This talk will build on Nitu$^\prime$'s talk to give some of the homotopy theory used in the Madsen-Weiss paper about the cohomology of the stable mapping class group $\Gamma_{\infty, 1+1}$. I will tell you about Quillen's plus construction and about operads as a recognition principle for infinite loop spaces, as well as at least hint at how they will be used in Tillmann's proof that ${\bf Z}\times{B}\Gamma_{\infty,1+1}^+$ is an infinite loop space.