  The Dowker complex is an abstract simplicial complex that is constructed from
a binary relation in a straightforward way. Although there are two ways to
perform this construction -- vertices for the complex are either the rows or
the columns of the matrix representing the relation -- the two constructions
are homotopy equivalent. This article shows that the construction of a Dowker
complex from a relation is a non-faithful covariant functor. Furthermore, we
show that this functor can be made faithful by enriching the construction into
a cosheaf on the Dowker complex. The cosheaf can be summarized by an integer
weight function on the Dowker complex that is a complete isomorphism invariant
for the relation. The cosheaf representation of a relation actually embodies
both Dowker complexes, and we construct a duality functor that exchanges the
two complexes. Finally, we explore a different cosheaf that detects the failure
of the Dowker complex itself to be a faithful functor.
