Symmetric powers and the Braid group

We will discuss a method for calculating Ext between polynomial functors from groups to abelian groups. It is based on the relationship between the homotopy theories of spaces and of simplicial groups. After reviewing the general result, I will tell about joint work in progress with Minkyu Kim and Christine Vespa: We consider the functor $S^n$, that sends a group $G$ to the $n$-th symmetric power of $\mathrm{ab}(G)$. Ext into this functor is trivial rationally, but has a lot of torsion. It turns out, perhaps surprisingly, that one can express Ext into $S^n$, in terms of the homology of Braid groups.