Structured and decorated cospans from the viewpoint of double category theory
Applied Category Theory 2023
2023-08-04
Open systems as cospans
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Mathematical modeling of open systems as cospans goes back 25+ years.
\[ a \rightarrow x \leftarrow b \]
- 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 \(\leadsto\) structured cospans
- extend the apex with extra data \(\leadsto\) 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 \(\mathsf{S}\) with pushouts, there is a category \(\mathsf{Csp}(\mathsf{S})\) having
- as objects, the objects of \(\mathsf{S}\);
- as morphisms, isomorphism classes of cospans in \(\mathsf{S}\).
Cospans compose by pushout in \(\mathsf{S}\).
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 \(\mathsf{S}\) with pushouts, there is bicategory \(\mathbf{Csp}(\mathsf{S})\) having
- as objects, the objects of \(\mathsf{S}\);
- as morphisms \(a \to b\), cospans \(a \rightarrow x \leftarrow b\) in \(\mathsf{S}\);
- as 2-morphisms \((a \rightarrow x \leftarrow b) \Rightarrow (a \rightarrow y \leftarrow b)\), maps \(h \colon x \to y\) in \(\mathsf{S}\) 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 \(\mathbf{Dbl}\)
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 \(\mathsf{S}\) with pushouts, there is a double category \(\mathbb{C}\mathsf{sp}(\mathsf{S})\) having
- as objects, objects of \(\mathsf{S}\);
- as arrows, morphisms of \(\mathsf{S}\);
- as proarrows \(m: a \mathrel{\mkern 3mu\vcenter{\hbox{$\scriptstyle+$}}\mkern-10mu{\to}}b\), cospans \(a \rightarrow x \leftarrow b\) in \(\mathsf{S}\);
- as cells, maps of cospans in \(\mathsf{S}\):
Is this double category really so different from the bicategory? Yes!
Double categories
Definition: A double category is a pseudocategory in \(\mathbf{Cat}\).
So, a double category \(\mathbb{D}\) consists of
- category of objects \(\mathbb{D}_0\) and category of morphisms \(\mathbb{D}_1\)
- source and target functors \(\mathop{\mathrm{src}}, \mathop{\mathrm{tgt}}: \mathbb{D}_1 \rightrightarrows \mathbb{D}_0\)
- external composition \(\odot: \mathbb{D}_1 \times_{\mathbb{D}_0} \mathbb{D}_1 \to \mathbb{D}_1\) and identity \(\mathop{\mathrm{id}}: \mathbb{D}_0 \to \mathbb{D}_1\)
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 \(\mathbb{D}\),
- objects of \(\mathbb{D}_0\) = objects of \(\mathbb{D}\)
- morphisms of \(\mathbb{D}_0\) = arrows of \(\mathbb{D}\)
- objects of \(\mathbb{D}_1\) = proarrows of \(\mathbb{D}\)
- morphisms of \(\mathbb{D}_1\) = cells of \(\mathbb{D}\)
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 \(\alpha: F \Rightarrow G\) between double functors \(F, G: \mathbb{D} \to \mathbb{E}\) consists of
- for each object \(x\) in \(\mathbb{D}\), an arrow \(\alpha_x: Fx \to Gx\) in \(\mathbb{E}\);
- for each proarrow \(m: x \mathrel{\mkern 3mu\vcenter{\hbox{$\scriptstyle+$}}\mkern-10mu{\to}}y\) in \(\mathbb{D}\), a cell in \(\mathbb{E}\) of form
that are natural with respect to arrows and cells in \(\mathbb{D}\), respectively, and functorial with respect to external composition and identity.
Fact: There is a 2-category \(\mathbf{Dbl}\) of double categories, double functors, and natural transformations.
Cocartesian double categories
Definition. A cocartesian double category is a cocartesian object in \(\mathbf{Dbl}\), i.e., a double category \(\mathbb{D}\) such that the diagonal \(\Delta: \mathbb{D} \to \mathbb{D} \times \mathbb{D}\) and unique map \(!: \mathbb{D} \to \mathbb{1}\) have left adjoints in \(\mathbf{Dbl}\).
Proposition: Cocartesian double categories, concretely (Grandis 2019, Corollary 4.3.7)
A double category \(\mathbb{D}\) is cocartesian if and only if
- the categories \(\mathbb{D}_0\) and \(\mathbb{D}_1\) are cocartesian (have finite coproducts)
- the functors \(\mathop{\mathrm{src}}, \mathop{\mathrm{tgt}}: \mathbb{D}_1 \to \mathbb{D}_0\) are cocartesian (preserve finite coproducts)
- the functors \(\odot: \mathbb{D}_1 \times_{\mathbb{D}_0} \mathbb{D}_1 \to \mathbb{D}_1\) and \(\mathop{\mathrm{id}}: \mathbb{D}_0 \to \mathbb{D}_1\) are also cocartesian, so that the canonical comparison cells are isomorphisms in \(\mathbb{D}_1\):
For cartesian double categories, see the PhD thesis (Aleiferi 2018).
Cocartesian double category of cospans
Example: cocartesian double category of cospans
For any category \(\mathsf{X}\) with finite colimits, \(\mathbb{C}\mathsf{sp}(\mathsf{X})\) is a cocartesian double category:
- finite coproducts in \(\mathbb{C}\mathsf{sp}(\mathsf{X})_0 = \mathsf{X}\) exist by assumption
- finite coproducts in \(\mathbb{C}\mathsf{sp}(\mathsf{X})_1 = \mathsf{X}^{\{\bullet \rightarrow \bullet \leftarrow \bullet\}}\) are computed pointwise in \(\mathsf{X}\)
- source and target functors \(\mathop{\mathrm{ft}}_L, \mathop{\mathrm{ft}}_R: \mathbb{C}\mathsf{sp}(\mathsf{X})_1 \rightrightarrows \mathsf{X}\) 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 \(L: \mathsf{A} \to \mathsf{X}\), an \(L\)-structured cospan consists of
- a pair of objects \(a,b \in \mathsf{A}\)
- a cospan in \(\mathsf{X}\) of the form \(La \rightarrow x \leftarrow Lb\).
Baez and Courser (2020) showed that when \(\mathsf{X}\) has pushouts,
- there is a double category \({}_{L}\mathbb{C}\mathsf{sp}(\mathsf{X})\) of \(L\)-structured cospans
- under extra assumptions, \({}_{L}\mathbb{C}\mathsf{sp}(\mathsf{X})\) 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 \(\mathbf{Dbl}\)) - Abstract nonsense: for any 2-category \(\mathbf{C}\) with finite 2-products, a cartesian object in \(\mathbf{C}\) can be given the structure of a symmetric pseudomonoid in \(\mathbf{C}\)
- Hence, structured cospans are a symmetric monoidal double category
(symmetric pseudomonoid in \(\mathbf{Dbl}\))
Cocartesian double cat. of structured cospans
Theorem
Let \(\mathsf{A}\) be a category with finite coproducts, \(\mathsf{X}\) be a category with finite colimits, and \(L: \mathsf{A} \to \mathsf{X}\) be a functor that preserves finite coproducts. Then \({}_{L}\mathbb{C}\mathsf{sp}(\mathsf{X})\) is a cocartesian double category.
Proof sketch
- By assumption, \({}_{L}\mathbb{C}\mathsf{sp}(\mathsf{X})_0 = \mathsf{A}\) has finite coproducts.
- Also by assumption, the canonical comparison maps are isomorphisms in \(\mathsf{X}\): \[ L_{a,b}: L(a)+L(b) \xrightarrow{\cong} L(a+b), \qquad L_0: 0_{\mathsf{X}} \xrightarrow{\cong} L(0_{\mathsf{A}}) \]
- Coproducts in \({}_{L}\mathbb{C}\mathsf{sp}(\mathsf{X})_1\) are given by restricting coproducts in \(\mathbb{C}\mathsf{sp}(\mathsf{X})_1\) 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 \(\mathsf{A}\) with finite colimits and a lax monoidal functor \(F: (\mathsf{A},+) \to (\mathbf{Cat},\times)\), an \(F\)-decorated cospan consists of
- a cospan \(a \rightarrow m \leftarrow b\) in \(\mathsf{A}\)
- 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
\[ F: \mathbb{A} \to \mathbb{S}\mathsf{pan}(\mathbf{Cat}), \]
there is a double category \(\int F\), the double Grothendieck construction of \(F\), with underlying categories
\[ \textstyle (\int F)_0 = \int F_0 \qquad\text{and}\qquad (\int F)_1 = \int (\mathop{\mathrm{apex}}\circ F_1). \]
In the double category \(\int F\),
- objects are pairs \((a,x)\), where \(a \in \mathbb{A}\) and \(x \in F(a)\);
- proarrows \((a,x) \mathrel{\mkern 3mu\vcenter{\hbox{$\scriptstyle+$}}\mkern-10mu{\to}}(b,y)\) are pairs \((m,s)\), where \(m: a \mathrel{\mkern 3mu\vcenter{\hbox{$\scriptstyle+$}}\mkern-10mu{\to}}b\) is in \(\mathbb{A}\) and \(s \in \mathop{\mathrm{apex}}F(m)\) such that \[ \textstyle (\mathop{\mathrm{leg}}_L F(m))(s) = x \qquad\text{and}\qquad (\mathop{\mathrm{leg}}_R F(m))(s) = y. \]
Moreover, there is a canonical projection, a strict double functor
\[ \textstyle \pi_F: \int F \to \mathbb{A}. \]
The double Grothendieck construction is due to Cruttwell, Lambert, Pronk, and Szyld (2022).
Generalized decorated cospans
Corollary/Definition
Given a category \(\mathsf{A}\) with pushouts and a lax double functor
\[ F: \mathbb{C}\mathsf{sp}(\mathsf{A}) \to \mathbb{S}\mathsf{pan}(\mathbf{Cat}), \]
there is a double category of \(F\)-decorated cospans given by \(\int F\).
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 \(\mathsf{A}\) with finite colimits and a lax monoidal functor \(F: (\mathsf{A},+) \to (\mathbf{Cat},\times)\), 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: \[ \mathbb{B}: \mathbf{MonCat}_{\mathrm{lax}} \to \mathbf{Dbl}_{\mathrm{lax}} \]
- Apply the double Grothendieck construction to the composite lax double functor \[ \mathbb{C}\mathsf{sp}(\mathsf{A}) \xrightarrow{\operatorname{Apex}} \mathbb{B}(\mathsf{A}, +) \xrightarrow{\mathbb{B}(F)} \mathbb{B}(\mathbf{Cat}, \times) \xrightarrow{\operatorname{Apex}_*} \mathbb{S}\mathsf{pan}(\mathbf{Cat}) \]
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