What space does "Loop infinity of the infinite dimensional sphere" refer to? I've been reading Hatcher's survey "A Short Exposition of the Madsen-Weiss Theorem".  In it, he discusses the Barratt-Priddy-Quillen theorem, which says that the homology of the infinite symmetric group is the same as the homology of one component of $\Omega^{\infty} S^{\infty}$.  My question : what space exactly is being referred to in this notation?  It would appear that he intends to take the limit of $\Omega^n S^n$ as $n$ goes to infinity; however, I don't see a natural inclusion $\Omega^n S^n \subset \Omega^{n+1} S^{n+1}$, so I don't know how to make sense of this.  Another possible interpretation would be to take the limit of $\Omega^n S^{\infty}$ as $n$ goes to infinity, but again I don't know a natural inclusion $\Omega^n S^{\infty} \subset \Omega^{n+1} S^{\infty}$.
 A: Your first statement is correct: it is taking a colimit of $\Omega^n S^n$.  The inclusion from one to the next is called the "suspension" map.  The points of $\Omega^n S^n$ are basepoint-preserving functions $S^n \to S^n$, and the suspension map takes such a function $f$ and sends it to $f \wedge id: S^n \wedge S^1 \to S^n \wedge S^1$.
This is easier to describe in terms of another model.  $S^n$ is homeomorphic to the space obtained by taking $[0,1]^n$ and identifying the boundary to a single point.  Under this identification, you could describe a function $S^n \to S^n$ using coordinates as a function
$$(x_1,\ldots,x_n) \mapsto f(x_1,\ldots,x_n),$$
and the suspension map sends this to the function
$$(x_1,\ldots,x_n,x_{n+1}) \mapsto (f(x_1,\ldots,x_n), x_{n+1})$$
Of course, you have to check that this preserves the equivalence relation if $f$ does and so on.
Most models of $S^\infty$ are contractible, and hence so are $\Omega^n S^\infty$; these are definitely not what you want.  The notation is not to be taken literally; at best, $S^\infty$ should denote an "infinite suspension" operator which doesn't actually produce spaces as output.
