The classical story for morphisms between systems as given in myers_Categorical_2023 requires a certain square to commute strictly. However, in the case of non-deterministic systems, which use morphims in the Kleisli category for the powerset monad, it makes sense to ask this square to be filled by a 2-cell.

Specifically, a closed nondeterministic discrete system is a set

However, the Kleisli category of the powerset monad is in fact a 2-category (or a poset-enriched category), because we can relate

How can we allow this in categorical systems theory?

Essentially, the idea is to generalize Definition 4.1.0.1 from myers_Categorical_2023 (see also Matteo's Categorical Systems Theory for more general context) in the following way. Instead of starting with an indexed category

Then, we build the double category of lenses and charts in the following way. The horizontal and vertical morphisms (i.e. the lenses and the charts) are precisely the horizontal and vertical morphisms that we would get if we postcomposed

For example, consider the following indexed 2-category.

Let

Then

We could also do the same thing for the non-empty powerset monad

Now, build a double category category of lenses and charts for the indexed 2-category for the non-empty powerset monad as described above. There is then a corresponding systems theory on this double category, built via the section

This systems theory allows us to have more morphisms between systems than we previously could. For instance, suppose that we have a closed system

On the other hand, a *lax* morphism (i.e. morphism in the systems theory for the indexed 2-category)

Systems are ubiquitous, in science as in life. People regularly deal with physical systems, political systems, economical systems, living systems, learning systems, writing systems, voting systems, computing systems, etc. As we zoom into a thing, we inevitably realize it is comprised of smaller interacting parts. As we zoom out, we realize it is itself part of an even more complex system.

Given the staggering variety systems come in, it is no surprise the existing scientific and mathematical frameworks to describe them are manifold and often incompatible with one another.
Studying each of these frameworks by themselves is surely a useful endeavour, but that doesn't mean there is nothing to learn from a generalist approach which aims to distill the common motifs underpinning each paradigm.
Hence in approaching the study of systems theory, the category theorist (a label that indicates a philosophy more than a subject of study) asks: **what are the formal structures underlying all the different approaches to systems?**

An important consequence of this approach is *removing opacity*.
In fact, each framework created to deal with systems makes certain assumptions regarding the structure of these systems, the way they compose and the way they relate, as well as what behaviour they are interested in and what even means to display a certain behaviour.
It's easy to get lost in these questions, and to miss important insight because of Blindness to Structure.
So an important contribution of categorical systems theory is to clarify, within each individual framework, what are the choices that have been made.

Categorical Systems Theory has an old tradition, probably starting with Rosen's 1958 paper. It lived in a pre-paradigmatic phase for a long time, until David Jaz Myers tied up some loose ends and produced an organized theory, expounded in his book Categorical Systems Theory.

In Greek, the word `system' means `composite'. Categorical systems theory takes this etymology very seriously, approaching the study of systems as the study of `things that compose'. Thankfully, the mathematical theory of composition is rich and has been abundantly studied before, under the guise of **operads**. The idea that systems are algebras of operads is more than a decade old now, being first proposed by Spivak in his 2013 paper.

This is the *algebraic* aspect of systems theory: it concerns the way systems are put together by operations (incidentally, also the word *algebra* is etymologically related to composition)!
There is also a *geometric* aspect to systems theory, if we might abuse the algebro-geometric duality.
Systems are objects with an internal structure, which can be probed by morphisms which compare systems to each other.
Having this extra geometric structure is quite important, albeit often overlooked. It is not overlooked in coalgebraic automata theory, where the algebraic aspect is neglected but the idea that systems shall be objects of a category is taken in great consideration.

The narratology of categorical systems theory can be organized in three levels of decreasing abstraction. It's easier to start from the topmost level:

- What kind of
*comparisons*between systems we want to ponder? - What kind of
*compositions*of systems we want to ponder?

The most familiar paradigms in applied category theory are the following:

One could conceive other paradigms. For instance, one might want to compare systems by quantifying their similarity with a number, a cohomology class, or some other extensive measurement. One could compose them in different ways, for instance by glueing them instead of wiring them.

Mathematically, the answers to the questions posed by a choice of paradigm correspond to the following:

From a paradigm, we can build a 2-category of theories, whose objects are theories of systems and whose maps are lax maps thereof.

However, the concept of theory at the minute is underspecified. Most times we describe a theory we are actually giving a description of class of theories all parametrized by some common data (e.g. a category with pullbacks, a category together with a monad, etc.). So a theory is often just some data we can use to get an operad and an algebra in a specified way. Informally, one defines a doctrine as follows (this one is straight from David's book Categorical Systems Theory):

- What does it mean to be a system? Does it have a notion of states, or of behaviors? Or is it a diagram describing the way some primitive parts are organized?
- What should the interface of a system be?
- How can interfaces be connected in composition patterns?
- How are systems composed through composition patterns between their interfaces?
- What is a map between systems, and how does it affect their interfaces?
- When can maps between systems be composed along the same composition patterns as the systems?

Thus a doctrine is a *uniform*, meaning *functorial*, *way of building theories*:

A list of examples.

The doctrine