Local states in string diagrams

ABSTRACT. We establish that the local store monad defined by Plotkin and Power is a monad with arities for an appropriate notion of arities in the category of presheaves over the category of injections. From this follows that the monad is associated to a graded Lawvere theory. We explain how to present this graded theory by generators and relations, which we then depict in the graphical language of string diagrams.