  We propose a categorification of the Dowker duality theorem for relations.
Dowker's theorem states that the Dowker complex of a relation $R \subseteq X
\times Y$ of sets $X$ and $Y$ is homotopy equivalent to the Dowker complex of
the transpose relation $R^T \subseteq Y \times X$. Given a relation $R$ of
small categories $\mathcal{C}$ and $\mathcal{D}$, that is, a functor of the
form $R \colon \mathcal{R} \to \mathcal{C} \times \mathcal{D}$, we define the
bisimplicial rectangle nerve $ER$ and the Dowker nerve $DR$. The diagonal
$d(ER)$ of the bisimplicial set $ER$ maps to the simplicial set $DR$ by a
natural projection $d(\pi_R) \colon d(ER) \to DR$.
  We introduce a criterion on relations of categories ensuring that the
projection from the diagonal of the bisimplicial rectangle nerve to the Dowker
nerve is a weak equivalence. Relations satisfying this criterion are called
Dowker relations. If both the relation $R$ of categories and its transpose
relation $R^T$ are Dowker relations, then the Dowker nerves $DR$ and $DR^T$ are
weakly equivalent simplicial sets.
  In order to justify the abstraction introduced by our categorification we
give two applications. The first application is to show that Quillen's Theorem
A can be considered as an instance of Dowker duality. In the second application
we consider a simplicial complex $K$ with vertex set $V$ and show that the
geometric realization of $K$ is naturally homotopy equivalent to the geometric
realization of the simplicial set with the set of $n$-simplices given by
functions $\{0,1,\dots,n\}\to V$ whose image is a simplex of $K$.
