CatColab: formal, interoperable, conceptual modeling
Applied Category Theory 2025
2025-06-02
Introducing CatColab
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This talk describes work in progress on CatColab, a collaborative environment for formal, interoperable, conceptual modeling.
- formal
- models are mathematical objects that can be critiqued with clarity
- interoperable
- models, and modeling languages, can be interoperated with each other
- conceptual
- modeling languages are well adapted to concepts used by practitioners
See also the blog post: Introducing CatColab (Carlson 2024)
Outline
- Demo
- Mathematics behind CatColab
Credits
As of June 2025, the core development team for CatColab is:
- Kris Brown
- Kevin Carlson
- Owen Lynch
- Jason Moggridge
- Evan Patterson
I am also grateful for support from colleagues, collaborators, and funders.
For more credits, see: https://catcolab.org/help/credits
Demo
Will a few of the logics currently available in CatColab:
- Causal loop diagrams
- System structure diagrams, aka primitive stock-flow diagrams
- Database schemas
Try it yourself:
- Stable: https://catcolab.org
- Bleeding edge: https://next.catcolab.org
Warning
CatColab is early alpha software under active development.
Mathematics
Mathematical foundation for CatColab is the framework of double theories:
- “Cartesian double theories: A double-categorical framework for categorical logic” (Lambert and Patterson 2024)
- “Products in double categories, revisited,” Section 10: “Finite-product double theories” (Patterson 2024)
This talks focuses on special cases already implemented in CatColab
Motivation
Formal languages are the syntactic counterpart to categorical structures:
| Logic/language | Categorical structure |
|---|---|
| Algebraic theories | Cartesian categories |
| Typed lambda theories | Cartesian closed categories |
| Resource theories | Symmetric monoidal categories |
| Statistical theories | Markov categories |
| … | … |
- Many (though not all) of these categorical structures are known to be models of double theories
- But we’re starting with simpler examples, often not regarded as logics at all
Simple double theories
Definition
A simple double theory is a strict double category.
This is a concept with an attitude, understood as a categorified theory:
- object = “object type”
- arrow (vertical morphism) = “operation on objects”
- proarrow (horizontal morphism) = “morphism type”
- cell = “operation on morphisms”
- \(f\) = object operation with input type \(x\) and output type \(w\)
- \(\alpha\) = morphism operation with input type \(m\) and output type \(n\)
Double theories are usually small double categories, but that is inessential.
Models of simple double theories
Definition
A model of a simple double theory \(\mathbb{T}\) is
- a lax double functor \(M: \mathbb{T} \to \mathbb{S}\mathsf{pan}\), or equivalently
- a normal lax double functor \(M: \mathbb{T} \to \mathbb{P}\mathsf{rof}\)
Such models are categorified copresheaves:
| Idea | 1-dimensional | 2-dimensional |
|---|---|---|
| Schema/theory | Category \(\mathsf{C}\) | Double category \(\mathbb{D}\) |
| Semantics | \(\mathsf{Set}\) | \(\mathbb{S}\mathsf{pan}\ (= \mathbb{S}\mathsf{et})\) |
| Instance/model | Functor \(\mathsf{C} \to \mathsf{Set}\) | Lax functor \(\mathbb{D} \to \mathbb{S}\mathsf{pan}\) |
Discrete double theories
Simplifying still further:
Definition
A discrete double theory is
- a simple double theory having only trivial arrows and cells, or equivalently
- a double category of the form \(\mathbb{D}\mathsf{isc}(\mathsf{B}) := \mathbb{H}(\mathsf{B})\) for some category \(\mathsf{B}\)
Such a double theory has only object and morphism types, no operations.
Models of discrete double theories are also known as “displayed categories” (Ahrens and Lumsdaine 2019).
Fact
A model of a discrete double theory \(\mathbb{D}\mathsf{isc}(\mathsf{B})\) is equivalent to a category sliced over \(\mathsf{B}\):
\[ \mathsf{Lax}(\mathbb{D}\mathsf{isc}(\mathsf{B}), \mathbb{S}\mathsf{pan}) \simeq \mathsf{Cat}/\mathsf{B}. \]
Discrete tabulator theories
We now extend discrete double theories with a kind of double limit.
Recall that the tabulator of a profunctor \(P: \mathsf{C} \mathrel{\mkern 3mu\vcenter{\hbox{$\scriptstyle+$}}\mkern-13mu{\to}}\mathsf{D}\) is its category of elements \(\int P\), having
- as objects, triples \(c \in \mathsf{C}\), \(d \in \mathsf{D}\), and \(p: c \dashrightarrow d\) in \(P\);
- as morphisms \((c,d,p) \to (c',d',p')\), morphisms \(f: c \to c'\) in \(\mathsf{C}\) and \(g: d \to d'\) in \(\mathsf{D}\) making the square commute.
Definition
A discrete tabulator theory is a double category with tabulators, having only trivial arrows and cells (identities and projections).
A model of a discrete tabulator theory \(\mathbb{T}\) is a normal lax functor \(M: \mathbb{T} \to \mathbb{P}\mathsf{rof}\) that preserves tabulators.
CatColab developer docs: tabulators
Examples
The mathematics behind our examples:
- Causal loop diagrams
- Database schemas
- System structure diagrams
Causal loop diagrams
Causal loop diagrams (and also regulatory networks) are
- signed graphs, or
- free signed categories
Let \(\mathsf{Sgn}:= \{\pm 1\} \cong \mathbb{Z}_2\) be the group of nonzero signs.
Definition
A signed category is a category \(\mathsf{C}\) equipped with a functor \(\mathsf{C} \to \mathsf{Sgn}\).
So, the category of signed categories is the slice
\[ \mathsf{SgnCat} := \mathsf{Cat}/\mathsf{Sgn}. \]
Aduddell et al. (2024), Section 2.1
Causal loop diagrams via double theories
Theory
The theory of signed categories is the discrete double theory generated by
- a object type \(x\)
- a morphism type \(n: x \mathrel{\mkern 3mu\vcenter{\hbox{$\scriptstyle+$}}\mkern-13mu{\to}}x\) (for “negative”)
subject to the equation \(n \odot n = \mathrm{id}_x\) (where \(\mathrm{id}_x: x \mathrel{\mkern 3mu\vcenter{\hbox{$\scriptstyle+$}}\mkern-13mu{\to}}x\) is “positive”).
Equivalently,
\[ \mathbb{T}_{\mathsf{SgnCat}} := \mathbb{D}\mathsf{isc}(\mathsf{Sgn}), \]
so
\[ \mathsf{Lax}(\mathbb{T}_{\mathsf{SgnCat}}, \mathbb{S}\mathsf{pan}) \simeq \mathsf{Cat}/\mathsf{Sgn} = \mathsf{SgnCat}. \]
CatColab developer docs: signed categories
Free signed categories
What’s the point about thinking of signed graphs as free signed categories?
Motifs are morphisms between free signed categories, e.g.,
Positive/reinforcing feedback loops are morphisms out of:
![]()
Negative/balancing feedback loops are morphisms out of:
![]()
Aduddell et al. (2024), Section 2.2
Diagrams as free categorical structures
This phenomenon is generic:
Slogan
Diagrammatic languages are free categorical structures whenever it makes sense for arrows to compose.
A famous example:
Example
Petri nets are free symmetric/commutative monoidal categories.
Coming in a future version of CatColab!
On Petri nets and SMCS: Baez et al. (2021)
Database schemas
A basic notion of database schema is a finitely presented profunctor
where \(\mathrm{Mapping} := \mathrm{Hom}_{\mathrm{Entity}}\) and \(\mathrm{AttrOp} := \mathrm{Hom}_{\mathrm{AttrType}}\).
In SQL jargon:
- “Entity” = table
- “Mapping” = foreign key
- “Attr” = data attribute
- “AttrType” = data type, such as
TEXTorREAL
Schemas via double theories
Theory
The theory of profunctors is the “walking proarrow” \(\mathbb{D}\mathsf{isc}(\mathsf{2})\), a discrete double theory freely generated by
- two objects \(0,1\)
- one proarrow \(0 \mathrel{\mkern 3mu\vcenter{\hbox{$\scriptstyle+$}}\mkern-13mu{\to}}1\).
A model is a profunctor, either directly or indirectly via “barrels”:
\[ \mathsf{Lax}(\mathbb{D}\mathsf{isc}(\mathsf{2}), \mathbb{S}\mathsf{pan}) \simeq \mathsf{Cat}/\mathsf{2}. \]
In future versions of CatColab:
- Database instances as modules over models of double theories
- Algebraic databases a la Schultz et al. (2017)
Distributors and barrels on Joyal’s CatLab, esp. Proposition 2.4
System structure diagrams
A system structure diagram, the qualitative counterpart to a stock-flow diagram, consists of:
- stocks, say of wheat or of people
- flows, going from an upstream stock to a downstream stock
- links, going from a source stock to a target flow
In categorical terms, links go from objects to morphisms!
Image: (Meadows 2008, fig. 21)
System structure diagrams via double theories
Theory
The theory of categories with links is the discrete tabulator theory freely generated by
- an object type \(x\)
- a morphism type \(\ell: x \mathrel{\mkern 3mu\vcenter{\hbox{$\scriptstyle+$}}\mkern-13mu{\to}}\top \mathrm{id}_x\)
where \(\top \mathrm{id}_x\) denotes the tabulator of the hom-type on \(x\)
A system structure diagram is then a free category with links, which implies:
- Links compose with flows in certain ways to give composite links
- A richer notion of morphism between system structure diagrams
CatColab developer docs: categories with links
References
