In the theory of Abelian envelopes, a pseudo-tensor category over a field is a rigid linear monoidal category that is an additive category, is idempotent complete, and has a one-dimensional endomorphism algebra for its tensor unit. A category is idempotent complete if every endomorphism that is an idempotent splits.
The definition replaces the Abelian category requirement for a tensor category in this setting by additivity and idempotent completeness (Coulembier et al. 2023). Every tensor category in this sense is therefore a pseudo-tensor category, but the converse need not hold. This categorical notion is unrelated to the pseudotensor of tensor analysis.
