String diagrams for regular logic

David I. Spivak (joint with Brendan Fong)

Presented on 2018/10/27

Octoberfest

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 0 / 19

Introduction

Outline

1 IntroductionApplication: playing with logicImplications for string diagramsString diagrams for regular logic

2 Regular categories and regular logic

3 Bringing it all together

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 0 / 19

Introduction Application: playing with logic

Minority Report

The 2002 movie Minority report showed detective Tom Cruise playingseamlessly with logic.

A computer database held relevant information.

Cruise could pull it up, and manipulate it, to solve crimes.

Let’s imagine such a detective scenario. The knowledge base says:

Any two people who work in the same tiny company are acquainted.

Categorical Informatics is a tiny company.

David works at Categorical Informatics.

Ryan works at Categorical Informatics.

We of course want to conclude that David and Ryan are acquainted.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 1 / 19

Introduction Application: playing with logic

Minority Report

The 2002 movie Minority report showed detective Tom Cruise playingseamlessly with logic.

A computer database held relevant information.

Cruise could pull it up, and manipulate it, to solve crimes.

Let’s imagine such a detective scenario. The knowledge base says:

Any two people who work in the same tiny company are acquainted.

Categorical Informatics is a tiny company.

David works at Categorical Informatics.

Ryan works at Categorical Informatics.

We of course want to conclude that David and Ryan are acquainted.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 1 / 19

Introduction Application: playing with logic

Sample scenario

Assume:

works workscompany

tiny

person personacquainted`

Ci

Ci

Ci

company

=

true David works Ci= Ryan works Ci= Ci tiny=

Show:

true David acquainted Ryan=

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 2 / 19

Introduction Application: playing with logic

Sample scenario

Assume:

works workscompany

tiny

person personacquainted`

Ci

Ci

Ci

company

=

true David works Ci= Ryan works Ci= Ci tiny=

Show:

true David acquainted Ryan=

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 2 / 19

Introduction Application: playing with logic

Sample scenario

Assume:

works workscompany

tiny

person personacquainted`

Ci

Ci

Ci

company

=

true David works Ci= Ryan works Ci= Ci tiny=

Show:

true David acquainted Ryan=

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 2 / 19

Introduction Application: playing with logic

Sample scenario

Assume:

works workscompany

tiny

person personacquainted`

Ci

Ci

Ci

company

=

true David works Ci= Ryan works Ci= Ci tiny=

Show:

true David acquainted Ryan=

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 2 / 19

Introduction Application: playing with logic

Picture proof

true David works Ci= Ryan works Ci= Ci tiny=

Combine!

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 3 / 19

Introduction Application: playing with logic

Picture proof

Combined:

David works Ci

Ryan works Ci

tiny Ci

true =

Group two Ci’s!

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 3 / 19

Introduction Application: playing with logic

Picture proof

Ci’s grouped:

David works

Ci

Ryan works Ci

tiny Ci

true =

Substitute!

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 3 / 19

Introduction Application: playing with logic

Picture proof

Substituted:

David works

Ryan works

Ci

Ci

tinytrue =

Group two Ci’s!

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 3 / 19

Introduction Application: playing with logic

Picture proof

Two Ci’s grouped:

David works

Ryan works

Ci

Ci

tinytrue =

Substitute!

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 3 / 19

Introduction Application: playing with logic

Picture proof

Substituted:

David works

Ryan works

Ci

tinytrue =

Group Ci!

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 3 / 19

Introduction Application: playing with logic

Picture proof

Ci grouped:

David works

Ryan works

Ci

tinytrue =

Discard group!

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 3 / 19

Introduction Application: playing with logic

Picture proof

Group discarded:

David works

Ryan works

tinytrue =

Group!

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 3 / 19

Introduction Application: playing with logic

Picture proof

Grouped:

David works

Ryan works

tinytrue =

Substitute!

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 3 / 19

Introduction Application: playing with logic

Picture proof

Substituted:

David

Ryan

acquaintedtrue =

Done!

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 3 / 19

Introduction Implications for string diagrams

Two-dimensional manipulation of string diagrams

In this talk we discuss a 2-dimensional language for wiring diagrams.

It includes all the sorts of operations shown above.

Together with operations like discarding and breaking wires:

`

`` etc...

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 4 / 19

Introduction Implications for string diagrams

Comparing to other string diagram languages

Let’s compare to string diagram calculus for traced SMCs and hypercats.

In traced SMCs, you can compose, tensor, swap, and trace.

You can do these anywhere in the diagram, with axioms.These can be considered generators and relations for an operad.Traced categories are algebras on the operad 1-Cob.

In hypergraph categories, add Frobenius maps, plus axioms.

Hypergraph categories are algebras on the operad Cospan.

In our picture proof, we had more operations and relations.

Order on elements of each arity, preserved by substitution.Meet-semilattice structures on elements of each arity.Top element (true) can be discarded; corresponding structure for ∧.Removing dots, breaking wires.

We will see that this is a 2-dimensional structure.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 5 / 19

Introduction Implications for string diagrams

Comparing to other string diagram languages

Let’s compare to string diagram calculus for traced SMCs and hypercats.

In traced SMCs, you can compose, tensor, swap, and trace.

You can do these anywhere in the diagram, with axioms.

These can be considered generators and relations for an operad.Traced categories are algebras on the operad 1-Cob.

In hypergraph categories, add Frobenius maps, plus axioms.

Hypergraph categories are algebras on the operad Cospan.

In our picture proof, we had more operations and relations.

Order on elements of each arity, preserved by substitution.Meet-semilattice structures on elements of each arity.Top element (true) can be discarded; corresponding structure for ∧.Removing dots, breaking wires.

We will see that this is a 2-dimensional structure.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 5 / 19

Introduction Implications for string diagrams

Comparing to other string diagram languages

Let’s compare to string diagram calculus for traced SMCs and hypercats.In traced SMCs, you can compose, tensor, swap, and trace.

You can do these anywhere in the diagram, with axioms.These can be considered generators and relations for an operad.Traced categories are algebras on the operad 1-Cob.

X1 X2

Y

X1a

X1b

X1c X2a

X2b

X2c

Ya

Yb

Yc

Yd

−X1a

−X1b

+X1c

−X2a

+X2b

+X2c

− Ya

− Yb

+ Yc

+ Yd

In hypergraph categories, add Frobenius maps, plus axioms.Hypergraph categories are algebras on the operad Cospan.

In our picture proof, we had more operations and relations.Order on elements of each arity, preserved by substitution.Meet-semilattice structures on elements of each arity.Top element (true) can be discarded; corresponding structure for ∧.Removing dots, breaking wires.

We will see that this is a 2-dimensional structure.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 5 / 19

Introduction Implications for string diagrams

Comparing to other string diagram languages

Let’s compare to string diagram calculus for traced SMCs and hypercats.In traced SMCs, you can compose, tensor, swap, and trace.

You can do these anywhere in the diagram, with axioms.These can be considered generators and relations for an operad.Traced categories are algebras on the operad 1-Cob.

In hypergraph categories, add Frobenius maps, plus axioms.Hypergraph categories are algebras on the operad Cospan.

a b

c2

t

1

1u3 2

v

4

3w

14

x1

2y

2

s

3

z

6

5

1 2 3 4 1 2 1 2 3

a b c

s t u v w x y z

1 2 3 4 5 6

outer

In our picture proof, we had more operations and relations.Order on elements of each arity, preserved by substitution.Meet-semilattice structures on elements of each arity.Top element (true) can be discarded; corresponding structure for ∧.Removing dots, breaking wires.

We will see that this is a 2-dimensional structure.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 5 / 19

Introduction Implications for string diagrams

Comparing to other string diagram languages

Let’s compare to string diagram calculus for traced SMCs and hypercats.

In traced SMCs, you can compose, tensor, swap, and trace.

You can do these anywhere in the diagram, with axioms.These can be considered generators and relations for an operad.Traced categories are algebras on the operad 1-Cob.

In hypergraph categories, add Frobenius maps, plus axioms.

Hypergraph categories are algebras on the operad Cospan.

In our picture proof, we had more operations and relations.

Order on elements of each arity, preserved by substitution.Meet-semilattice structures on elements of each arity.Top element (true) can be discarded; corresponding structure for ∧.Removing dots, breaking wires.

We will see that this is a 2-dimensional structure.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 5 / 19

Introduction String diagrams for regular logic

Formal presentation of the calculus I.

The graphical calculus shown above can be understood as follows.

Fix a set Λ (elements will be string labels).

Consider the monoidal bicategory CospancoΛ .

Objects: arities nv−→ Λ, i.e. lists (v(1), . . . , v(n)) ∈ Λn.

1-morphisms:n1 n12 n2

Λv1 v2

2-morphisms: opposite of usual direction (hence −co)Monoidal structure: (0,+).

Consider the (locally posetal) monoidal bicategory Poset.

Obj: posets; 1-morphisms: monotone maps; 2-morphisms: nat. trans.Monoidal structure: (1,×).

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 6 / 19

Introduction String diagrams for regular logic

Formal presentation of the calculus I.

The graphical calculus shown above can be understood as follows.

Fix a set Λ (elements will be string labels).

Consider the monoidal bicategory CospancoΛ .

Objects: arities nv−→ Λ, i.e. lists (v(1), . . . , v(n)) ∈ Λn.

1-morphisms:n1 n12 n2

Λv1 v2

2-morphisms: opposite of usual direction (hence −co)Monoidal structure: (0,+).

Consider the (locally posetal) monoidal bicategory Poset.

Obj: posets; 1-morphisms: monotone maps; 2-morphisms: nat. trans.Monoidal structure: (1,×).

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 6 / 19

Introduction String diagrams for regular logic

Formal presentation of the calculus I.

The graphical calculus shown above can be understood as follows.

Fix a set Λ (elements will be string labels).

Consider the monoidal bicategory CospancoΛ .

Objects: arities nv−→ Λ, i.e. lists (v(1), . . . , v(n)) ∈ Λn.

1-morphisms:n1 n12 n2

Λv1 v2

2-morphisms: opposite of usual direction (hence −co)Monoidal structure: (0,+).

Consider the (locally posetal) monoidal bicategory Poset.

Obj: posets; 1-morphisms: monotone maps; 2-morphisms: nat. trans.Monoidal structure: (1,×).

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 6 / 19

Introduction String diagrams for regular logic

Formal presentation of the calculus II.

We have monoidal bicategories Cospan and Poset.

Definition

A regular hypergraph category is a lax monoidal 2-functor

T : CospancoΛ → Poset

such that the laxators are right adjoints.

Silly terminology: ajax monoidal functors: the laxators

1 T (0)ρ1

and T (v)× T (v ′) T (v + v ′)

ρv,v′

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 7 / 19

Introduction String diagrams for regular logic

Formal presentation of the calculus II.

We have monoidal bicategories Cospan and Poset.

Definition

A regular hypergraph category is a lax monoidal 2-functor

T : CospancoΛ → Poset

such that the laxators are right adjoints.

Silly terminology: ajax monoidal functors: the laxators

1 T (0)ρ1

and T (v)× T (v ′) T (v + v ′)

ρv,v′

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 7 / 19

Introduction String diagrams for regular logic

Formal presentation of the calculus II.

We have monoidal bicategories Cospan and Poset.

Definition

A regular hypergraph category is a lax monoidal 2-functor

T : CospancoΛ → Poset

such that the laxators are right adjoints.

Silly terminology: ajax monoidal functors: the laxators are adjoints

1 T (0)ρ1

>λ1

and T (v)× T (v ′) T (v + v ′)

ρv,v′

>λv,v′

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 7 / 19

Introduction String diagrams for regular logic

Aside: we’re pushing this notation for adjunctions

Throughout this talk, I’ll use a new notation for adjunctions.

Usual notation: C DR

>L

C DR

⊥L

D C .L

⊥R

Note that > is sometimes used as the name of a monad, but...... it really doesn’t indicate where the monad is (it’s on D).

Our notation: C DR⇐L

C DR

⇐L

D CL⇒R

The 2-arrow points in the direction of the left adjoint.Reason: it tells you the direction of the unit and counit.

C

D

C

R

⇐L

D

C

D

R

⇐L

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 8 / 19

Introduction String diagrams for regular logic

Aside: we’re pushing this notation for adjunctions

Throughout this talk, I’ll use a new notation for adjunctions.

Usual notation: C DR

>L

C DR

⊥L

D C .L

⊥R

Note that > is sometimes used as the name of a monad, but...... it really doesn’t indicate where the monad is (it’s on D).

Our notation: C DR⇐L

C DR

⇐L

D CL⇒R

The 2-arrow points in the direction of the left adjoint.Reason: it tells you the direction of the unit and counit.

C

D

C

R

⇐L

D

C

D

R

⇐L

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 8 / 19

Introduction String diagrams for regular logic

Aside: we’re pushing this notation for adjunctions

Throughout this talk, I’ll use a new notation for adjunctions.

Usual notation: C DR

>L

Note that > is sometimes used as the name of a monad, but...... it really doesn’t indicate where the monad is (it’s on D).

Our notation: C DR⇐L

The 2-arrow points in the direction of the left adjoint.Reason: it tells you the direction of the unit and counit.

C

D

C

R

⇐L

D

C

D

R

⇐L

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 8 / 19

Introduction String diagrams for regular logic

Regular hypergraph categories and regular categories

Denote by Cospan-Alg the category of regular hypergraph categories, i.e.sets Λ and ajax 2-functors

T : CospancoΛ → Poset.

Theorem

There is an adjunction

Cospan-Alg RegCatΦ⇒Ψ

,

such that for any regular category R, the counit Ψ(Φ(R))→ R is anequivalence of categories.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 9 / 19

Introduction String diagrams for regular logic

Regular hypergraph categories and regular categories

Denote by Cospan-Alg the category of regular hypergraph categories, i.e.sets Λ and ajax 2-functors

T : CospancoΛ → Poset.

Theorem

There is an adjunction

Cospan-Alg RegCatΦ⇒Ψ

,

such that for any regular category R, the counit Ψ(Φ(R))→ R is anequivalence of categories.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 9 / 19

Introduction String diagrams for regular logic

Plan

We’ll return to the theorem shortly.

First we want to recall the definition of regular categories.

We also want to make the connection to regular logic.

The Cospan-algebra story is a graphical representation of the logic.This will be evident, but one can take the theorem as justification.

Then we’ll unpack the theorem and conclude.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 10 / 19

Regular categories and regular logic

Outline

1 Introduction

2 Regular categories and regular logicRegular categoriesRegular logic

3 Bringing it all together

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 10 / 19

Regular categories and regular logic Regular categories

Regular categories

Definition

A regular category is a category for which

all finite limits exist,

the kernel pair of any morphism admits a coequalizer, and

coequalizers are stable under pullback.

Examples of regular categories:

Set, and more generally any topos;

Setop, opposite of any topos, TopSpop;

The category of models of any Lawvere theory (Groups, Rings, ...);

The slice (also the coslice) of any regular category over any object;

Exponential ideal: if R regular and C a category, then RC is regular.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 11 / 19

Regular categories and regular logic Regular categories

Regular categories

Definition

A regular category is a category for which

all finite limits exist,

the kernel pair of any morphism admits a coequalizer, and

coequalizers are stable under pullback.

Examples of regular categories:

Set, and more generally any topos;

Setop, opposite of any topos, TopSpop;

The category of models of any Lawvere theory (Groups, Rings, ...);

The slice (also the coslice) of any regular category over any object;

Exponential ideal: if R regular and C a category, then RC is regular.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 11 / 19

Regular categories and regular logic Regular categories

How to think of regular categories

Regular categories are those with a good bicategory of relations.

A relation in R is a subobject S ⊆ A× B.

When R is regular, pullbacks and images play nicely...

... so that relations form a posetal bicategory RelR.

That is, relations can be composed and compared.One can recover the morphisms in R as the adjunctions in RelR !

Every young category theorist should prove to themselves that Set isthe category of adjunctions in Rel.

Regular categories have enough structure to do regular logic.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 12 / 19

Regular categories and regular logic Regular categories

How to think of regular categories

Regular categories are those with a good bicategory of relations.

A relation in R is a subobject S ⊆ A× B.

When R is regular, pullbacks and images play nicely...

... so that relations form a posetal bicategory RelR.

That is, relations can be composed and compared.One can recover the morphisms in R as the adjunctions in RelR !

Every young category theorist should prove to themselves that Set isthe category of adjunctions in Rel.

Regular categories have enough structure to do regular logic.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 12 / 19

Regular categories and regular logic Regular categories

How to think of regular categories

Regular categories are those with a good bicategory of relations.

A relation in R is a subobject S ⊆ A× B.

When R is regular, pullbacks and images play nicely...

... so that relations form a posetal bicategory RelR.

That is, relations can be composed and compared.One can recover the morphisms in R as the adjunctions in RelR !

Every young category theorist should prove to themselves that Set isthe category of adjunctions in Rel.

Regular categories have enough structure to do regular logic.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 12 / 19

Regular categories and regular logic Regular categories

How to think of regular categories

Regular categories are those with a good bicategory of relations.

A relation in R is a subobject S ⊆ A× B.

When R is regular, pullbacks and images play nicely...

... so that relations form a posetal bicategory RelR.

That is, relations can be composed and compared.One can recover the morphisms in R as the adjunctions in RelR !

Every young category theorist should prove to themselves that Set isthe category of adjunctions in Rel.

Regular categories have enough structure to do regular logic.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 12 / 19

Regular categories and regular logic Regular logic

Regular logic and regular categories

In regular logic, one has

A set of types Λ

A set of relation symbols `a1:A1,...,ak :AkR1(a1, . . . , ak) : Prop

Operations ∧, true , =, and ∃, from which to build up formulas ϕ,ψ.

A notion of entailment: ϕ `a:A,b:B ψ.

A set of axioms involving entailment.

Example: the regular theory of “two sets and a function”:

Λ = {A,B}, one relation symbol: `a:A,b:B f (a, b) : Prop

Axioms:

f is “total”: true `a:A ∃(b : B). f (a, b)f is “deterministic”: ∃(a : A). f (a, b) = f (a, b′) `b,b′:B b = b′

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 13 / 19

Regular categories and regular logic Regular logic

Regular logic and regular categories

In regular logic, one has

A set of types Λ

A set of relation symbols `a1:A1,...,ak :AkR1(a1, . . . , ak) : Prop

Operations ∧, true , =, and ∃, from which to build up formulas ϕ,ψ.

A notion of entailment: ϕ `a:A,b:B ψ.

A set of axioms involving entailment.

Example: the regular theory of “two sets and a function”:

Λ = {A,B}, one relation symbol: `a:A,b:B f (a, b) : Prop

Axioms:

f is “total”: true `a:A ∃(b : B). f (a, b)f is “deterministic”: ∃(a : A). f (a, b) = f (a, b′) `b,b′:B b = b′

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 13 / 19

Regular categories and regular logic Regular logic

Regular logic and regular categories

In regular logic, one has

A set of types Λ

A set of relation symbols `a1:A1,...,ak :AkR1(a1, . . . , ak) : Prop

Operations ∧, true , =, and ∃, from which to build up formulas ϕ,ψ.

A notion of entailment: ϕ `a:A,b:B ψ.

A set of axioms involving entailment.

Example: the regular theory of “two sets and a function”:

Λ = {A,B}, one relation symbol: `a:A,b:B f (a, b) : Prop

Axioms:

f is “total”: true `a:A ∃(b : B). f (a, b)f is “deterministic”: ∃(a : A). f (a, b) = f (a, b′) `b,b′:B b = b′

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 13 / 19

Regular categories and regular logic Regular logic

Regular logic and regular categories

In regular logic, one has

A set of types Λ

A set of relation symbols `a1:A1,...,ak :AkR1(a1, . . . , ak) : Prop

Operations ∧, true , =, and ∃, from which to build up formulas ϕ,ψ.

A notion of entailment: ϕ `a:A,b:B ψ.

A set of axioms involving entailment.

Example: the regular theory of “two sets and a function”:

Λ = {A,B}, one relation symbol: `a:A,b:B f (a, b) : Prop

Axioms:

f is “total”: true `a:A ∃(b : B). f (a, b)f is “deterministic”: ∃(a : A). f (a, b) = f (a, b′) `b,b′:B b = b′

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 13 / 19

Regular categories and regular logic Regular logic

Regular logic and regular categories

In regular logic, one has

A set of types Λ

A set of relation symbols `a1:A1,...,ak :AkR1(a1, . . . , ak) : Prop

Operations ∧, true , =, and ∃, from which to build up formulas ϕ,ψ.

A notion of entailment: ϕ `a:A,b:B ψ.

A set of axioms involving entailment.

Example: the regular theory of “two sets and a function”:

Λ = {A,B}, one relation symbol: `a:A,b:B f (a, b) : Prop

Axioms:

f is “total”: true `a:A ∃(b : B). f (a, b)f is “deterministic”: ∃(a : A). f (a, b) = f (a, b′) `b,b′:B b = b′

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 13 / 19

Regular categories and regular logic Regular logic

Regular logic and regular categories

In regular logic, one has

A set of types Λ

A set of relation symbols `a1:A1,...,ak :AkR1(a1, . . . , ak) : Prop

Operations ∧, true , =, and ∃, from which to build up formulas ϕ,ψ.

A notion of entailment: ϕ `a:A,b:B ψ.

A set of axioms involving entailment.

Example: the regular theory of “two sets and a function”:

Λ = {A,B}, one relation symbol: `a:A,b:B f (a, b) : Prop

Axioms:

f is “total”: true `a:A ∃(b : B). f (a, b)f is “deterministic”: ∃(a : A). f (a, b) = f (a, b′) `b,b′:B b = b′

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 13 / 19

Regular categories and regular logic Regular logic

Regular logic and cospan-algebras

truea fb

a`a:A

true `a:A ∃(b : B). f (a, b)

f

f

ab1

b2

b1

b2

`b1,b2:B

∃(a : A). f (a, b1) = f (a, b2) `b1,b2:B b1 = b2

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 14 / 19

Regular categories and regular logic Regular logic

Regular logic and cospan-algebras

truea fb

a`a:A

true `a:A ∃(b : B). f (a, b)

f

f

ab1

b2

b1

b2

`b1,b2:B

∃(a : A). f (a, b1) = f (a, b2) `b1,b2:B b1 = b2

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 14 / 19

Bringing it all together

Outline

1 Introduction

2 Regular categories and regular logic

3 Bringing it all togetherWhere are we?Recalling and justifying the theoremConcluding

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 14 / 19

Bringing it all together Where are we?

Where are we?

We have regular categories, regular logic, and cospan-algebras.

They are three different perspectives on the same subject.

Regular logic is an “internal language” for regular categories.

The bicategory of cospans is a “string diagram language” for regcats.

Next we’ll recall the theorem, give one slide of justification, and conclude.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 15 / 19

Bringing it all together Where are we?

Where are we?

We have regular categories, regular logic, and cospan-algebras.

They are three different perspectives on the same subject.

Regular logic is an “internal language” for regular categories.

The bicategory of cospans is a “string diagram language” for regcats.

Next we’ll recall the theorem, give one slide of justification, and conclude.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 15 / 19

Bringing it all together Where are we?

Where are we?

We have regular categories, regular logic, and cospan-algebras.

They are three different perspectives on the same subject.

Regular logic is an “internal language” for regular categories.

The bicategory of cospans is a “string diagram language” for regcats.

Next we’ll recall the theorem, give one slide of justification, and conclude.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 15 / 19

Bringing it all together Recalling and justifying the theorem

Recalling the theorem

Recall that a Cospan-algebra is an ajax 2-functor T : CospancoΛ → Poset.

Theorem

There is an adjunction Cospan-Alg RegCat⇒ , such that

Ψ(Φ(R))→ R is an equivalence for any regular category R.

Comments:

We can beef this up to a 2-reflection RegCat ⊆ Cospan-Alg.

Cospan algebras and regular categories look different on the surface.

Remember how complicated the def. of regcats was?Finite limits, coequalizers of kernel pairs, pullback stability.Cospan-Alg is certain functors Cospan→ Poset.

Easier to see posets and adjunctions in RegCat: subobject lattices.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 16 / 19

Bringing it all together Recalling and justifying the theorem

Recalling the theorem

Recall that a Cospan-algebra is an ajax 2-functor T : CospancoΛ → Poset.

Theorem

There is an adjunction Cospan-Alg RegCat⇒ , such that

Ψ(Φ(R))→ R is an equivalence for any regular category R.

Comments:

We can beef this up to a 2-reflection RegCat ⊆ Cospan-Alg.

Cospan algebras and regular categories look different on the surface.

Remember how complicated the def. of regcats was?Finite limits, coequalizers of kernel pairs, pullback stability.Cospan-Alg is certain functors Cospan→ Poset.

Easier to see posets and adjunctions in RegCat: subobject lattices.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 16 / 19

Bringing it all together Recalling and justifying the theorem

Recalling the theorem

Recall that a Cospan-algebra is an ajax 2-functor T : CospancoΛ → Poset.

Theorem

There is an adjunction Cospan-Alg RegCat⇒ , such that

Ψ(Φ(R))→ R is an equivalence for any regular category R.

Comments:

We can beef this up to a 2-reflection RegCat ⊆ Cospan-Alg.

Cospan algebras and regular categories look different on the surface.

Remember how complicated the def. of regcats was?Finite limits, coequalizers of kernel pairs, pullback stability.Cospan-Alg is certain functors Cospan→ Poset.

Easier to see posets and adjunctions in RegCat: subobject lattices.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 16 / 19

Bringing it all together Recalling and justifying the theorem

Recalling the theorem

Recall that a Cospan-algebra is an ajax 2-functor T : CospancoΛ → Poset.

Theorem

There is an adjunction Cospan-Alg RegCat⇒ , such that

Ψ(Φ(R))→ R is an equivalence for any regular category R.

Comments:

We can beef this up to a 2-reflection RegCat ⊆ Cospan-Alg.

Cospan algebras and regular categories look different on the surface.

Remember how complicated the def. of regcats was?Finite limits, coequalizers of kernel pairs, pullback stability.Cospan-Alg is certain functors Cospan→ Poset.

Easier to see posets and adjunctions in RegCat: subobject lattices.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 16 / 19

Bringing it all together Recalling and justifying the theorem

Why it works

One can form the syntactic category RT of T : CospanΛ → Poset.

Ob(RT ) := {(v , ϕ) | n v−→ Λ, ϕ ∈ T (v)}. ϕ

RT ((v , ϕ), (v ′, ϕ′)) := {θ ∈ T (v+v ′) | θ ` ϕ, θ ` ϕ′, θ is functional}

θ ϕ` θ ψ` + another logical condition

One shows that this syntactic category is regular.

E.g. for each v , the poset T (v) is automatically a meet-semilattice.

Why? Any function vf−→ w is an adjoint in Cospan...

... so T (f ) will be an adjoint in Poset. Thus we get adjunctions:

1 T (0) T (v)ρ1

⇐ ⇐

T (v)× T (v) T (v + v) T (v)ρv,v

⇐ ⇐

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 17 / 19

Bringing it all together Recalling and justifying the theorem

Why it works

One can form the syntactic category RT of T : CospanΛ → Poset.

Ob(RT ) := {(v , ϕ) | n v−→ Λ, ϕ ∈ T (v)}. ϕ

RT ((v , ϕ), (v ′, ϕ′)) := {θ ∈ T (v+v ′) | θ ` ϕ, θ ` ϕ′, θ is functional}

θ ϕ` θ ψ` + another logical condition

One shows that this syntactic category is regular.

E.g. for each v , the poset T (v) is automatically a meet-semilattice.

Why? Any function vf−→ w is an adjoint in Cospan...

... so T (f ) will be an adjoint in Poset. Thus we get adjunctions:

1 T (0) T (v)ρ1

⇐ ⇐

T (v)× T (v) T (v + v) T (v)ρv,v

⇐ ⇐

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 17 / 19

Bringing it all together Recalling and justifying the theorem

Why it works

One can form the syntactic category RT of T : CospanΛ → Poset.

Ob(RT ) := {(v , ϕ) | n v−→ Λ, ϕ ∈ T (v)}. ϕ

RT ((v , ϕ), (v ′, ϕ′)) := {θ ∈ T (v+v ′) | θ ` ϕ, θ ` ϕ′, θ is functional}

θ ϕ` θ ψ` + another logical condition

One shows that this syntactic category is regular.

E.g. for each v , the poset T (v) is automatically a meet-semilattice.

Why? Any function vf−→ w is an adjoint in Cospan...

... so T (f ) will be an adjoint in Poset. Thus we get adjunctions:

1 T (0) T (v)ρ1

⇐ ⇐

T (v)× T (v) T (v + v) T (v)ρv,v

⇐ ⇐

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 17 / 19

Bringing it all together Recalling and justifying the theorem

Why it works

One can form the syntactic category RT of T : CospanΛ → Poset.

Ob(RT ) := {(v , ϕ) | n v−→ Λ, ϕ ∈ T (v)}. ϕ

RT ((v , ϕ), (v ′, ϕ′)) := {θ ∈ T (v+v ′) | θ ` ϕ, θ ` ϕ′, θ is functional}

θ ϕ` θ ψ` + another logical condition

One shows that this syntactic category is regular.

E.g. for each v , the poset T (v) is automatically a meet-semilattice.

Why? Any function vf−→ w is an adjoint in Cospan...

... so T (f ) will be an adjoint in Poset. Thus we get adjunctions:

1 T (0) T (v)ρ1

⇐ ⇐

T (v)× T (v) T (v + v) T (v)ρv,v

⇐ ⇐

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 17 / 19

Bringing it all together Recalling and justifying the theorem

Why it works

One can form the syntactic category RT of T : CospanΛ → Poset.

Ob(RT ) := {(v , ϕ) | n v−→ Λ, ϕ ∈ T (v)}. ϕ

RT ((v , ϕ), (v ′, ϕ′)) := {θ ∈ T (v+v ′) | θ ` ϕ, θ ` ϕ′, θ is functional}

θ ϕ` θ ψ` + another logical condition

One shows that this syntactic category is regular.

E.g. for each v , the poset T (v) is automatically a meet-semilattice.

Why? Any function vf−→ w is an adjoint in Cospan...

... so T (f ) will be an adjoint in Poset. Thus we get adjunctions:

1 T (0) T (v)ρ1

⇐ ⇐

T (v)× T (v) T (v + v) T (v)ρv,v

⇐ ⇐

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 17 / 19

Bringing it all together Recalling and justifying the theorem

Why it works

One can form the syntactic category RT of T : CospanΛ → Poset.

Ob(RT ) := {(v , ϕ) | n v−→ Λ, ϕ ∈ T (v)}. ϕ

RT ((v , ϕ), (v ′, ϕ′)) := {θ ∈ T (v+v ′) | θ ` ϕ, θ ` ϕ′, θ is functional}

θ ϕ` θ ψ` + another logical condition

One shows that this syntactic category is regular.

E.g. for each v , the poset T (v) is automatically a meet-semilattice.

Why? Any function vf−→ w is an adjoint in Cospan...

... so T (f ) will be an adjoint in Poset. Thus we get adjunctions:

1 T (0) T (v)ρ1

⇐ ⇐

T (v)× T (v) T (v + v) T (v)ρv,v

⇐ ⇐

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 17 / 19

Bringing it all together Concluding

Conjecture and outlook

We conjecture that this story extends to coherent and geometric logic.

Conjecture

The 2-category of categories is reflective in that of: lax monoidal2-functors Cospanco → whose composite with → Poset is ajax.

Dropping the ajax condition may give something like quantaloids.

Landing in categories other than Poset gives “fuzzy regcats.”

E.g. Cospan→ LawvMetSp: “distance to entailment” ϕ `17 ψ.Other quantales (e.g. powerset of a monoid) give other fuzz.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 18 / 19

Bringing it all together Concluding

Conjecture and outlook

We conjecture that this story extends to coherent and geometric logic.

Conjecture

The 2-category of coherent categories is reflective in that of:

lax monoidal2-functors Cospanco → J Lat whose composite with J Lat→ Poset is ajax.

Dropping the ajax condition may give something like quantaloids.

Landing in categories other than Poset gives “fuzzy regcats.”

E.g. Cospan→ LawvMetSp: “distance to entailment” ϕ `17 ψ.Other quantales (e.g. powerset of a monoid) give other fuzz.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 18 / 19

Bringing it all together Concluding

Conjecture and outlook

We conjecture that this story extends to coherent and geometric logic.

Conjecture

The 2-category of coherent categories is reflective in that of: lax monoidal2-functors Cospanco → J Lat

whose composite with J Lat→ Poset is ajax.

Dropping the ajax condition may give something like quantaloids.

Landing in categories other than Poset gives “fuzzy regcats.”

E.g. Cospan→ LawvMetSp: “distance to entailment” ϕ `17 ψ.Other quantales (e.g. powerset of a monoid) give other fuzz.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 18 / 19

Bringing it all together Concluding

Conjecture and outlook

We conjecture that this story extends to coherent and geometric logic.

Conjecture

The 2-category of coherent categories is reflective in that of: lax monoidal2-functors Cospanco → J Lat whose composite with J Lat→ Poset is ajax.

Dropping the ajax condition may give something like quantaloids.

Landing in categories other than Poset gives “fuzzy regcats.”

E.g. Cospan→ LawvMetSp: “distance to entailment” ϕ `17 ψ.Other quantales (e.g. powerset of a monoid) give other fuzz.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 18 / 19

Bringing it all together Concluding

Conjecture and outlook

We conjecture that this story extends to coherent and geometric logic.

Conjecture

The 2-category of geometric categories is reflective in that of: laxmonoidal 2-functors Cospanco → SupLat whose composite withSupLat → Poset is ajax.

Dropping the ajax condition may give something like quantaloids.

Landing in categories other than Poset gives “fuzzy regcats.”

E.g. Cospan→ LawvMetSp: “distance to entailment” ϕ `17 ψ.Other quantales (e.g. powerset of a monoid) give other fuzz.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 18 / 19

Bringing it all together Concluding

Conjecture and outlook

We conjecture that this story extends to coherent and geometric logic.

Conjecture

The 2-category of geometric categories is reflective in that of: laxmonoidal 2-functors Cospanco → SupLat whose composite withSupLat → Poset is ajax.

Dropping the ajax condition may give something like quantaloids.

Landing in categories other than Poset gives “fuzzy regcats.”

E.g. Cospan→ LawvMetSp: “distance to entailment” ϕ `17 ψ.Other quantales (e.g. powerset of a monoid) give other fuzz.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 18 / 19

Bringing it all together Concluding

Conjecture and outlook

We conjecture that this story extends to coherent and geometric logic.

Conjecture

The 2-category of geometric categories is reflective in that of: laxmonoidal 2-functors Cospanco → SupLat whose composite withSupLat → Poset is ajax.

Dropping the ajax condition may give something like quantaloids.

Landing in categories other than Poset gives “fuzzy regcats.”

E.g. Cospan→ LawvMetSp: “distance to entailment” ϕ `17 ψ.Other quantales (e.g. powerset of a monoid) give other fuzz.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 18 / 19

Bringing it all together Concluding

Summary

Formulas in regular logic looks like this:

∃b. f (a, b) ∧ g(b, a′) `a,a′ ∃c . h(c , a) ∧ a = a′.

Such things can be represented pictorially in a regular hypercat:

f ga a′b

h

c

a a′`a,a′

i.e. as an inequality of elements in an ajax monoidal 2-functor

T : Cospanco → Poset.

We have 2-reflectivity, suggesting that the diagram language is robust.

Thanks! Comments and questions welcome.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 19 / 19

Bringing it all together Concluding

Summary

Formulas in regular logic looks like this:

∃b. f (a, b) ∧ g(b, a′) `a,a′ ∃c . h(c , a) ∧ a = a′.

Such things can be represented pictorially in a regular hypercat:

f ga a′b

h

c

a a′`a,a′

i.e. as an inequality of elements in an ajax monoidal 2-functor

T : Cospanco → Poset.

We have 2-reflectivity, suggesting that the diagram language is robust.

Thanks! Comments and questions welcome.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 19 / 19

Bringing it all together Concluding

Summary

Formulas in regular logic looks like this:

∃b. f (a, b) ∧ g(b, a′) `a,a′ ∃c . h(c , a) ∧ a = a′.

Such things can be represented pictorially in a regular hypercat:

f ga a′b

h

c

a a′`a,a′

i.e. as an inequality of elements in an ajax monoidal 2-functor

T : Cospanco → Poset.

We have 2-reflectivity, suggesting that the diagram language is robust.

Thanks! Comments and questions welcome.

David I. Spivak String diagrams for regular logic Presented on 2018/10/27 19 / 19