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Almost. A Petri net can be thought of as a strictly symmetric strict
monoidal category in which the monoid of objects is free. (The monoid
of arrows is determined by the transition relation in the obvious way;
read tensor as par.) In this way, [1], a petri net generates a
tensor theory with axioms given by the transition relation. In so far
as it is possible to think of models of linear logic as symmetric 
monoidal categories (and they are, perhaps, [2], best thought of
as *-autonomous categories, i.e. closed symmetric monoidal categories
with involution) there is some connection between these two ideas.
This area has been explored Meseguer and Montanari, Winskel, Brown,
Asperti, and probably others.

[1] From Petri Nets to Linear Logic, Mart\'\i-Oliet and Meseguer,
Category Theory and Computer Science 1989, Springer-Verlag LNCS 389.

[2] Linear Logic, *-autonomous categories and cofree algebras. 
Proc AMS conference on categories in computer science (ed. Gray
& Scedrov).

David Murphy,               | JANET: dvjm@uk.ac.glasgow.cs
Dept. of Computing Science, | UUCP:  ..!mcsun!ukc!uk.ac.glasgow.cs!dvjm
University of Glasgow,      | ARPA:  dvjm%cs.glasgow.ac.uk@nsfnet-relay.ac.uk
Glasgow G12 8QQ, SCOTLAND   + The Superfluous is very necessary