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The Dowker complex is an abstract simplicial complex that is constructed froma binary relation in a straightforward way. Although there are two ways toperform this construction -- vertices for the complex are either the rows orthe columns of the matrix representing the relation -- the two constructionsare homotopy equivalent. This article shows that the construction of a Dowkercomplex from a relation is a non-faithful covariant functor. Furthermore, weshow that this functor can be made faithful by enriching the construction intoa cosheaf on the Dowker complex. The cosheaf can be summarized by an integerweight function on the Dowker complex that is a complete isomorphism invariantfor the relation. The cosheaf representation of a relation actually embodiesboth Dowker complexes, and we construct a duality functor that exchanges thetwo complexes. Finally, we explore a different cosheaf that detects the failureof the Dowker complex itself to be a faithful functor.

Cosheaf Representations of Relations and Dowker Complexes