An Abelian Ambient Category for Behaviors in Algebraic Systems Theory

We describe an abelian category \(\textbf{ab}(M)\) in which the solution sets of finitely many linear equations over an arbitrary ring R with values in an arbitrary left R-module M reside as objects. Such solution sets are also called behaviors in algebraic systems theory. We both characterize \(\textbf{ab}(M)\) by a universal property and give a construction of \(\textbf{ab}(M)\) as a Serre quotient of the free abelian category generated by R. We discuss features of \(\textbf{ab}(M)\) relevant in the context of algebraic systems theory: if R is left coherent and M is an fp-injective fp-cogenerator, then \(\textbf{ab}(M)\) is antiequivalent to the category of finitely presented left R-modules. This provides an alternative point of view to the important module-behavior duality in algebraic systems theory. We also obtain a dual statement: if R is right coherent and M is fp-faithfully flat, then \(\textbf{ab}(M)\) is equivalent to the category of finitely presented right R-modules. As an example application, we discuss delay-differential systems with constant coefficients and a polynomial signal space. Moreover, we propose definitions of controllability and observability in our setup.