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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