It is shown how a category of Petri nets can be viewed as a subcategory of two sorted algebras over multisets. This casts Petri nets in a familiar framework and provides a useful idea of morphism on nets different from the conventional definition – the morphisms here respect the behaviour of nets. The categorical constructions with result provide a useful way to synthesise nets and reason about nets in terms of their components; for example various forms of parallel composition of Petri nets arise naturally from the product in the category. This abstract setting makes plain a useful functor from the category of Petri nets to a category of spaces of invariants and provides insight into the generalisations of the basic definition of Petri nets – for instance the coloured and higher level nets of Kurt Jensen arise through a simple modification of the sorts of the algebras underlying nets. Further it provides a smooth formal relation with other models of concurrency such as Milner’s Calculus of Communicating Systems (CCS) and Hoare’s Communicating Sequential Processes (CSP).