An extension of Brown functor to cospans of spaces

Let $mathcal{A}$ be a small abelian category. The purpose of this paper is to introduce and study a category $overline{mathcal{A}}$ which implicitly appears in construction of some TQFTs where $overline{mathcal{A}}$ is determined by $mathcal{A}$. If $mathcal{A}$ 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 $overline{mathcal{A}}$ up to a scaling. In this paper, we go further by giving a sufficient condition for an $mathcal{A}$-valued Brown functor to extend to a homotopy-theoretic analogue of $overline{mathcal{A}}$-valued TQFT for arbitrary $mathcal{A}$. The results of this paper and our subsequent paper reproduces TQFTs obtained by DW theory and TV theory.