We define a semi-infinite analogue of the Weil algebra associated an infinite-dimensional Lie algebra. It can be used for the definition of semi-infinite characteristic classes by analogy with the Chern-Weil construction. The second term of a spectral sequence of this Weil complex consists of the semi-infinite cohomology of the Lie algebra with coefficients in its “adjoint semi-infinite symmetric powers.” We compute this cohomology for the Virasoro algebra. This is just the BRST cohomology of the bosonic βγ-system with central charge 26. We give a complete description of the Fock representations of this bosonic system as modules over the virasoro algebra, using Friedan-Martinec-Shenker bosonization. We derive a combinatorial identity from this result.