Dowker Duality for Relations of Categories

Abstract

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 $K$.