The Homotopy Hypothesis asserts that topological n-types and n-groupoids have equivalent homotopy categories. We consider the stable analog of this hypothesis, comparing stable n-types and group-like symmetric monoidal n-groupoids. In this talk we will concentrate on the cases n=1 and 2. In particular, we will establish the hypothesis in both cases, and outline some aspects of the proof for n=2. We will also indicate how to transfer homotopical information between the topological and categorical contexts. This is based on joint work with Nick Gurski and Niles Johnson.