Group completions and the homotopical monadicity theorem

This paper is divided into three parts.

In the first part, we give the general abstract axiomatic theory and treat the classical examples of infinite loop space machines with structured spaces or $G$-spaces as input and spectra or $G$-spectra as output.

The new prequel paper gives a logically compelling and more general but less useful analog that does not involve group completion.

In the second part, we develop a general context of composite adjunctions that feeds into the first.

It specializes to give infinite loop space machines that take either orbital presheaves or algebras over categories of operators as input.

The new sequel focuses on new constructions and applications when the starting category is that of orbital presheaves of spaces and the output is $G$-spectra.

In the brief third part, we show how the multiplicative theory fits into the axiomatic frameworks of the first and second parts.