Concrete Dualities and Essential Arities
Many dualities arise in the same way: they are induced by dualizing objects. We show that these dualities are connected to a question occurring in universal algebra. Indeed, they cause a strong interplay between the essential arity of finitary operations in one category and the concrete form of the copowers in the other. We elaborate on this connection and its usefulness for universal algebra and clone theory in particular. As the paper’s main result we show that, under some mild assumptions, the essential arity of finitary operations from an object A to a finite object B in one category is bounded if and only if the concrete form of the copowers of the dual of A has a certain (easily verifiable) set-theoretic property.
Keywords: Concrete dualities; essential variables; dualizing objects; natural dualities; finitary operations; dual operations; clones