  In this paper, we propose a new approach towards the classification of
spherical fusion categories by their Frobenius-Schur exponents. We classify
spherical fusion categories of Frobenius-Schur exponent 2 up to monoidal
equivalence. We also classify modular categories of Frobenius-Schur exponent 2
up to braided monoidal equivalence. It turns out that the Gauss sum is a
complete invariant for modular categories of Frobenius-Schur exponent 2. This
result can be viewed as a categorical analog of Arf's theorem on the
classification of non-degenerate quadratic forms over fields of characteristic
2.
