4:21 AM
@skd e.g. mahowald used this kind of construction a lot. a typical application might be to relate certain algebraic power operations, in Ext over the Steenrod algebra, to geometric power operations that arise from this quadratic construction S^1 ^_{Sigma_2} (X^X). often the goal is to deduce the existence of new homotopy elements or new differentials from old
these kinds of constructions play a role in his work on showing that if one Kervaire invariant element exists and satisfies certain properties, then the next Kervaire invariant element exists
these kinds of extended powers also show up a lot in proofs of nilpotence theorems, like the Nishida nilpotence theorem, which @JeremyHahn would explain much better than I
i'm not sure the universal Q_1-algebra is homologically equivalent to this 2-fold loop space, though. the Q_1-algebra structure doesn't seem to force the homology product to be associative, or the Adem relations to hold, or the bracket to do ...
i'm at a loss for words. the bracket is always awful. it can be 2^{awful}?
anyway. these kinds of extended powers are also used by people like bruner, or turner & hackney, to construct operations in spectral sequences &c &c. the operad is just a convenient encoding of where they come from
