Let be a connected pointed space. The spaces are very large and complicated, but a fairly concrete model for their homotopy type can be given using the theory of operads. As a toy example, let’s take . In this case, the **James construction** gives a homotopy equivalence

where is the free topological monoid on , subject to the relation that the basepoint be the identity. In other words, one can describe by taking the disjoint union (the free topological monoid on ) and making identifications to make be the identity. The space comes with a canonical filtration

where denotes elements of which can be expressed as products of elements of . The “associated graded” quotients of are given by the smash powers .

More generally, let be the little -cubes operad; then comes with a canonical action on any -fold loop space. For a connected space , it is a theorem of May (related to delooping machinery) that is homotopy equivalent to the free -algebra on , again subject to the condition that the base-point be the unit element. In other words, we can model via the space

where the relation is precisely the one that makes the identity element. This space, too, has a filtration via the subspaces

where consists of the image of the first pieces of the disjoint union above. The associated graded pieces are given by the quotients

in other words, we take the coinvariants of in the category of pointed spaces (rather than in spaces). It turns out that this filtration actually becomes much nicer stably.

Theorem 1 (Snaith)The above filtration splits after taking suspension spectra. That is, we have an equivalence