By a theorem of Majid, every monoidal category with a neutral quasi-monoidal functor to finitely generated and projective \(\Bbbk \)-modules gives rise to a coquasi-bialgebra. We prove that if the category is also rigid, then the associated coquasi-bialgebra admits a preantipode, providing in this way an analogue for coquasi-bialgebras of Ulbrich’s reconstruction theorem for Hopf algebras. When \(\Bbbk \) is a field, this allows us to characterize coquasi-Hopf algebras as well in terms of rigidity of finite-dimensional corepresentations.

中文翻译：

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前准肢和刚性单曲面类的准准双代数

根据马吉德定理，每个具有中立拟单调函子到有限生成的射影\（\ Bbbk \） -模的单项曲面都产生了一个准双代数。我们证明，如果类别也是刚性的，则相关的准双代数接纳一个前对映体，从而以这种方式为Ulbrich霍普夫代数重构定理的准双代数提供一个类似物。当\（\ Bbbk \）是一个字段时，这使我们能够根据有限维核心表示的刚度来刻画拟霍夫代数。