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Let A be a small abelian category. The purpose of this paper is to introduce and study a category A which implicitly appears in construction of some TQFTs where A is determined by A. If A is the category of abelian groups, then the TQFTs obtained by Dijkgraaf-Witten theory of abelian groups or Turaev-Viro theory of bicommutative Hopf algebras factor through A up to a scaling. In this paper, we go further by giving a sufficient condition for an A-valued Brown functor to extend to aarXiv:2005.10621v1 fatcat:6skooobzangjlkok5dxgkqvh5i