This proof shows that every digraph functor in the homotopy category is representable, indicating potential applications in algebraic topology.
We prove that any digraph Brown functor -- i.e. a contravariant functor from the homotopy category of finite directed graphs to the category of abelian groups, satisfying the triviality axiom, the additivity axiom, and the Mayer-Vietoris axiom -- is representable. Furthermore, we show that the first path cohomology functor is a digraph Brown functor.
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Liao et al. (2025) studied this question.
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