On the Construction of Pullbacks for Safe Petri Nets

The product of safe Petri nets is a well known operation : it generalizes to concurrent systems the usual synchronous product of automata. In this paper, we consider a more general way of combining nets, called a pullback. The pullback operation generalizes the product to nets which interact both by synchronized transitions and/or by a shared sub-net (i.e. shared places and transitions). To obtain all pullbacks, we actually show that all equalizers can be defined in the category of safe nets. Combined to the known existence of products in this category, this gives more than what we need : we actually obtain that all small limits exist, i.e. that safe nets form a complete category.