People interested in category theory as applied to systems may wish to check out the following article, reporting work I carried out while engaged in a systems engineering program at Oakland University.
The problem addressed is a longstanding one, that of building bridges to negotiate the gap between qualitative and quantitative descriptions of complex phenomena, like those we meet in analyzing and engineering systems, especially intelligent systems endowed with a capacity for processing information and acquiring knowledge of objective reality.
One of the ways this problem arises has to do with describing change in logical, qualitative, or symbolic terms, long before we grasp the reality beneath the appearances firmly enough to cast it in measured, quantitative, real number form.
Development on the quantitative shore got no further than a Sisyphean beachhead until the discovery/invention of differential calculus by Leibniz and Newton, after which things advanced by leaps and bounds.
And there’s our clue what we need to do on the qualitative shore, namely, to discover/invent the missing logical analogue of differential calculus.
With that preamble …
Differential Logic and Dynamic Systems
This article develops a differential extension of propositional calculus and applies it to a context of problems arising in dynamic systems. The work pursued here is coordinated with a parallel application that focuses on neural network systems, but the dependencies are arranged to make the present article the main and the more self-contained work, to serve as a conceptual frame and a technical background for the network project.
Resources
- Differential Propositional Calculus • Part 1 • Part 2
- Differential Logic • Part 1 • Part 2 • Part 3
- Differential Logic and Dynamic Systems
• Part 1 • Part 2 • Part 3 • Part 4 • Part 5
cc: Cybernetics (1) (2) • Ontolog • Peirce List • Structural Modeling • Systems Science
where 
is the object, 
is the sign, and 
is the interpretant sign in view.
This set is called a sign relation.
as a subset of a cartesian product, 
where 
is the set of objects under consideration in a given context, 
is the set of signs, and 
is the set of interpretant signs being considered in the same context.
Among other things, this is important because sets have properties their elements do not and it amounts to a category mistake to confuse the two levels. In particular, the properties of reducibility and irreducibility are defined at the level of whole sign relations, not their individual elements.
is probably not the best. It tends to mislead us into thinking we already have 
in hand, in other words, that we already have perfect information about 
is all the information we have at a given moment of investigation and 
That might lead us to notate it as 
This brings to mind the way we speak of observables in physics, as operators on the wave function that represents the total state of the system observed.
whenever I ran across it. I suppose that small arc of revolution had been building for years but it struck me as crossing a threshold to a more explicit, self‑conscious stage about that time.
Inquiry Into Inquiry 