Triangular Prism Equations and Categorification

Abstract: We present the triangular prism equations for fusion categories, demonstrating their equivalence to the pentagon equations in the spherical case, modulo a change of basis. These equations offer valuable insights for managing complexity through localization. Additionally, we establish an enhanced version of Z. Wang's conjecture regarding the second Frobenius-Schur indicator in pivotal fusion categories. As applications, we introduce new categorification criteria and complete the classification of unitary 1-Frobenius simple integral fusion categories up to rank 8 and up to FPdim 20000.