Let us define a modulated Cauchy algebra to be a set with a function , where is the set of Cauchy approximations in and is the set of functions with domain and codomain , such that
A modulated Cauchy algebra homomorphism is a function between modulated Cauchy algebras and such that
The category of modulated Cauchy algebras is the category whose objects are modulated Cauchy algebras and whose morphisms are modulated Cauchy algebra homomorphisms. The set of modulated Cauchy real numbers, denoted , is defined as the initial object in the category of modulated Cauchy algebras.
The modulated Cauchy real numbers are sometimes defined using sequences , with modulus of Cauchy convergence , respectively. However, the composition of a sequence and a modulus of Cauchy convergence yields a Cauchy approximation, so one could define Cauchy approximations as and as , with and following from function evaluation. In fact, Cauchy approximations and sequences with a modulus of Cauchy convergence are inter-definable.