Problem

Solution(maybe)

(Non)compositionalityin

categorical systems theory

Jules Hedges1

1Max Planck Institute for Mathematics in the Sciences, Leipzig

ACT 2020, 6th July 2020, via quarantine

Problem

Solution(maybe)

Categorical systems theory

Consider a category where:

• Morphisms are open systems• Objects are boundaries

left boundary xf−→ y right boundary

N.b. Why a category?

• i.e. why a strict separation into left and right boundaries?We don’t need a category! Other operad algebras work too!Categories are just convenient!

Problem

Solution(maybe)

Complex systems

The composition fg is coupling along a common boundary, andyields a complex system, i.e. a system that is a complex ofsmaller parts 1

Often C also admits a monoidal structure for disjoint(non-coupling) composition

1Not to be confused with a complicated system

Problem

Solution(maybe)

Systems vs. processes

Side remark:As well as systems theories we also have process theories

Most of this talk still applies, replace “left boundary” and“right boundary” with input and output

Problem

Solution(maybe)

Behaviour

To each boundary x we associate a set B(x) of possiblebehaviours that can be observed on that boundary

To each open system f : x → y we associate a setB(f ) ⊆ B(x)× B(y)• (a, b) ∈ B(f ) means “it is possible to simultaneously

observe a on the left boundary and b on the rightboundary of f ”

No observations can be made except on the boundary

Problem

Solution(maybe)

Behaviours compose

Suppose f and g share a common behaviour on their commonboundary:

(a, b) ∈ B(f ) and (b, c) ∈ B(g) for some b ∈ B(y)

In most situations, this implies that (a, c) ∈ B(fg) 2So:

B(f )B(g) ⊆ B(fg)

(LHS composition in Rel)

2If your behaviour doesn’t satisfy this, you should probably trysomething else

Problem

Solution(maybe)

In fancy terminology:

B is a lax functor3 C → Rel• C is locally discrete 2-category (exactly one 2-cell)• Rel is a locally thin 2-category (at most one 2-cell)

3Or lax pseudofunctor if being pedantic

Problem

Solution(maybe)

Emergent behaviours

In many practical situations, the converse fails:

fg can exhibit “emergent” behaviours that do not arise fromindividual behaviours of f and g

Definition. An emergent behaviour of fg (w.r.t. thedecomposition (f , g)) is an element of B(fg) \ B(f )B(g)

So we do not have a functor B : C → Rel

In practice: This is much less interestingFunctoriality sometimes fails very badly in real examples

Problem

Solution(maybe)

Grand challenge for ACT

Given a lax functor to Rel 4 associate some mathematicalobject (cohomology?) that ‘describes’ how it fails to be afunctor (i.e. how the laxator fails to be an iso), in a useful way

“Useful” = encodes something worth knowing about emergentbehaviour

4Or linear/additive relations etc. as convenient

Problem

Solution(maybe)

Example: Open graph reachability5

OGph = structured cospan category of open graphs

• Objects: finite sets• Morphisms: cospans of graph homomorphisms

L(x)ι1−→ g ι2←− L(y)

L(−) = discrete graph on a set

5Probably not the best example, but the easiest to draw pictures of

Problem

Solution(maybe)

Reachability

Define B : OGph→ Rel by:• On objects: B(x) = x• On morphisms: B(x ι1−→ g ι2←− y) =

{(a, b) ∈ (x , y) | ι1(a) and ι2(b) are connected in g}

Proposition. B is a lax functor

Problem

Solution(maybe)

Reachability is not a functor

Minimal counterexample: the zig-zag

B(f )B(g) = ∅ ( {(∗, ∗)} = B(fg)

Problem

Solution(maybe)

ComorphismsIdea: It doesn’t make sense to ask what is the behaviour of anopen system in isolation - only in context

Call a possible context for a morphism x → y a comorphism

Write C(x , y) for the set of comorphisms x → y

Draw them as inside-out diagram elements6

6Or alternatively as combs

Problem

Solution(maybe)

C is a functor C × Cop → Set

Notation: Write C(f , id)(c) = f∗c and C(id, f )(c) = f ∗c(so C(f , g)(c) = f∗g∗c = g∗f∗c)

Problem

Solution(maybe)

Contexts in a monoidal category

/x1,x2,y1,y2 : C(x1 ⊗ x2, y1 ⊗ y2)× C(x1, y1)→ C(x2, y2)\x1,x2,y1,y2 : C(x1 ⊗ x2, y1 ⊗ y2)× C(x2, y2)→ C(x1, y1)

Problem

Solution(maybe)

General purpose contexts 1

For C a monoidal category,

C(x , y) =∫ a∈C

C(I , a⊗ x)× C(a⊗ y , I )

aka. a state in the category of optics, C(x , y) ∼= OptC(I ,(xy

))

Problem

Solution(maybe)

General purpose contexts 2

For C a traced monoidal category, C(x , y) = C(y , x)

For C a compact closed category, C(x , y) = C(I , x ⊗ y)

Problem

Solution(maybe)

Augmented semantic category

Fix a semantic category D (e.g. D = Rel)and a mapping on objects B : Ob(C)→ Ob(D)

Definition. Define a category Ĉ by:• Ob(Ĉ) = Ob(C)• Ĉ(x , y) = C(x , y)×

(C(x , y)→ D(B(x),B(y))

)i.e. morphisms are pairs of

1 A system

2 Its behaviour in every context

• Identity: (idx , îdx) where îdx(c) = idB(x) for allc ∈ C(x , x)• Composition: (f , f̂ )(g , ĝ) = (fg , f̂g) wheref̂g(c) = f̂ (g∗c)ĝ(f∗c)

Problem

Solution(maybe)

Ĉ as a semantic category

There is forgetful functor Ĉ → C 7

We would like to find sections of itand view Ĉ as our semantic category instead of D

Highly dubious central claim: This is workable framework forfunctorial semantics, in settings with emergent effects

N.b. This is the essence of how open games work

7Unproven guess: Some reasonable extra hypotheses on C make it abifibration

Problem

Solution(maybe)

Augmented semantic functors

Proposition. Let Bx ,y : C(x , y)× C(x , y)→ D(B(x),B(y)) bea family of functions satisfying

• Bx ,x(idx , c) = idB(x) for all c ∈ C(x , x)• Bx ,z(fg , c) = Bx ,y (f , g∗c)By ,z(g , f∗c)

This induces a section B : C → Ĉ by f 7→ (f ,B(f ,−)).

Problem

Solution(maybe)

Example: Context for reachability

We could take OGph to be e.g. the general purpose contextfor compact closed categories

Instead let’s try something ad-hoc tailored to reachability

Let OGph(x , y) to be the set of open graphs x → y whose setof nodes is exactly x + y

Idea: Edges in c ∈ OGph(x , y) represent reachability in the“real” context

Problem

Solution(maybe)

Reachability in context

Problem

Solution(maybe)

Reachability in context

ProblemSolution (maybe)