Suppose that G is a reductive group over a local field F. Brylinski-Deligne constructed topological coverings of G(F) starting with K-theoretic data. These coverings are a generalization of Weil's metaplectic group. We will explain that the Brylinski-Deligne construction factors through a 2-groupoid having to do with the étale cohomology of BG. In the function field setting, the passage from K-theory to étale cohomology has a vivid geometric interpretation in terms of the Beilinson-Drinfeld affine Grassmannian. This is a survey of recent works by Gaitsgory, Lysenko, James Tao, and myself.