Double-categorical logic
in theory and practice

Second Virtual Workshop on Double Categories

Evan Patterson

Topos Institute

2024-10-29

Why double categories?

My personal favorite answers:

Double categories unify functional and relational thinking
- Understand relations, spans, profunctors, etc via universal properties
Double categories are the setting for open systems:
- An open system is both an object and a morphism
- Hence is a proarrow in a double category
Double category theory is an organizing framework for categorical logic

This talk is mostly about (3).

Categorical logic

Originates with Lawvere’s functorial semantics of algebraic theories:

Logic	Category theory
algebraic theory	category $\mathsf{T}$ with finite products
model of theory	functor $M: \mathsf{T} \to \mathsf{Set}$ preserving finite products
model homomorphism	natural transformation $M \Rightarrow N$

Logic	Category theory
schema	small category $\mathsf{C}$
instance of schema	functor $X: \mathsf{C} \to \mathsf{Set}$
morphism of instances	natural transformation $X \Rightarrow Y$

Standard dictionary of categorical logic

Logic	Category theory
theory $\mathsf{T}$	category $\mathsf{T}$ with extra structure
model of $\mathsf{T}$ in $\mathsf{S}$	structure-preserving functor $M: \mathsf{T} \to \mathsf{S}$
model morphism	structure-preserving natural transformation $M \Rightarrow N$

Categorical logic is two-dimensional:

Theories, models, and model morphisms form a 2-category
Such 2-categories are sometimes called “doctrines”
E.g., categories, monoidal categories, categories with finite products, … are objects of different doctrines

Another dictionary of categorical logic?

A sleight of hand in that story:

I replaced $\mathsf{Set}$ with an arbitrary “semantics category” $\mathsf{S}$
That flexibility is useful, but $\mathsf{Set}$ still plays a distinguished role

If we take this seriously, we might propose a different dictionary:

Logic	Category theory
theory $\mathsf{T}$	category $\mathsf{T}$ with extra structure
model of $\mathsf{T}$	structure-preserving module (copresheaf) over $\mathsf{T}$
model morphism	structure-preserving module homomorphism

Part 1: Double theories

Joint work with Michael Lambert.

Main references:

“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)

Toward two-dimensional functorial semantics

Generalizing from the terminal double category:

Proposition

Let $\mathbb{D}$ be a strict double category and let $F: \mathbb{D} \to \mathbb{S}\mathsf{pan}$ be a lax functor.

For each object $x \in \mathbb{D}$ , the data $(Fx, F\mathrm{id}_x, F_{x,x}, F_x)$ is a category
For each arrow $f: x \to y$ in $\mathbb{D}$ , the data $(Ff, F\mathrm{id}_f)$ is a functor
For each special cell $\alpha$ in $\mathbb{D}$ , the image $F\alpha$ is a natural transformation:

Toward two-dimensional functorial semantics

Proposition

Let $\mathbb{D}$ be a strict double category and let $F: \mathbb{D} \to \mathbb{S}\mathsf{pan}$ be a lax functor.

For each proarrow $m: x \mathrel{\mkern 3mu\vcenter{\hbox{$\scriptstyle+$}}\mkern-13mu{\to}}y$ in $\mathbb{D}$ , the data $(Fm, F_{x,m}, F_{m,y})$ is a bimodule, i.e., a profunctor
For each cell $\alpha$ in $\mathbb{D}$ , the image $F\alpha$ is a bimodule homomorphism, i.e., a natural transformation

These are the basic notions of category theory! Moreover, replacing $\mathbb{S}\mathsf{pan}$ with:

$\mathbb{S}\mathsf{pan}(\mathsf{C})$ gives notions internal to $\mathsf{C}$
$\mathbb{M}\mathsf{at}(\mathcal{V})$ gives notions enriched in $\mathcal{V}$

Double theories

With this motivation, we propose that:

A “double theory” is a strict double category $\mathbb{T}$ , possibly equipped with extra structure
A model of a double theory is structure-preserving lax functor $\mathbb{T} \to \mathbb{S}\mathsf{pan}$

For a fixed double theory $\mathbb{T}$ , there will be a (possibly virtual) double category, even a (virtual) equipment, with:

objects = models of $\mathbb{T}$
arrows = morphisms of models (possibly lax, oplax, or pseudo)
proarrows = bimodules between models
cells = transformations/bimodule homomorphisms

Meta-logic	Double category theory
logic	double theory
theory	model of double theory
interpretation between theories	model morphism
model of theory	module/instance over model
model homomorphism	module/instance homomorphism

Simple double theories

Definition

A simple double theory is a small, strict double category.

This is a concept with an attitude, understood as a categorified theory:

object = “object type”
arrow = “operation on objects”
proarrow = “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 $n$

Simple double theories, the big picture

Concept	Definition
Simple double theory	Small, strict double category $\mathbb{T}$
Model of simple double theory	Lax functor $F: \mathbb{T} \to \mathbb{S}\mathsf{pan}$
Morphism between models (strict, pseudo, op/lax)	(Strict, pseudo, op/lax) natural transformation $\alpha: F \Rightarrow G$
Bimodule between models	Module $M: F \mathrel{\mkern 3mu\vcenter{\hbox{$\scriptstyle\mid$}}\mkern-8mu{\Rightarrow}}G$
Cell between morphisms/bimodules	Modulation

Cartesian double theories, the big picture

Concept	Definition
Cartesian double theory	Small, cartesian, strict double category $\mathbb{T}$
Model of theory	Cartesian lax functor $F: \mathbb{T} \to \mathbb{S}\mathsf{pan}$
Model morphism (strict, pseudo, op/lax)	Cartesian (strict, pseudo, op/lax) natural transformation $\alpha: F \Rightarrow G$
Model bimodule	Cartesian module $M: F \mathrel{\mkern 3mu\vcenter{\hbox{$\scriptstyle\mid$}}\mkern-8mu{\Rightarrow}}G$
Cell	Modulation

Discrete double theories

Simplifying still further from simple double theories:

Definition

A discrete double theory is

a simple double theory having only trivial arrows and cells, or equivalently
a double category of the form $\mathbb{D}\mathsf{isc}(\mathsf{B}) := \mathbb{H}(\mathsf{B})$ , where $\mathsf{B}$ is a small category

Such a double theory has only object and morphism types, no operations.

Fact

A model of a discrete double theory $\mathbb{D}\mathsf{isc}(\mathsf{B})$ is equivalent to a category sliced over $\mathsf{B}$ :

L a x (D i s c (B), S p a n) ≃ C a t / B .

$\mathsf{Lax}(\mathbb{D}\mathsf{isc}(\mathsf{B}), \mathbb{S}\mathsf{pan}) \simeq \mathsf{Cat}/\mathsf{B}.$

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

S g n C a t := C a t / S g n .

$\mathsf{SgnCat} := \mathsf{Cat}/\mathsf{Sgn}.$

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,

T_{S g n C a t} := D i s c (S g n),

$\mathbb{T}_{\mathsf{SgnCat}} := \mathbb{D}\mathsf{isc}(\mathsf{Sgn}),$

L a x (T_{S g n C a t}, S p a n) ≃ C a t / S g n = S g n C a t .

$\mathsf{Lax}(\mathbb{T}_{\mathsf{SgnCat}}, \mathbb{S}\mathsf{pan}) \simeq \mathsf{Cat}/\mathsf{Sgn} = \mathsf{SgnCat}.$

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 = column that is not a foreign key

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”:

L a x (D i s c (2), S p a n) ≃ C a t / 2 .

$\mathsf{Lax}(\mathbb{D}\mathsf{isc}(\mathsf{2}), \mathbb{S}\mathsf{pan}) \simeq \mathsf{Cat}/\mathsf{2}.$

In future versions of CatColab:

Database instances as modules over models of double theories
Algebraic databases a la Schultz et al. (2017)

Double-categorical logic in theory and practice Second Virtual Workshop on Double Categories Evan Patterson 2024-10-29

Double-categorical logic in theory and practice
Why double categories?
Categorical logic
Categorical logic, the minimal setting
Standard dictionary of categorical logic
Another dictionary of categorical logic?
Categorical logics are double categories
Outline
Part 1: Double theories
Toward two-dimensional functorial semantics
Toward two-dimensional functorial semantics
Toward two-dimensional functorial semantics
Double theories
Double-categorical logic
Flavors of double theory
Simple double theories
Models of simple double theories
Example: theory of monads
Example: theory of monads
Simple double theories, the big picture
Cartesian double theories, the big picture
Example: theory of pseudomonoids
Interlude: Demo
Part 2: CatColab
Discrete double theories
Causal loop diagrams
Causal loop diagrams via double theories
Free signed categories
Diagrams as free categorical structures
Database schemas
Schemas via double theories
Design of CatColab
Desiderata
Structure editing
User interface
References