Let V be a symplectic vector space, so that the symplectic group Sp(V) acts on V. If L is a Lagrangian subspace of V, and k is a commutative ring, then the k-module Map(L, k) is a “geometric quantization” of V. However, it does not admit an action of Sp(V); but there is a central extension of Sp(V) by the group k* of units in k, called the metaplectic group, which *does* act on Map(L, k). This is called the Weil representation, and it plays an important role in many areas of mathematics. I will explain a categorification of the Weil representation, where the metaplectic group is replaced by an extension of Sp(V) by the classifying space Bk*. I will also allow k to be a ring spectrum, and explain how the cocycle defining the extension of Sp(V) by Bk* is computed by the J-homomorphism. This is motivated by 1) applications to Langlands duality with coefficients in ring spectra (cf my thesis); 2) work of Teleman and Braverman-Dhillon-Finkelberg-Raskin-Travkin; and 3) Maslov data in Floer homotopy theory; I hope to explain these relationships in the talk.