  We explain how homotopical information of two composeable relations can be
organized in two simplicial categories that augment the relations row and
column complexes. We show that both of these categories realize to weakly
equivalent spaces, thereby extending Dowker's duality theorem. We also prove a
functorial version of this result. Specializing the above construction a
bifiltration of Dowker complexes that coherently incorporates the total weights
of a relation's row and column complex into one single object is introduced.
This construction is motivated by challenges in data analysis that necessitate
the simultaneous study of a data matrix rows and columns. To illustrate the
applicability of our constructions for solving those challenges we give an
appropriate reconstruction result.
