String diagrams for regular logic David I. Spivak (joint with - - PowerPoint PPT Presentation

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 Introduction

Application: playing with logic Implications for string diagrams String 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 playing seamlessly with logic. A computer database held relevant information. Cruise could pull it up, and manipulate it, to solve crimes.

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 playing seamlessly 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 works

company

tiny

person person acquainted

⊢

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

Introduction Application: playing with logic

Sample scenario

Assume:

works works

company

tiny

person person acquainted

⊢ Ci Ci Ci

company

=

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

Introduction Application: playing with logic

Sample scenario

Assume:

works works

company

tiny

person person acquainted

⊢ Ci Ci Ci

company

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

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

Introduction Application: playing with logic

Sample scenario

Assume:

works works

company

tiny

person person acquainted

⊢ 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 tiny 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

Two Ci’s grouped:

David works Ryan works Ci Ci tiny 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 tiny true =

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 tiny true =

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 tiny true =

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 tiny 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 Ryan acquainted true =

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.

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.

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

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 c

2

t

1 1

u

3 2

v

4 3

w

1 4

x

1 2

y

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

• uter

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 Cospanco

Λ .

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 Cospanco

Λ .

Objects: arities n

v

− → Λ, i.e. lists (v(1), . . . , v(n)) ∈ Λn. 1-morphisms: n1 n12 n2 Λ

v1 v2

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

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 Cospanco

Λ .

Objects: arities n

v

− → Λ, i.e. lists (v(1), . . . , v(n)) ∈ Λn. 1-morphisms: n1 n12 n2 Λ

v1 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 : Cospanco

Λ → Poset

such that the laxators are right adjoints.

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 : Cospanco

Λ → 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 : Cospanco

Λ → 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 D

R ⊤ L

C D

R ⊥ 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).

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 D

R ⊤ L

C D

R ⊥ 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 D

R

⇐

L

C D

R

⇐

L

D C

L

⇒

R

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

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 D

R ⊤ 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 D

R

⇐

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 : Cospanco

Λ → Poset.

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 : Cospanco

Λ → Poset.

Theorem There is an adjunction Cospan-Alg RegCat

Φ

⇒

Ψ

, such that for any regular category R, the counit Ψ(Φ(R)) → R is an equivalence 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 logic

Regular categories Regular 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.

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...

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 !

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 is the category of adjunctions in Rel.

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 is the 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

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:Ak R1(a1, . . . , ak) : Prop

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:Ak R1(a1, . . . , ak) : Prop Operations ∧, true , =, and ∃, from which to build up formulas ϕ, ψ.

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:Ak R1(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.

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:Ak R1(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”:

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:Ak R1(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

true a f b a ⊢a:A

true ⊢a:A ∃(b : B). f (a, b)

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

true a f b a ⊢a:A

true ⊢a:A ∃(b : B). f (a, b)

f f a b1 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 together

Where are we? Recalling and justifying the theorem Concluding

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.

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.

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 : Cospanco

Λ → Poset.

Theorem There is an adjunction Cospan-Alg RegCat ⇒ , such that Ψ(Φ(R)) → R is an equivalence for any regular category R.

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 : Cospanco

Λ → 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.

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 : Cospanco

Λ → 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.

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 : Cospanco

Λ → 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 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 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.

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 v

f

− → w is an adjoint in Cospan... ... so T(f ) will be an adjoint in Poset. Thus we get adjunctions:

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 v

f

− → 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

⇐ ⇐

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 v

f

− → 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.

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:

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 monoidal 2-functors Cospanco → J Lat

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 monoidal 2-functors Cospanco → J Lat whose composite with J Lat → Poset is ajax.

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: lax monoidal 2-functors Cospanco → SupLat whose composite with SupLat → Poset is ajax.

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: lax monoidal 2-functors Cospanco → SupLat whose composite with SupLat → Poset is ajax. Dropping the ajax condition may give something like quantaloids.

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: lax monoidal 2-functors Cospanco → SupLat whose composite with SupLat → 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′.

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 g a a′ b h c a a′ ⊢a,a′

i.e. as an inequality of elements in an ajax monoidal 2-functor T : Cospanco → Poset.

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 g a 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