  The categorical modeling of Petri nets has received much attention recently.
The Dialectica construction has also had its fair share of attention. We
revisit the use of the Dialectica construction as a categorical model for Petri
nets generalising the original application to suggest that Petri nets with
different kinds of transitions can be modelled in the same categorical
framework. Transitions representing truth-values, probabilities, rates or
multiplicities, evaluated in different algebraic structures called lineales are
useful and are modelled here in the same category. We investigate (categorical
instances of) this generalised model and its connections to more recent models
of categorical nets.
