The canonical delooping machine

We suggest a new delooping machine, which is based on recognizing an nn-fold loop space by a collection of operations acting on it, like the traditional delooping machines of James, Stasheff, May, Boardman–Vogt, Segal, and Bousfield. Unlike the traditional delooping machines, which carefully select a nice space of such operations, we consider all natural operations on nn-fold loop spaces, resulting in the algebraic theory Map∗(⋁•Sn,⋁•Sn). The advantage of this new approach is that the delooping machine is universal in a certain sense, the proof of the recognition principle is more conceptual, it works the same way for all values of nn, and it does not need the test space to be connected.