Elgot theories: a new perspective on the equational properties of iteration

Bloom and ??sik's concept of iteration theory summarises all equational properties that iteration has in common applications, for example, in domain theory, where to every system of recursive equations, the least solution is assigned. This paper shows that in the coalgebraic approach to iteration, the more appropriate concept is that of a functorial iteration theory (called Elgot theory). These theories have a particularly simple axiomatisation, and all well-known examples of iteration theories are functorial. Elgot theories are proved to be monadic over the category of sets in context (or, more generally, the category of finitary endofunctors of a locally finitely presentable category). This demonstrates that functoriality is an equational property from the perspective of sets in context. In contrast, Bloom and ??sik worked in the base category of signatures rather than sets in context, and there iteration theories are monadic but Elgot theories are not. This explains why functoriality was not included in the definition of iteration theories.