2179ocl-001Vocl-001V.xmlGrothendieck lenses for functors into 2Cat2024413Owen LynchThe 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 with a function , where is the powerset monad. A morphism from to is a function such that the following commutes
However, the Kleisli category of the powerset monad is in fact a 2-category (or a poset-enriched category), because we can relate via if for all , . If we write the above square as a square in the Kleisli category of powerset, it looks like
and we can ask that rather than commuting, we have a filler 2-cell given by
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 , start with an indexed 2-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 with the forgetful functor . However, a 2-cell filling the outer boundary
consists of the assertion that
commutes (we call the composite ), and a 2-morphism
=6pt, Rightarrow, from=0, to=1]
\end {tikzcd}
]]>For example, consider the following indexed 2-category.2187ocl-001Wocl-001W.xmlThe two-categorical nondeterministic systems theoryDefinition2024413Owen LynchLet and consider the indexed category defined in the following way. is the biKleisli category (see Kleisli category, Proposition 2.7) of the powerset monad and the comonad . Said more concretely, let be the category where the objects are sets, and a morphism from to is a function .Then has a natural poset enrichment, where iff for all , .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 defined by .This systems theory allows us to have more morphisms between systems than we previously could. For instance, suppose that we have a closed system
Then for a subset , the existence of a strict morphism (i.e. morphism in the systems theory for the indexed 1-category)
implies that all possible paths according to the dynamics of that start in must stay in for all time. This is because the update function on must send a state to the same subset of that the update function on sends it to, which is only possible if this subset is a subset of .On the other hand, a lax morphism (i.e. morphism in the systems theory for the indexed 2-category)
just implies that for each state in , it is possible to stay in in the future. This is because we have more freedom in our choice of dynamics for , we only have to send to a subset of the possible states that the update function for would send it to. Because we chose the non-empty powerset monad, this subset must be non-empty, so the existence of this morphism implies that there is always a possibility to stay in when we update, assuming that we start in .Related2193st-0001st-0001.xmlCategorical Systems TheoryMatteo Capucci
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.
2195st-historyst-history.xmlHistory
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.
2196st-0002st-0002.xmlA quick tour of the ideas
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.
2197#952unstable-952.xmlRemarkst-0002
The word operad is quite overloaded, and, in some sense, not overloaded enough. An operad, traditionally, is a structure encoding formal operations of arbitrary finite arity which compose associatively and have a unit. In fact, the idea can be easily generalized much further, by having "arities" being structured objects.
In this generalized form, operads are usually called multicategories, but I'd like to keep calling them operads because (a) morally, they still are and (b) operad is a much nicer and less scary word than multicategory.
This translates to their even-more-generalized form, -multicategories, which I call -operads.
2199#953unstable-953.xmlIdea.st-0002
Operads are theories of composition.
Theories of systems should be algebras of theories of compositions.
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.
2200#954unstable-954.xmlIdea.st-0002
Theories of systems should be algebras of double operads, i.e. operads in categories.
2201#955unstable-955.xmlRemarkst-0002
This 'definition' is preemptively general.
While the ideal, for both me and David, is to eventually work in terms of general -operads (that being the 'morally right' setting), at the minute most of categorical systems theory is done for , where is a made-up notation for the free symmetric monoidal category 2-monad on (Example 4.1.16 in Higher operads, higher categories).
Concretely, this means that a double -operad is a symmetric monoidal double category.
2202st-0003st-0003.xmlTheories, Doctrines, ParadigmsMatteo Capucci
The narratology of categorical systems theory can be organized in three levels of decreasing abstraction.
It's easier to start from the topmost level:
2204#940unstable-940.xmlParadigmPreliminary definitionMatteo Capuccist-0003
A paradigm of systems theory is a way to answer the following questions:
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:
2206#941unstable-941.xmlParadigm of setsExampleMatteo Capuccist-0003
In this paradigm, systems are organized in sets, thus can only be compared for equality.
Composition is described by symmetric operads, thus basically symmetric monoidal categories.
This is a fairly common paradigm in the literature, e.g. Spivak's paper on wiring diagrams can be considered to work in the paradigm of sets.
2208#942unstable-942.xmlParadigm of categoriesExampleMatteo Capuccist-0003
In this paradigm, systems are organized in categories, thus can be compared with morphisms
Composition is described by symmetric double operads, thus basically symmetric monoidal double categories.
This is the default paradigm in categorical systems theory.
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:
2210#943unstable-943.xmlParadigmDefinitionMatteo Capuccist-0003
A paradigm is an equipment along with a monad , i.e. a way to define what 'operad' and 'algebra' mean.
From a paradigm, we can build a 2-category of theories, whose objects are theories of systems and whose maps are lax maps thereof.
2212#944unstable-944.xml2-Category of theoriesDefinitionMatteo Capuccist-0003
Let be a paradigm.
The associated 2-category of theories is the 2-category of -operads and right algebras thereof, with lax maps and 2-cells.
Objects are thus pairs where is a -operad and a right algebra thereof.
2214#945unstable-945.xmlTheoryPreliminary definitionMatteo Capuccist-0003
A theory for a paradigm is an object of .
2216#946unstable-946.xmlTheories in the set paradigmExampleMatteo Capuccist-0003
The 2-category of theories for the paradigm is the 2-category whose objects are pairs were the first is a symmetric monoidal category and the latter is a symmetric monoidal copresheaf .
A map of theories is given by a symmetric lax monoidal functor between the base categories and a natural transformation.
2218#947unstable-947.xmlTheories in the categories paradigmExampleMatteo Capuccist-0003
The 2-category of theories for the paradigm is the 2-category whose objects are pairs were the first is a symmetric monoidal double category and the latter is a symmetric monoidal lax copresheaf , also known as a doubly indexed category.
A map of theories is given by a symmetric lax monoidal lax double functor between the base double categories and a lax natural transformation.
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):
2220#948unstable-948.xmlDoctrinePreliminary definitionMatteo Capuccist-0003
A doctrine of systems is a particular way to answer the following questions about it means to be a 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:
2222#949unstable-949.xmlDoctrineDefinitionMatteo Capuccist-0003
A doctrine in the paradigm is a 2-functor
The objects of are called theories for the doctrine .
2224#950unstable-950.xmlRemarkMatteo Capuccist-0003
The reason we already called the 2-category of theories is easily seen: clearly the identity functor of is a doctrine, and in fact the 'universal one', since it is terminal among doctrines over .
Thus all right algebras for -operads in are theories for the universal doctrine for the paradigm .
The definitive definition of theory mentions directly the doctrine:
2226#951unstable-951.xmlTheoryDefinitionMatteo Capuccist-0003
A theory for a doctrine is an object in .
2228st-examplesst-examples.xmlA zoo of theories of systems
A list of examples.
2229st-ex-0001st-ex-0001.xmlFully observable open dynamical systemsMatteo Capucci
The doctrine is parametrized by cartesian categories.
Given , the theory has as its theory of compositions the cartesian double category of lenses whose forward part is an identity.
Its indexing part is the doubly indexed functor sending to the discrete category of maps
which are functorially acted upon by forward-trivial lenses as follows:
Indexing by charts is works as usual, sending a chart to the discrete profunctor