Structured and decorated cospans from the viewpoint of double category theory
Applied Category Theory 2023
2023-08-04
Open systems as cospans
% https://tex.stackexchange.com/a/269194
Mathematical modeling of open systems as cospans goes back 25+ years.
a→x←b
- Apex x of cospan is the system itself
- Feet a,b of cospan are boundary or interface of system
Usually the system itself is “more structured” than its boundary:
- restrict the feet relative to the apex ⇝ structured cospans
- extend the apex with extra data ⇝ decorated cospans
Some early references: (Katis, Sabadini, and Walters 1997), (Rosebrugh, Sabadini, and Walters 2005)
Evolution of open systems as cospans (1)
Open systems should form a categorical structure, most obviously a category:
Example: Category of cospans
Given a category S with pushouts, there is a category Csp(S) having
- as objects, the objects of S;
- as morphisms, isomorphism classes of cospans in S.
Cospans compose by pushout in S.
Problems:
- In software, we work with representatives of isomorphism classes
- What happened to the morphisms between systems?
Evolution of open systems as cospans (2)
Both problems are solved by moving to a 2-dimensional categorical structure, traditionally a bicategory:
Example: Bicategory of cospans
Given a category S with pushouts, there is bicategory Csp(S) having
- as objects, the objects of S;
- as morphisms a→b, cospans a→x←b in S;
- as 2-morphisms (a→x←b)⇒(a→y←b), maps h:x→y in S making the diagram commute:
Evolution of open systems as cospans (3)
New problem: We need a monoidal structure for “parallel composition.”
Monoidal categories are straightforward enough to work with
Monoidal bicategories are much more complicated, perhaps “irreducibly” three-dimensional:
monoidal bicategory = tricategory with one object
However, monoidal double categories can be defined using only 2-categorical concepts:
monoidal double category = pseudomonoid in Dbl
Moreover, monoidal bicategories can sometimes be obtained from monoidal double categories (Shulman 2010)
Evolution of open systems as cospans (4)
So, for technical reasons, people started using double categories.
Example: Double category of cospans
Given a category S with pushouts, there is a double category Csp(S) having
- as objects, objects of S;
- as arrows, morphisms of S;
- as proarrows m:a+→b, cospans a→x←b in S;
- as cells, maps of cospans in S:
Is this double category really so different from the bicategory? Yes!
Double categories
Definition: A double category is a pseudocategory in Cat.
So, a double category D consists of
- category of objects D0 and category of morphisms D1
- source and target functors src,tgt:D1⇉D0
- external composition ⊙:D1×D0D1→D1 and identity id:D0→D1
obeying the category laws up to coherent isomorphism.
For us, double categories and double functors are pseudo by default.
Philosophy behind double category theory
Terminology: In a double category D,
- objects of D0 = objects of D
- morphisms of D0 = arrows of D
- objects of D1 = proarrows of D
- morphisms of D1 = cells of D
Interpretation: First and foremost, proarrows are objects, not morphisms.
Slogan:
Proarrows are “objects that happen to have sources and targets.”
Also: open systems are systems that happen to have boundaries!
Transformations between double functors
Proarrows play the role of objects in all the main concepts. For example:
Definition: A natural transformation α:F⇒G between double functors F,G:D→E consists of
- for each object x in D, an arrow αx:Fx→Gx in E;
- for each proarrow m:x+→y in D, a cell in E of form
that are natural with respect to arrows and cells in D, respectively, and functorial with respect to external composition and identity.
Fact: There is a 2-category Dbl of double categories, double functors, and natural transformations.
Cocartesian double categories
Definition. A cocartesian double category is a cocartesian object in Dbl, i.e., a double category D such that the diagonal Δ:D→D×D and unique map !:D→1 have left adjoints in Dbl.
Proposition: Cocartesian double categories, concretely (Grandis 2019, Corollary 4.3.7)
A double category D is cocartesian if and only if
- the categories D0 and D1 are cocartesian (have finite coproducts)
- the functors src,tgt:D1→D0 are cocartesian (preserve finite coproducts)
- the functors ⊙:D1×D0D1→D1 and id:D0→D1 are also cocartesian, so that the canonical comparison cells are isomorphisms in D1:
For cartesian double categories, see the PhD thesis (Aleiferi 2018).
Cocartesian double category of cospans
Example: cocartesian double category of cospans
For any category X with finite colimits, Csp(X) is a cocartesian double category:
- finite coproducts in Csp(X)0=X exist by assumption
- finite coproducts in Csp(X)1=X{∙→∙←∙} are computed pointwise in X
- source and target functors ftL,ftR:Csp(X)1⇉X preserve coproducts
- external composition and identity functors preserve coproducts because:
- colimits commute with colimits
- specifically, pushouts commute with coproducts
- We expect “nice” double categories of open systems to be cocartesian
- Double categories of structured cospans are cocartesian!
Structured cospans
Structured cospans are easy to use and include open systems such as:
- open graphs
- open Petri nets and reaction networks
- open parameterized dynamical systems (Aduddell et al. 2023)
Structured cospans are implemented in Catlab.jl.
Structured cospans
Given a functor L:A→X, an L-structured cospan consists of
- a pair of objects a,b∈A
- a cospan in X of the form La→x←Lb.
Baez and Courser (2020) showed that when X has pushouts,
- there is a double category LCsp(X) of L-structured cospans
- under extra assumptions, LCsp(X) can be given the structure of a symmetric monoidal double category
Fact (1) is easy; (2) is more involved and has been given three different proofs.
Idea: Bypass monoidal structure using double-categorical universal property.
Monoidal structure from cocartesian property
Familiar fact: Any cocartesian category can be given the structure of symmetric monoidal category by making a choice of coproducts.
Works for double categories too! Outline of argument:
- Structured cospans are a cocartesian double category
(cocartesian object in Dbl) - Abstract nonsense: for any 2-category C with finite 2-products, a cartesian object in C can be given the structure of a symmetric pseudomonoid in C
- Hence, structured cospans are a symmetric monoidal double category
(symmetric pseudomonoid in Dbl)
Cocartesian double cat. of structured cospans
Theorem
Let A be a category with finite coproducts, X be a category with finite colimits, and L:A→X be a functor that preserves finite coproducts. Then LCsp(X) is a cocartesian double category.
Proof sketch
- By assumption, LCsp(X)0=A has finite coproducts.
- Also by assumption, the canonical comparison maps are isomorphisms in X: La,b:L(a)+L(b)≅→L(a+b),L0:0X≅→L(0A)
- Coproducts in LCsp(X)1 are given by restricting coproducts in Csp(X)1 along inverse comparisons:
Decorated cospans
Not all double categories of open systems are cocartesian.
- Example: double category of open dynamical systems
Can use decorated cospans instead, which is:
- A more elaborate theory
- But does not assume cocartesianess
Decorated cospans
Given a category A with finite colimits and a lax monoidal functor F:(A,+)→(Cat,×), an F-decorated cospan consists of
- a cospan a→m←b in A
- an object x in F(m), the decoration.
Baez, Courser, and Vasilakopoulou (2022) showed that there is a symmetric monoidal double category of F-decorated cospans.
Question: Recover decorated cospans using the Grothendieck construction?
- Grothendieck construction decorates objects, not morphisms
- Fortunately, cospans are objects according to double category theory!
- So, use a double-categorical Grothendieck construction?
Double Grothendieck construction
Theorem (partial statement) (Cruttwell et al. 2022)
Given a lax double functor
F:A→Span(Cat),
there is a double category ∫F, the double Grothendieck construction of F, with underlying categories
(∫F)0=∫F0and(∫F)1=∫(apex∘F1).
In the double category ∫F,
- objects are pairs (a,x), where a∈A and x∈F(a);
- proarrows (a,x)+→(b,y) are pairs (m,s), where m:a+→b is in A and s∈apexF(m) such that (legLF(m))(s)=xand(legRF(m))(s)=y.
Moreover, there is a canonical projection, a strict double functor
πF:∫F→A.
The double Grothendieck construction is due to Cruttwell, Lambert, Pronk, and Szyld (2022).
Generalized decorated cospans
Corollary/Definition
Given a category A with pushouts and a lax double functor
F:Csp(A)→Span(Cat),
there is a double category of F-decorated cospans given by ∫F.
These decorated cospans are more general than the usual ones:
- Decorations of a cospan depend on the whole cospan, not just its apex
- Feet of cospans are also decorated, compatibly with cospan decorations
Modular reconstruction of decorated cospans
Corollary
Given a category A with finite colimits and a lax monoidal functor F:(A,+)→(Cat,×), there is a double category of F-decorated cospans, as defined in the literature.
Proof sketch
- Monoidal categories are double categories with trivial categories of objects, 2-functorially: B:MonCatlax→Dbllax
- Apply the double Grothendieck construction to the composite lax double functor Csp(A)Apex→B(A,+)B(F)→B(Cat,×)Apex∗→Span(Cat)
Caveat
The symmetric monoidal structure on the double category is not recovered.
Conclusion
This talk is based on my paper:
Patterson, 2023. Structured and decorated cospans from the viewpoint of double category theory. arXiv:2304.00447.
Other topics covered:
- structured cospans as cocartesian equipments
- cocartesian maps between cocartesian equipments of structured cospans
- application of generalized decorated cospans to theories of processes