Duality and tangles of set separations

Abstract: Applications of tangles of connectivity systems suggest a duality between these, in which for two sets $X$ and $Y!$ the elements $x$ of $X$ map to subsets $Y_x$ of $Y!$, and the elements $y$ of $Y!$ map to subsets $X_y$ of $X$, so that $x\in X_y$ if and only if $y\in Y_x$ for all $x\in X$ and $y\in Y!$. We explore this duality, and relate the tangles arising from the dual systems to each other.