories, are monadic over sets in context. Shortly: from the new perspective functoriality is equational.