In his first efforts after his doctoral thesis, my doctoral advisor, Sufian Husseini, developed a collection of ideas aimed at generalizing the James construction $J(X)$. His purpose was to model the (Moore) loop spaces of CW complexes that are not necessarily suspensions. By the mid 1960s he had successfully given a construction that applied (like James' construction) to `special' CW complexes: countable CW complexes $X$ with $X_1 = *$.

One of Husseini's main ideas was the observation that the monoid product $\mu: J(X)\times J(X)\to J(X)$ carries products of open cells in $J(X)\times J(X)$ homeomorphically to open cells in $J(X)$. He then defined and studied CW complexes which were topological monoids satisfying this reduced product type cell product property, calling such objects complexes of reduced product type (RPT complexes for short).

This theory never caught on: a search of MathSciNet reveals exactly one paper that cites Husseini's work on RPT complexes (apart from papers by his students expanding on the theory). One may speculate about the reasons for this, but certainly one problem is that Husseini's ideas evolved rapidly from paper to paper and there is no single complete and coherent account of all his ideas; another culprit is the `old-fashioned' approach that he took, with plenty of point-set topology and very little category theory.

In the course of a completely different project I found myself facing difficult questions about loop spaces and homotopy fibers, and I thought that a simple adaptation of Husseini's theory might hold the answers. As it turned out, the adaptation has been far from simple: to carry it out, I have had to completely redevelop the theory from the ground up. In this talk, I'll describe these redeveloped foundations and some of the main theorems that build on it.