In this paper, we introduce the notion of the pivotal cover Cpiv of a left rigid monoidal category C to develop a theoretical foundation for the theory of Frobenius-Schur (FS) indicators in "non-pivotal" settings. For an object V∈Cpiv, the (n, r)-th FS indicator νn,r(V) is defined by generalizing that of an object of a pivotal monoidal category. This notion gives a categorical viewpoint to some recent results on generalizations of FS indicators.Based on our framework, we also study the FS indicators of the "adjoint object" in a finite tensor category, which can be considered as a generalization of the adjoint representation of a Hopf algebra. The indicators of this object closely relate to the space of endomorphisms of the iterated tensor product functor.
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