An abelian ambient category for behaviors in algebraic systems theory

Upon Owen’s request, I will post an email I sent a couple of days ago to colleagues I met at Topos:

This is the first time I am advertising a paper, however, I believe the following paper by Sebastian Posur is worth an exception, especially since I now know that some of you are interested in dynamical systems and systems theory.

His paper

An abelian ambient category for behaviors in algebraic systems theory

establishes both the so-called module-behavior-duality and the controllability-observability-duality in systems theory in a previously unreached level of generality.

The idea is strikingly simple and extremely elegant:

(0) It starts by observing that an R-module S is equally encoded by a functor S:R → Ab, where the ring R is now viewed as a preadditive category on one object, call it •. The R-module S is typically called the signal space in systems theory.

(1) This functor lifts by the universal property of the Abelian closure Abel(R) (which I mentioned in my talk at Topos) to an exact functor Eval_S: Abel(R) → Ab, which he calls the evaluation functor for reasons explained in the paper.

(2) The kernel of the evaluation functor is a Serre subcategory of Abel(R) and the Serre quotient category ab(S) := Abel(R) / ker(Eval_S) is the correct category of abstract behaviors, or "ambient category for behaviors”.

This concludes the construction. Furthermore:

(a) The exact functor Eval_S factors over the faithful exact functor F: ab(S) → Ab. The latter can be interpreted as a forgetful functor mapping an abstract behavior in ab(S) to its underlying Abelian group in Ab.

(b) A system module N = coker(A) ∈ R-mod is a finitely presented R-module on n generators. The R-matrix A with n columns is called its presentation matrix and viewed as a morphism between free R-modules of finite rank.

(c) The Abelian group Hom(N,S) = \{x \in S^n | Ax = 0\} ∈ Ab of solutions over S (= S-trajectories) can now be interpreted as an object in the highly more structured subcategory ab(S) ⊂ Ab of behaviors. In other words, Hom(-,S): R-mod → Ab factors over Hom(-,S): R-mod → ab(S). Post-composing with F yields back the classical, less structured behavior functor Hom(-,S): R-mod → Ab.

(d) So a typical object in ab(S) is the space Hom(N,S) = \{x \in S^n | Ax = 0\} of solutions of a system module N. When an object in ab(S) is not a space of solutions it is then an “abstract” behavior, justifying the name of ab(S).

(e) The single object • of R is mapped by the composition R ↪ Abel( R ) → ab(S) to the signal space S, viewed as the “unrestricted behavior” in ab(S). It is the space of S-trajectories of the free (system) module R of rank 1, with presentation matrix being the 1x1 (or 0x1) zero matrix.

Some remarks:

  • The ring R is only assumed unital. The above construction requires no further restrictions.

  • Abel(R) is called R-mod-mod in the paper: The category of finitely presented Ab-valued functors from the category of finitely presented modules.

  • The most popular example of a signal space is the R-module S = C^∞(ℝⁿ,ℝ) of real-valued smooth functions for the commutative polynomial ring R = ℝ[∂₁, …, ∂ₙ] of partial differential operators. This algebraic setup describes linear PDE-systems with constant coefficients. In this setup, the Serre quotient ab(S) is equivalent to a module category. A generalization to an arbitrary unital ring R and an arbitrary signal space S was missing. The category ab(S) of abstract behaviors is what experts in the field (Willems, Oberst, Wood, …) were only able to construct (up to equivalence) for special rings or special signal spaces. In retrospect, the obstacle that needed to be overcome was to give up on (finitely presented) modules and adopt finitely presented functors.

  • Even though S is typically far from being finitely generated, the Abelian category ab(S) of behaviors is (as a Serre quotient of Abel(R)) computable under reasonable conditions on R and S. In even more favorable situations the exact localization functor Abel(R) → ab(S) has a right adjoint, i.e., ab(S) is even a reflective subcategory of Abel(R).

  • Section 8 describes the duality between controllability and observability along these lines. Observability generally only makes sense for morphisms between behaviors.

The minor criticism I have regarding the paper is that it departs from what I believe is the notational convention used in algebraic systems theory. The finitely presented system module is typically denoted by M and the signal space is typically denoted by F or \mathcal{F}. Sebastian denotes the signal space by M, maybe since the letter F is reserved for the forgetful functor. As above, he denotes a finitely presented system module by cok(A). In the above summary, I thus use S for what he calls M.