Debiasing word embeddings has been largely limited to individual and independent social categories. However, real-world corpora typically present multiple social categories that possibly correlate or intersect with each other. For instance, “hair weaves” is stereotypically associated with African American females, but neither African American nor females alone. Therefore, this work studies biases associated with multiple social categories: joint biases induced by the union of different categories and intersectional biases that do not overlap with the biases of the constituent categories. We first empirically observe that individual biases intersect non-trivially (i.e., over a one-dimensional subspace). Drawing from the intersectional theory in social science and the linguistic theory, we then construct an intersectional subspace to debias for multiple social categories using the nonlinear geometry of individual biases. Empirical evaluations corroborate the efficacy of our approach.