Than Abstract Nonsense: a Category-Theoretic Sketch of the Syntactic Category System

Introduction Methodology Category Theory Syntactic Category System Conclusion

More than abstract nonsense: A Category-theoretic sketch of the syntactic category system

Theoretical and Applied Linguistics University of Cambridge

SyntaxLab, 29 January 2019

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Overview Introduction Methodology Category Theory Category Functor and Natural Transformation Adjunction Syntactic Category System Cross-functional-hierarchy parallelism Global interconnection Entire SCS Conclusion

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

The Syntactic Category System (SCS)

V K Q C P Cl n Foc v √ N Asp T A ....

Figure 1: The universe of syntactic categories.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system #parallelism

#granularity

I Parallel hierarchies: V–v–T–C N–n–Num–D. . .

I Stacked hierarchies: v–C >> V–v–T–C >> V–Appl–Voice–Asp–Tns–Mod–Fin–Foc–Top >> ... N–n–Num–D vs. N–n–Cl–D Flexible hierarchies: I V–v–T–C vs. V–v–Asp–C RSCS has an intuitively rich ontological structure.

Introduction Methodology Category Theory Syntactic Category System Conclusion

The SCS is not just a set

I Functional hierarchies aka. extended projections, hierarchies of projections, etc.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system #granularity

#parallelism

Introduction Methodology Category Theory Syntactic Category System Conclusion

The SCS is not just a set

I Functional hierarchies aka. extended projections, hierarchies of projections, etc.

I Parallel hierarchies: V–v–T–C N–n–Num–D. . .

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system #parallelism

#granularity

N–n–Num–D vs. N–n–Cl–D Flexible hierarchies: I V–v–T–C vs. V–v–Asp–C RSCS has an intuitively rich ontological structure.

Introduction Methodology Category Theory Syntactic Category System Conclusion

The SCS is not just a set

I Functional hierarchies aka. extended projections, hierarchies of projections, etc.

I Parallel hierarchies: V–v–T–C N–n–Num–D. . .

I Stacked hierarchies: v–C >> V–v–T–C >> V–Appl–Voice–Asp–Tns–Mod–Fin–Foc–Top >> ...

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system #parallelism

#granularity

RSCS has an intuitively rich ontological structure.

Introduction Methodology Category Theory Syntactic Category System Conclusion

The SCS is not just a set

I Functional hierarchies aka. extended projections, hierarchies of projections, etc.

I Parallel hierarchies: V–v–T–C N–n–Num–D. . .

I Stacked hierarchies: v–C >> V–v–T–C >> V–Appl–Voice–Asp–Tns–Mod–Fin–Foc–Top >> ... N–n–Num–D vs. N–n–Cl–D Flexible hierarchies: I V–v–T–C vs. V–v–Asp–C

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system #parallelism

#granularity

Introduction Methodology Category Theory Syntactic Category System Conclusion

The SCS is not just a set

I Functional hierarchies aka. extended projections, hierarchies of projections, etc.

I Parallel hierarchies: V–v–T–C N–n–Num–D. . .

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

The SCS is not just a set

I Functional hierarchies aka. extended projections, hierarchies of projections, etc.

I Parallel hierarchies: #parallelism V–v–T–C N–n–Num–D. . .

I Stacked hierarchies: #granularity v–C >> V–v–T–C >> V–Appl–Voice–Asp–Tns–Mod–Fin–Foc–Top >> ... N–n–Num–D vs. N–n–Cl–D Flexible hierarchies: I V–v–T–C vs. V–v–Asp–C RSCS has an intuitively rich ontological structure.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system I What category is parallel with what? #parallelism V–Appl–Voice–Asp–Tns–Mod–Fin–Foc–Top N–Gen–n–Cl–Num–Q–Det–K I Which hierarchy is stacked on which? #granularity Res–Proc–Init–v–T–C >> vs. V–v–Asp–Tns–Mod–C >> V–v–Asp–Tns–Mod–C Res–Proc–Init–v–T–C

I More subtle than ﬁrst impression

RCross-hierarchy relations are rather intricate.

Introduction Methodology Category Theory Syntactic Category System Conclusion

Ontological structure of the SCS

I Independent of concrete derivations (hence part of lexicon)

Introduction Methodology Category Theory Syntactic Category System Conclusion

Ontological structure of the SCS

I Independent of concrete derivations (hence part of lexicon) I More subtle than ﬁrst impression

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system I Which hierarchy is stacked on which? #granularity Res–Proc–Init–v–T–C >> vs. V–v–Asp–Tns–Mod–C >> V–v–Asp–Tns–Mod–C Res–Proc–Init–v–T–C RCross-hierarchy relations are rather intricate.

Introduction Methodology Category Theory Syntactic Category System Conclusion

Ontological structure of the SCS

I Independent of concrete derivations (hence part of lexicon) I More subtle than ﬁrst impression

I What category is parallel with what? #parallelism V–Appl–Voice–Asp–Tns–Mod–Fin–Foc–Top N–Gen–n–Cl–Num–Q–Det–K

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system RCross-hierarchy relations are rather intricate.

Introduction Methodology Category Theory Syntactic Category System Conclusion

Ontological structure of the SCS

I Independent of concrete derivations (hence part of lexicon) I More subtle than ﬁrst impression

I What category is parallel with what? #parallelism V–Appl–Voice–Asp–Tns–Mod–Fin–Foc–Top N–Gen–n–Cl–Num–Q–Det–K I Which hierarchy is stacked on which? #granularity Res–Proc–Init–v–T–C >> vs. V–v–Asp–Tns–Mod–C >> V–v–Asp–Tns–Mod–C Res–Proc–Init–v–T–C

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Ontological structure of the SCS

I Independent of concrete derivations (hence part of lexicon) I More subtle than ﬁrst impression

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system I The ontological structure is layered: individual category → individual hierarchy → individual granularity level → multiple granularity levels → entire SCS #“ladder of abstraction”

I A crucial concept: order relation

Introduction Methodology Category Theory Syntactic Category System Conclusion

More abstract aspects of the SCS ontological structure

I Parallelism and granularity stacking are complementary

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system #“ladder of abstraction”

I A crucial concept: order relation

Introduction Methodology Category Theory Syntactic Category System Conclusion

More abstract aspects of the SCS ontological structure

I Parallelism and granularity stacking are complementary

I The ontological structure is layered: individual category → individual hierarchy → individual granularity level → multiple granularity levels → entire SCS

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system I A crucial concept: order relation

Introduction Methodology Category Theory Syntactic Category System Conclusion

More abstract aspects of the SCS ontological structure

I Parallelism and granularity stacking are complementary

I The ontological structure is layered: individual category → individual hierarchy → individual granularity level → multiple granularity levels → entire SCS #“ladder of abstraction”

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

More abstract aspects of the SCS ontological structure

I Parallelism and granularity stacking are complementary

I The ontological structure is layered: individual category → individual hierarchy → individual granularity level → multiple granularity levels → entire SCS #“ladder of abstraction”

I A crucial concept: order relation

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system I It reﬂects acquisition: (Biberauer & Roberts 2015)

Figure 2: Successive category division results in stacked hierarchies.

I It links cognitive domains: “universal spine” Classiﬁcation–PoV–Anchoring–Linking (Wiltschko 2014)

Introduction Methodology Category Theory Syntactic Category System Conclusion

Why is the SCS ontological structure worth studying?

I It underlies derivation: ﬁrst-Merge position T–C on f-hierarchy ⇒ T–C in concrete derivation

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system I It links cognitive domains: “universal spine” Classiﬁcation–PoV–Anchoring–Linking (Wiltschko 2014)

Introduction Methodology Category Theory Syntactic Category System Conclusion

Why is the SCS ontological structure worth studying?

I It underlies derivation: ﬁrst-Merge position T–C on f-hierarchy ⇒ T–C in concrete derivation I It reﬂects acquisition: (Biberauer & Roberts 2015)

Figure 2: Successive category division results in stacked hierarchies.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Why is the SCS ontological structure worth studying?

I It underlies derivation: ﬁrst-Merge position T–C on f-hierarchy ⇒ T–C in concrete derivation I It reﬂects acquisition: (Biberauer & Roberts 2015)

Figure 2: Successive category division results in stacked hierarchies.

I It links cognitive domains: “universal spine” Classiﬁcation–PoV–Anchoring–Linking (Wiltschko 2014)

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

In this talk, I will

I Explore the SCS ontological structure

I Formalize hierarchies and cross-hierarchy relations

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system I It suits our task: functional hierarchies are ordered sets

I A branch of math is dedicated to studying abstract structures:

Category theory is [. . . ] unmatched in its ability to organize and layer abstractions, to ﬁnd commonalities between structures of all sorts, and [. . . ] it has also been branching out into science, informatics, and industry. We believe that it has the potential to be a major cohesive force in the world, building rigorous bridges between disparate worlds, both theoretical and practical. (Fong & Spivak 2018)

Introduction Methodology Category Theory Syntactic Category System Conclusion

Mathematics is the professional tool to study structures

I It is rigorous and can make intuitions explicit

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system I A branch of math is dedicated to studying abstract structures:

Introduction Methodology Category Theory Syntactic Category System Conclusion

Mathematics is the professional tool to study structures

I It is rigorous and can make intuitions explicit

I It suits our task: functional hierarchies are ordered sets

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Mathematics is the professional tool to study structures

I It is rigorous and can make intuitions explicit

I It suits our task: functional hierarchies are ordered sets

I A branch of math is dedicated to studying abstract structures:

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system RNB the layered levels of abstraction.

Level 4: Adjunction (Category comparison).

Introduction Methodology Category Theory Syntactic Category System Conclusion

Category Theory Roadmap

“Big 3”: Category, Functor, and Natural Transformation.

I A Category is like a universe of discourse.

I A Functor connects two such universes.

I A Natural Transformation connects two such Functors.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system RNB the layered levels of abstraction.

Level 4: Adjunction (Category comparison).

Introduction Methodology Category Theory Syntactic Category System Conclusion

Category Theory Roadmap

“Big 3”: Category, Functor, and Natural Transformation.

I A Category is like a universe of discourse.

I A Functor connects two such universes.

I A Natural Transformation connects two such Functors.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system RNB the layered levels of abstraction.

Level 4: Adjunction (Category comparison).

Introduction Methodology Category Theory Syntactic Category System Conclusion

Category Theory Roadmap

“Big 3”: Category, Functor, and Natural Transformation.

I A Category is like a universe of discourse.

I A Functor connects two such universes.

I A Natural Transformation connects two such Functors.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system RNB the layered levels of abstraction.

Level 4: Adjunction (Category comparison).

Introduction Methodology Category Theory Syntactic Category System Conclusion

Category Theory Roadmap

“Big 3”: Category, Functor, and Natural Transformation.

I A Category is like a universe of discourse.

I A Functor connects two such universes.

I A Natural Transformation connects two such Functors.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Level 4: Adjunction (Category comparison).

Introduction Methodology Category Theory Syntactic Category System Conclusion

Category Theory Roadmap

“Big 3”: Category, Functor, and Natural Transformation.

I A Category is like a universe of discourse.

I A Functor connects two such universes.

I A Natural Transformation connects two such Functors. RNB the layered levels of abstraction.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Category Theory Roadmap

“Big 3”: Category, Functor, and Natural Transformation.

I A Category is like a universe of discourse.

I A Functor connects two such universes.

I A Natural Transformation connects two such Functors. RNB the layered levels of abstraction.

Level 4: Adjunction (Category comparison).

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system RAnything that satisﬁes this deﬁnition is a Category.

Introduction Methodology Category Theory Syntactic Category System Conclusion

Category Category Theory “Big 3” A Category C has objects and arrows (aka. morphisms), where

I Each object C has an identity arrow idC

I Arrows compose I Composition obeys two coherence conditions

I Associativity: h ◦ (g ◦ f) = (h ◦ g) ◦ f I Unit law: idB ◦ f = f, g ◦ idB = g #commutative diagram A f B A f B ) f f

◦ h◦g ◦ ) g g g g idB ( ◦

◦ f h ( h g◦f g D C B C h

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Category Category Theory “Big 3” A Category C has objects and arrows (aka. morphisms), where

I Each object C has an identity arrow idC

I Arrows compose I Composition obeys two coherence conditions

I Associativity: h ◦ (g ◦ f) = (h ◦ g) ◦ f I Unit law: idB ◦ f = f, g ◦ idB = g #commutative diagram A f B A f B ) f f

◦ h◦g ◦ ) g g g g idB ( ◦

◦ f h ( h g◦f g D C B C h RAnything that satisﬁes this deﬁnition is a Category.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system #monoid

#poset

A one-object-many-arrow Category M

f • id

...

A two-object-three-arrow Category 2

id • • id

Introduction Methodology Category Theory Syntactic Category System Conclusion

Category Examples A one-object-one-arrow Category 1

• id

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system #poset

#monoid

A two-object-three-arrow Category 2

id • • id

Introduction Methodology Category Theory Syntactic Category System Conclusion

Category Examples A one-object-one-arrow Category 1

• id

A one-object-many-arrow Category M

f • id

...

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system #monoid

#poset

Introduction Methodology Category Theory Syntactic Category System Conclusion

Category Examples A one-object-one-arrow Category 1

• id

A one-object-many-arrow Category M

f • id

...

A two-object-three-arrow Category 2

id • • id

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Category Examples A one-object-one-arrow Category 1

• id

A one-object-many-arrow Category M #monoid

f • id

...

A two-object-three-arrow Category 2 #poset

id • • id

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Category Examples A one-object-one-arrow Category 1

• id

A one-object-many-arrowPartially ordered set (poset) Category M #monoid

A set equipped with a partialg order relation v reﬂexive, transitive, antisymmetric I f • id

...

A two-object-three-arrow Category 2 #poset

id • • id

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system #discrete Category

I Every set is also a Category

id • id • id •

id • id • id • ...

The Category Pos of all posets and monotone functions

id id id id id id

• • • • • • ... 1 2 3 ...

I Every poset is also a Category #poset Category

Introduction Methodology Category Theory Syntactic Category System Conclusion

Category Examples The Category Set of all (small) sets and functions

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system #discrete Category

The Category Pos of all posets and monotone functions

id id id id id id

• • • • • • ... 1 2 3 ...

I Every poset is also a Category #poset Category

Introduction Methodology Category Theory Syntactic Category System Conclusion

Category Examples The Category Set of all (small) sets and functions

I Every set is also a Category

id • id • id •

id • id • id • ...

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system The Category Pos of all posets and monotone functions

id id id id id id

• • • • • • ... 1 2 3 ...

I Every poset is also a Category #poset Category

Introduction Methodology Category Theory Syntactic Category System Conclusion

Category Examples The Category Set of all (small) sets and functions

I Every set is also a Category #discrete Category

id • id • id •

id • id • id • ...

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system I Every poset is also a Category #poset Category

Introduction Methodology Category Theory Syntactic Category System Conclusion

Category Examples The Category Set of all (small) sets and functions

I Every set is also a Category #discrete Category

id • id • id •

id • id • id • ...

The Category Pos of all posets and monotone functions

id id id id id id

• • • • • • ... 1 2 3 ...

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Category Examples The Category Set of all (small) sets and functions

I Every set is also a Category #discrete Category

id • id • id •

id • id • id • ...

The Category Pos of all posets and monotone functions

id id id id id id

• • • • • • ... 1 2 3 ...

I Every poset is also a Category #poset Category

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Category Examples The Category Set of all (small) sets and functions

I Every set is also a Category #discrete Category

id • id • id •

id • id • id • ...

The Category Pos of all posets and monotone functions Monotone function id id id id id id A function f: A → B that is order-preserving

•I ∀x, y •∈ A, x v•y ⇒ f(x)• v f(y) • • ... 1 2 3 ...

I Every poset is also a Category #poset Category

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Category A functional hierarchy is a poset Category

I Objects: individual functional categories

I Arrows: instances of partial order relation

I Composition: by transitivity

I Identities: by reﬂexivity

id id id id id id id id

V v T C N n Num D

id id id id id id id

V Appl Voice Asp Tns Foc Top

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system RThis describes a speaker’s functional category inventory.

Introduction Methodology Category Theory Syntactic Category System Conclusion

Category Several f-hierarchies form a Category of posets NB this in itself is not a poset Category (but a preorder Category).

I Objects: individual f-hierarchies AN, AV, etc.

I Arrows: monotone functions (tbc)

I Composition: by monotone function composition

I Identities: identity monotone functions

id AN AV id

id AP AA id ...

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Category Several f-hierarchies form a Category of posets NB this in itself is not a poset Category (but a preorder Category).

I Objects: individual f-hierarchies AN, AV, etc.

I Arrows: monotone functions (tbc)

I Composition: by monotone function composition

I Identities: identity monotone functions

id AN AV id

id AP AA id ...

RThis describes a speaker’s functional category inventory.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Category Several f-hierarchies form a Category of posets NB this in itself is not a poset Category (but a preorder Category).

I Objects: individual f-hierarchies AN, AV, etc.

I Arrows: monotone functions (tbc)

I Composition: by monotone function composition

I Identities: identity monotone functions

id AN AV id

id AP AA id ...

RThis describes a speaker’s functional category inventory.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Category Several f-hierarchies form a Category of posets NB this in itself is not a poset Category (but a preorder Category). Preordered set (preorder) I Objects: individual f-hierarchies AN, AV, etc. A set equipped with a preorder relation I Arrows: monotone functions (tbc) I reﬂexive and transtive I Composition: by monotone function composition I partial order without antisymmetry I Identities: identity monotone functions

id AN AV id

id AP AA id ...

RThis describes a speaker’s functional category inventory.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

id AN AV id Lmerely expository

id AP AA id ...

RThis describes a speaker’s functional category inventory.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Functor and Natural Transformation Category Theory “Big 3”

A Functor F: C → D is an arrow between two Categories.

I It maps objects to objects and arrows to arrows

I It preserves composition and identities C F D

f idB F(f) idA A B F(A) F(B) idF(B) idF(A) g F(g) h F(h)

C idC F(C) idF(C)

RA Functor produces an image of one Category in another.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system I The Functor F: Set → Pos sends sets to the smallest posets built on them (ordered by =). #free Functor

I The Functor M · : Syn → Sem sends syntactic expressions (atomic or phrasal)J K to their meanings. #interpretation Functor

Introduction Methodology Category Theory Syntactic Category System Conclusion

Functor and Natural Transformation Examples

I The Functor U: Pos → Set sends posets to their underlying sets by forgetting the partial orders. #forgetful Functor

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system I The Functor M · : Syn → Sem sends syntactic expressions (atomic or phrasal)J K to their meanings. #interpretation Functor

Introduction Methodology Category Theory Syntactic Category System Conclusion

Functor and Natural Transformation Examples

I The Functor U: Pos → Set sends posets to their underlying sets by forgetting the partial orders. #forgetful Functor

I The Functor F: Set → Pos sends sets to the smallest posets built on them (ordered by =). #free Functor

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Functor and Natural Transformation Examples

I The Functor U: Pos → Set sends posets to their underlying sets by forgetting the partial orders. #forgetful Functor

I The Functor F: Set → Pos sends sets to the smallest posets built on them (ordered by =). #free Functor

I The Functor M · : Syn → Sem sends syntactic expressions (atomic or phrasal)J K to their meanings. #interpretation Functor

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Functor and Natural Transformation Monotone functions qua Functors (identities omitted)

Figure 3: A random monotone function (Fong & Spivak 2018).

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Functor and Natural Transformation Monotone functions qua Functors The Functor R: Ani → Tax sends animals to taxonomic ranks.

Figure 4: A Functor for biological taxonomy (Fong & Spivak 2018).

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Functor and Natural Transformation Functors between f-hierarchy Categories

The Functor F: AV → AN maps verbal categories to nominal ones

AV V v T C

AN N n Num D

B Don’t read too much into the speciﬁc mapping (yet).

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system What is a meaningful Functor between f-hierarchies? (tbc)

Introduction Methodology Category Theory Syntactic Category System Conclusion

Functor and Natural Transformation Functors between f-hierarchy Categories

Another Functor between AV and AN

AV V v T C

AN N n Num D

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system What is a meaningful Functor between f-hierarchies? (tbc)

Introduction Methodology Category Theory Syntactic Category System Conclusion

Functor and Natural Transformation Functors between f-hierarchy Categories

Another Functor between AV and AN

AV V v T C

G #constant Functor

AN N n Num D

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Functor and Natural Transformation Functors between f-hierarchy Categories

Another Functor between AV and AN

AV V v T C

G #constant Functor

AN N n Num D

What is a meaningful Functor between f-hierarchies? (tbc)

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system What is a stably meaningful Functor between f-hierarchies?

REssentially a question about cross-hierarchy parallelism.

Introduction Methodology Category Theory Syntactic Category System Conclusion

Functor and Natural Transformation Functors between f-hierarchy Categories

And this one?

BV V Appl v Asp Tns Mod Fin Foc Top

BN N Gen n Cl Num Q D

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system REssentially a question about cross-hierarchy parallelism.

Introduction Methodology Category Theory Syntactic Category System Conclusion

Functor and Natural Transformation Functors between f-hierarchy Categories

And this one?

BV V Appl v Asp Tns Mod Fin Foc Top

BN N Gen n Cl Num Q D

What is a stably meaningful Functor between f-hierarchies?

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Functor and Natural Transformation Functors between f-hierarchy Categories

And this one?

BV V Appl v Asp Tns Mod Fin Foc Top

BN N Gen n Cl Num Q D

What is a stably meaningful Functor between f-hierarchies?

REssentially a question about cross-hierarchy parallelism.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Functor and Natural Transformation Category Theory “Big 3” A Natural Transformation α: F ⇒ G is an arrow between Functors.

I It is a transformation between two Functorial images of C

I It is a family of maps αA in D 0 I For any f: A → A in C there is a naturality square in D

F C α D G F(f) A f A0 F(A) F(A0)

αA αA0 G(f) G(A) G(A0)

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system there is a B-arrow F(A) → B iﬀ there is an A-arrow A → G(B)

RThis describes a weak similarity between Categories.

Introduction Methodology Category Theory Syntactic Category System Conclusion

Adjunction Adjunction (aka. Adjointness or Adjoint Situation)

F In the conﬁguration A B, we say that F is left adjoint to G G and G is right adjoint to F, and write F a G, if

B(F(A), B) =∼ A(A, G(B)) (θ)

naturally in A ∈ A and B ∈ B.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system RThis describes a weak similarity between Categories.

Introduction Methodology Category Theory Syntactic Category System Conclusion

Adjunction Adjunction (aka. Adjointness or Adjoint Situation)

F In the conﬁguration A B, we say that F is left adjoint to G G and G is right adjoint to F, and write F a G, if

B(F(A), B) =∼ A(A, G(B)) (θ)

naturally in A ∈ A and B ∈ B.

there is a B-arrow F(A) → B iﬀ there is an A-arrow A → G(B)

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Adjunction Adjunction (aka. Adjointness or Adjoint Situation)

F In the conﬁguration A B, we say that F is left adjoint to G G and G is right adjoint to F, and write F a G, if

B(F(A), B) =∼ A(A, G(B)) (θ)

naturally in A ∈ A and B ∈ B.

there is a B-arrow F(A) → B iﬀ there is an A-arrow A → G(B)

RThis describes a weak similarity between Categories.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Adjunction Adjunction (aka. Adjointness or Adjoint Situation)

F InAdjunction the conﬁguration via hom-setA isomorphismB, we say that F is left adjoint to G G and G is right adjoint to F, and write F a G, if F g B(F(A), B) ∼ A(A, G(B))F(A) −→ B (θ) a = A B ↓θ ↑θ−1 f naturally in A ∈GA and B ∈ B. A −→ G(B) A F(A)

I Hom-set C(X, Y) theref is a B-arrow F(A) → gB iﬀ there is an A-arrow A →,G(B) ∼ θ = set of all C-arrows X → Y RG(ThisB) describes a weak similarityB between⇒ Isomorphism Categories.≡ bijection

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Adjunction Adjunction (aka. Adjointness or Adjoint Situation) Since the isomorphism( θ) is natural in A and B, we can

I let A = G(B) and get

g F(G(B)) → B F(A) −→ B ↓θ ↑θ−1 f f=id A −→ G(B) G(B) −−−−−−→G(B) G(B)

I let B = F(A) and get

g=id F(A) −−−−−−→F(A) F(A) A → G(F(A))

i.e. we use( θ) to map identity arrows.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Adjunction Adjunction (aka. Adjointness or Adjoint Situation) Since the isomorphism( θ) is natural in A and B, we can

AdjunctionI let A = viaG( unitB) and and get co-unit

εB g=idF(A) F(G(B)) → B F(G(B)) −→ B F(A) −−−−−−→ F(A) ↑ ↓ F f=id ηA f=idG(B) G(B)

a G(B) −−−−−−→ G(B) A −→ G(F(A)) AG(B) −−−−−−→ GB(B)

G F(f) I let B = F(A) and getA F(A) F(G(B)) idF(A) ηA f g=idF(A) g ε F(A)id−−−−−−→ F(A) B G(F(A)) G(B) G(B) B G(g) A → G(F(A))

i.e. we use( θ) to map identity arrows.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system RThese are the unit and co-unit of an Adjunction.

Introduction Methodology Category Theory Syntactic Category System Conclusion

Adjunction Adjunction (aka. Adjointness or Adjoint Situation)

The two special arrows

ηA : A → G(F(A))

εB : F(G(B)) → B

extend to two Natural Transformations (between two endofunctors)

η: IdA ⇒ G ◦ F

ε: F ◦ G ⇒ IdB

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system RThese are the unit and co-unit of an Adjunction.

Introduction Methodology Category Theory Syntactic Category System Conclusion

Adjunction Adjunction (aka. Adjointness or Adjoint Situation)

The two special arrows

ηA : A → G(F(A))

εB : F(G(B)) → B

extend to two Natural Transformations (between two endofunctors)

IdA F◦G η: IdA ⇒ G ◦ F A η A B ε B ε: F ◦ G ⇒ IdB G◦F IdB

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Adjunction Adjunction (aka. Adjointness or Adjoint Situation)

The two special arrows

ηA : A → G(F(A))

εB : F(G(B)) → B

extend to two Natural Transformations (between two endofunctors)

IdA F◦G η: IdA ⇒ G ◦ F A η A B ε B ε: F ◦ G ⇒ IdB G◦F IdB

RThese are the unit and co-unit of an Adjunction.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system I Two natural transformations: weak (Adjunction)

η: IdA ⇒ G ◦ F, ε: F ◦ G ⇒ IdB

I Two natural isomorphisms: intermediate (equivalence)

η: IdA ⇔ G ◦ F, ε: F ◦ G ⇔ IdB

I Two equalities: strong (isomorphism)

η: IdA = G ◦ F, ε: F ◦ G = IdB

Introduction Methodology Category Theory Syntactic Category System Conclusion

Adjunction Category comparison via unit and co-unit Three levels of similarity between Categories

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system I Two natural isomorphisms: intermediate (equivalence)

η: IdA ⇔ G ◦ F, ε: F ◦ G ⇔ IdB

I Two equalities: strong (isomorphism)

η: IdA = G ◦ F, ε: F ◦ G = IdB

Introduction Methodology Category Theory Syntactic Category System Conclusion

Adjunction Category comparison via unit and co-unit Three levels of similarity between Categories

I Two natural transformations: weak (Adjunction)

η: IdA ⇒ G ◦ F, ε: F ◦ G ⇒ IdB

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system I Two equalities: strong (isomorphism)

η: IdA = G ◦ F, ε: F ◦ G = IdB

Introduction Methodology Category Theory Syntactic Category System Conclusion

Adjunction Category comparison via unit and co-unit Three levels of similarity between Categories

I Two natural transformations: weak (Adjunction)

η: IdA ⇒ G ◦ F, ε: F ◦ G ⇒ IdB

I Two natural isomorphisms: intermediate (equivalence)

η: IdA ⇔ G ◦ F, ε: F ◦ G ⇔ IdB

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Adjunction Category comparison via unit and co-unit Three levels of similarity between Categories

I Two natural transformations: weak (Adjunction)

η: IdA ⇒ G ◦ F, ε: F ◦ G ⇒ IdB

I Two natural isomorphisms: intermediate (equivalence)

η: IdA ⇔ G ◦ F, ε: F ◦ G ⇔ IdB

I Two equalities: strong (isomorphism)

η: IdA = G ◦ F, ε: F ◦ G = IdB

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Adjunction Category comparison via unit and co-unit Three levels of similarity between Categories

AdjunctionI Two natural (weak transformations: similarity) weak (Adjunction)

For poset Categories η: IdA ⇒ G ◦ F, ε: F ◦ G ⇒ IdB After a round-trip A grows and F B shrinks, in the depicted way. I Two natural isomorphisms:A intermediatea B (equivalence)

G F(f) η: IdAA ⇔ G ◦ F, ε: FF◦(AG)⇔ IdB F(G(B)) idF(A) ηA I Two equalities: strongf (isomorphism) g ε id B G(F(A)) G(B) G(B) B G(gη) : IdA = G ◦ F, ε: F ◦ G = IdB

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Adjunction Category comparison via unit and co-unit Three levels of similarity between Categories

EquivalenceI Two natural (intermediate transformations: similarity) weak (Adjunction)

For poset Categories η: IdA ⇒ G ◦ F, ε: F ◦ G ⇒ IdB After a round-trip A and B land F somewhere isomorphic to I Two natural isomorphisms: intermediatea (equivalence) themselves. A B

G F(f) η: IdAA ⇔ G ◦ F, ε: FF◦(AG)⇔ IdB F(G(B)) idF(A) ηA I Two equalities: strongf (isomorphism) g ε id B G(F(A)) G(B) G(B) B G(gη) : IdA = G ◦ F, ε: F ◦ G = IdB

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Adjunction Category comparison via unit and co-unit Three levels of similarity between Categories

IsomorphismI Two natural (strong transformations: similarity) weak (Adjunction)

For poset Categories η: IdA ⇒ G ◦ F, ε: F ◦ G ⇒ IdB After a round-trip A and B go F back to themselves. I Two natural isomorphisms:A intermediatea B (equivalence)

G F(f) η: IdAA ⇔ G ◦ F, ε: FF◦(AG)⇔ IdB F(G(B)) idF(A) ηA I Two equalities: strongf (isomorphism) g ε id B G(F(A)) G(B) G(B) B G(gη) : IdA = G ◦ F, ε: F ◦ G = IdB

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Example Given an inclusion function ι: Z → R from integers to reals. Its − intuitive inverse is another inclusion ι 1 : R → Z, which does not exist (e.g. ι(2.5) ∈/ Z). But we have a potential second-best solution: the ceiling function d−e: R → Z which maps a real number to the smallest integer above (or equal to) it. We can verify that d−e and ι form a Galois connection.

Introduction Methodology Category Theory Syntactic Category System Conclusion

Adjunction Adjunction for poset Categories: Galois connection

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Adjunction Adjunction for poset Categories: Galois connection

Typical scenario: given a monotone function f, its inverse f−1, if existent, will allow us to say the f-paired elements are parallel. When there is no such f−1 yet we still want some reasonably “parallel” pairing, a Galois connection, if existent, is our friend. Example Given an inclusion function ι: Z → R from integers to reals. Its − intuitive inverse is another inclusion ι 1 : R → Z, which does not exist (e.g. ι(2.5) ∈/ Z). But we have a potential second-best solution: the ceiling function d−e: R → Z which maps a real number to the smallest integer above (or equal to) it. We can verify that d−e and ι form a Galois connection.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system RHere ε is an equality, so we have an “enhanced” Adjunction.

Introduction Methodology Category Theory Syntactic Category System Conclusion

Adjunction Adjunction for poset Categories: Galois connection

C,d−e R Z In the left conﬁguration: ι

idC(x) I (i) Since ι is monotone, C(f) x C(x) C(ι(n)) C(x) 6 n ⇒ ι(C(x)) 6 ι(n), ηx but x ι(C(x)) because f g 6 εn ι(C(x)) = dxe, therefore ι(C(x)) ι(n) n x ι(n) and so g ⇒ f. ι(g) 6 idι(n)

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system RHere ε is an equality, so we have an “enhanced” Adjunction.

Introduction Methodology Category Theory Syntactic Category System Conclusion

Adjunction Adjunction for poset Categories: Galois connection

C,d−e R Z In the left conﬁguration: ι

idC(x) (ii) Since C is monotone, C(f) I x C(x) C(ι(n)) x 6 ι(n) ⇒ C(x) 6 C(ι(n)), ηx but C(ι(n)) = dne = n, f g εn therefore C(x) 6 n and so ι(C(x)) ι(n) n f ⇒ g. ι(g) idι(n)

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system RHere ε is an equality, so we have an “enhanced” Adjunction.

Introduction Methodology Category Theory Syntactic Category System Conclusion

Adjunction Adjunction for poset Categories: Galois connection

C,d−e R Z ι In the left conﬁguration:

idC(x) C(f) Combining (i) and (ii), we x C(x) C(ι(n)) I obtain f ⇔ g, whence C a ι, ηx f g i.e. C is left adjoint to ι and ι εn is right adjoint to C. ι(C(x)) ι(n) n ι(g) idι(n)

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Adjunction Adjunction for poset Categories: Galois connection

C,d−e R Z ι In the left conﬁguration:

idC(x) C(f) Combining (i) and (ii), we x C(x) C(ι(n)) I obtain f ⇔ g, whence C a ι, ηx f g i.e. C is left adjoint to ι and ι εn is right adjoint to C. ι(C(x)) ι(n) n ι(g) idι(n)

RHere ε is an equality, so we have an “enhanced” Adjunction.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Adjunction Adjunction for poset Categories: Galois connection

Enhanced weak similarity (cf. Erne´ 2004)

In general, ifC, thed−e co-unit of an Adjunction is an isomorphism (whereofR equality isZ a special case), typically when an inclusion ι In the left conﬁguration: Functor has a left adjoint, the situation is called an idC(x) C(f) Combining (i) and (ii), we epi-Adjunctionx (orC(xright) perfectC(ι(n)) GaloisI connection for posets). obtain f ⇔ g, whence C a ι, ηx f g i.e. C is left adjoint to ι and ι εn is right adjoint to C. ι(C(x)) ι(n) n ι(g) idι(n)

RHere ε is an equality, so we have an “enhanced” Adjunction.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Adjunction Adjunction for poset Categories: Galois connection

Enhanced weak similarity (cf. Erne´ 2004)

In general, ifC, thed−e co-unit of an Adjunction is an isomorphism (whereofR equality isZ a special case), typically when an inclusion ι In the left configuration: Functor has a left adjoint, the situation is called an idC(x) C(f) Combining (i) and (ii), we epi-Adjunctionx (orC(xright) perfectC(ι(n)) GaloisI connection for posets). obtain f ⇔ g, whence C a ι, ηx f g i.e. C is left adjoint to ι and ι Alternatively, in this situationεn the “smaller” of the two Categories is right adjoint to C. ι(C(x(here)) )ι( isn) called anreflective Subcategory of the “bigger” one ι(gZ) (here R) andid theι(n) left adjoint (here d−e) called a reflector.

RHere ε is an equality, so we have an “enhanced” Adjunction.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system I Direct Functors between f-hierarchies are either unstable or not meaningful.

I But a stably meaningful Functorial connection exists between I Thiseach isf-hierarchy in addition and is an a “universal epi-Adjunction. spine” (Wiltschko 2014), e.g.

Introduction Methodology Category Theory Syntactic Category System Conclusion

Cross-functional-hierarchy parallelism Cross-f-hierarchy parallelism via epi-Adjunction

What is a stably meaningful Functor between f-hierarchies?

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system I But a stably meaningful Functorial connection exists between I Thiseach isf-hierarchy in addition and is an a “universal epi-Adjunction. spine” (Wiltschko 2014), e.g.

Introduction Methodology Category Theory Syntactic Category System Conclusion

Cross-functional-hierarchy parallelism Cross-f-hierarchy parallelism via epi-Adjunction

What is a stably meaningful Functor between f-hierarchies?

I Direct Functors between f-hierarchies are either unstable or not meaningful.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system I This is in addition is an epi-Adjunction.

Introduction Methodology Category Theory Syntactic Category System Conclusion

Cross-functional-hierarchy parallelism Cross-f-hierarchy parallelism via epi-Adjunction

What is a stably meaningful Functor between f-hierarchies?

I Direct Functors between f-hierarchies are either unstable or not meaningful.

I But a stably meaningful Functorial connection exists between each f-hierarchy and a “universal spine” (Wiltschko 2014), e.g.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system I This is in addition is an epi-Adjunction.

Introduction Methodology Category Theory Syntactic Category System Conclusion

Cross-functional-hierarchy parallelism Cross-f-hierarchy parallelism via epi-Adjunction

What is a stably meaningful Functor between f-hierarchies?

I Direct Functors between f-hierarchies are either unstable or not meaningful.

I But a stably meaningful Functorial connection exists between each f-hierarchy and a “universal spine” (Wiltschko 2014), e.g.

BV V Appl v Asp Tns Mod Fin Foc Top

G F

B0 Cls PoV Anc Lnk expository

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Cross-functional-hierarchy parallelism Cross-f-hierarchy parallelism via epi-Adjunction

What is a stably meaningful Functor between f-hierarchies?

I Direct Functors between f-hierarchies are either unstable or not meaningful.

I But a stably meaningful Functorial connection exists between each f-hierarchy and a “universal spine” (Wiltschko 2014), e.g.

BV V Appl v Asp Tns Mod Fin Foc Top G a F

B0 Cls PoV Anc Lnk expository

I This is in addition is an epi-Adjunction.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Cross-functional-hierarchy parallelism Cross-f-hierarchy parallelism via epi-Adjunction

expository

Let’s examine the f-hierarchy epi-Adjunction more closely:

BV V Appl v Asp Tns Mod Fin Foc Top G a F

B0 Cls PoV Anc Lnk

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system I Hence, the right adjoint G: B0 → BV is also unique, a` la Freyd Adjoint Functor Theorem (FAFT).

Introduction Methodology Category Theory Syntactic Category System Conclusion

Cross-functional-hierarchy parallelism Cross-f-hierarchy parallelism via epi-Adjunction

expository Let’s examine the f-hierarchy epi-Adjunction more closely:

BV V Appl v Asp Tns Mod Fin Foc Top G a F

B0 Cls PoV Anc Lnk

I The left adjoint F: BV → B0 is unique for any f-hierarchy B.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Cross-functional-hierarchy parallelism Cross-f-hierarchy parallelism via epi-Adjunction

expository Let’s examine the f-hierarchy epi-Adjunction more closely:

BV V Appl v Asp Tns Mod Fin Foc Top G a F

B0 Cls PoV Anc Lnk

I The left adjoint F: BV → B0 is unique for any f-hierarchy B.

I Hence, the right adjoint G: B0 → BV is also unique, a` la Freyd Adjoint Functor Theorem (FAFT).

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Cross-functional-hierarchy parallelism Cross-f-hierarchy parallelism via epi-Adjunction

Freyd Adjoint Functor Theorem (poset version) expository

Let’s examine the f-hierarchy epi-AdjunctionF more closely: P a Q P In a Galois connection 6 v where 6 has all joins BV V Appl v Asp G Tns Mod Fin Foc Top and Qv has all meets, adjoint Functors in a pair F a G

G a Funiquely determine each other by the formulae:

B Cls PoV Anc Lnk 0 F(p) = min{q ∈ Qv | p 6 G(q)} G(q) = max{p ∈ P | F(p) v q} 6

p ∈ P q ∈ Q for any 6, v.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system I The mathematically determined representatives coincide with our linguistically special categories, i.e. core functional or phase categories.

Introduction Methodology Category Theory Syntactic Category System Conclusion

Cross-functional-hierarchy parallelism Cross-f-hierarchy parallelism via epi-Adjunction Let’s examine the f-hierarchy epi-Adjunction more closely: expository

BV V Appl v Asp Tns Mod Fin Foc Top G a F

B0 Cls PoV Anc Lnk

I The right adjoint chooses a “representative” for each f-domain. FAFT says this is the highest category in each f-domain.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Cross-functional-hierarchy parallelism Cross-f-hierarchy parallelism via epi-Adjunction Let’s examine the f-hierarchy epi-Adjunction more closely: expository

BV V Appl v Asp Tns Mod Fin Foc Top G a F

B0 Cls PoV Anc Lnk

I The right adjoint chooses a “representative” for each f-domain. FAFT says this is the highest category in each f-domain.

I The mathematically determined representatives coincide with our linguistically special categories, i.e. core functional or phase categories.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Global interconnection Cross-f-hierarchy parallelism via epi-Adjunction

Let BV vary (e.g. BN, BP, or any other f-hierarchy a language variety may have) and we obtain a ﬂower-shaped conﬁguration

BP BA

a a a

BV B0 a BN a a

BW BZ where the center is the u-spine and the petals are the f-hierarchies.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Global interconnection Cross-f-hierarchy parallelism via epi-Adjunction

Let BV vary (e.g. BN, BP, or any other f-hierarchy a language variety may have) and we obtain a ﬂower-shaped conﬁguration

BP BA

a a

a 1 ﬂower = 1 f-category inventory

BV B0 a BN a a

BW BZ where the center is the u-spine and the petals are the f-hierarchies.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Global interconnection Cross-f-hierarchy parallelism via epi-Adjunction

Depending on one’s assumption about the major parts of speech, the “categorial ﬂower” may have more or fewer petals

a BP a a a

BV B0 a BN BV B0 a BN B a 0 a a BV BN BA

but all such ﬂowers are constructed by joint epi-Adjunction.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Global interconnection Cross-f-hierarchy parallelism via epi-Adjunction

We can also change B to P, Q, CFC, Cart . . . and obtain a garden of categorial ﬂowers, e.g. a a a

PV P0 a PN QV Q0 a QN SV S0 a SN

a a

a a PhV Ph0 a PhN CFCV CFC0 CFCN CartV Cart0 CartN

There is further global interconnection at the garden level.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system 1 ﬂower = 1 granularity level Both are beyond individual f-hierarchies and even individual adult speakers, since the f-category inventory may

I change over time in an individual

I vary across language varieties The garden is a collection of all theoretically possible f-category inventories, call it the Granularity Level Space (GLS).

Introduction Methodology Category Theory Syntactic Category System Conclusion

Global interconnection Global interconnection

Recall:

I Category division hierarchy

I Granularity level stacking

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system The garden is a collection of all theoretically possible f-category inventories, call it the Granularity Level Space (GLS).

Introduction Methodology Category Theory Syntactic Category System Conclusion

Global interconnection Global interconnection

Recall:

I Category division hierarchy

I Granularity level stacking 1 ﬂower = 1 granularity level Both are beyond individual f-hierarchies and even individual adult speakers, since the f-category inventory may

I change over time in an individual

I vary across language varieties

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Global interconnection Global interconnection

Recall:

I Category division hierarchy

I Granularity level stacking 1 ﬂower = 1 granularity level Both are beyond individual f-hierarchies and even individual adult speakers, since the f-category inventory may

I change over time in an individual

I vary across language varieties The garden is a collection of all theoretically possible f-category inventories, call it the Granularity Level Space (GLS).

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system I Parallel: via universal spine (as we have seen)

I Stackable: direct and composable epi-Adjunctions, e.g.

I Incomparable: neither parallel nor stackable, e.g.

VCFC v CFC CPh v Ph T CCFC

Introduction Methodology Category Theory Syntactic Category System Conclusion

Global interconnection Global interconnection

Three global relations between f-hierarchies X and Y

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system I Stackable: direct and composable epi-Adjunctions, e.g.

I Incomparable: neither parallel nor stackable, e.g.

VCFC v CFC CPh v Ph T CCFC

Introduction Methodology Category Theory Syntactic Category System Conclusion

Global interconnection Global interconnection

Three global relations between f-hierarchies X and Y

I Parallel: via universal spine (as we have seen)

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system I Incomparable: neither parallel nor stackable, e.g.

VCFC v CFC CPh v Ph T CCFC

Introduction Methodology Category Theory Syntactic Category System Conclusion

Global interconnection Global interconnection

Three global relations between f-hierarchies X and Y

I Parallel: via universal spine (as we have seen)

I Stackable: direct and composable epi-Adjunctions, e.g.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system I Incomparable: neither parallel nor stackable, e.g.

VCFC v CFC CPh v Ph T CCFC

Introduction Methodology Category Theory Syntactic Category System Conclusion

Global interconnection Global interconnection

Epi-Adjunctions across granularity levels

ThreeF a G global∧ F0 a relationsG0 ⇒ F between0 ◦ F a G ◦ f-hierarchiesG0 X and Y I Parallel: via universal spine (as we have seen) PhV v Ph CPh I Stackable: direct and composable epi-Adjunctions, e.g. F0 a G0

CFCV VCFC v CFC T CCFC F a G

WV Proc Init Voice Asp Tns Mod Force Top

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system I Incomparable: neither parallel nor stackable, e.g.

VCFC v CFC CPh v Ph T CCFC

Introduction Methodology Category Theory Syntactic Category System Conclusion

Global interconnection Global interconnection

Three global relations between f-hierarchies X and Y

I Parallel: via universal spine (as we have seen)

I Stackable: direct and composable epi-Adjunctions, e.g.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Global interconnection Global interconnection

Three global relations between f-hierarchies X and Y

I Parallel: via universal spine (as we have seen)

I Stackable: direct and composable epi-Adjunctions, e.g.

I Incomparable: neither parallel nor stackable, e.g.

VCFC v CFC CPh v Ph T CCFC

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Global interconnection Global interconnection

Parallel + stackable ⇒ a fully connected corner in the GLS a

PhV Ph0 a PhN a a a a

CFCV CFC0 a CFCN a a a a

CartV Cart0 a CartN

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Global interconnection Global interconnection

Abstract away from the internal details of ﬂowers a

PhV Ph0 a PhN Ph a a a f−1 f a

CFCV CFC0 a CFCN CFC a a a f0−1 f0 a

CartV Cart0 a CartN Cart

and we get an isomorphism Ph =∼ CFC =∼ Cart.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Global interconnection Global interconnection

zoom out Parallel + stackable + incomparable =====⇒ GLS poset ○

EP1 EP2 ....

Ph Ph1 2 Ph3 Ph4 ....

CFC1 CFC2 CFC3 CFC4 CFC5 ......

Figure 5: Granularity Level Space (GLS) ordered by inheritance. RThis is the highest abstraction layer for functional hierarchies.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Global interconnection Global interconnection

zoom out Parallel + stackable + incomparable =====⇒ GLS poset

○ 1 path = 1 division hierarchy

EP1 EP2 ....

Ph Ph1 2 Ph3 Ph4 ....

CFC1 CFC2 CFC3 CFC4 CFC5 ......

Figure 5: Granularity Level Space (GLS) ordered by inheritance. RThis is the highest abstraction layer for functional hierarchies.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Entire SCS Entire SCS Finally we add in acategorial categories

P √DOG GLS √RUN Foc V ¬ T √ON Asp √GOOD Num .... Acategorial ⋁ ⋀ ....

Figure 6: The entire Syntactic Category System (SCS).

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system RSyn as a Category has more structures than partial orders.

Introduction Methodology Category Theory Syntactic Category System Conclusion

Entire SCS Entire SCS And put the SCS in a larger universe. . .

Syn Sem

P E×E V-√RUN E DP⋀DP S √GOOD T Root V Asp FocP ¬AspP E→T→T ¬ Num TP 1 √ON .... ⋀ E→T Cat ⋁ N-√DOG T→T ....

Figure 7: Syntax and Semantics connected by interpretation Functor.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Entire SCS Entire SCS And put the SCS in a larger universe. . .

#monoidal Category Syn Sem # topos Category

P E×E V-√RUN E DP⋀DP S √GOOD T Root V Asp FocP ¬AspP E→T→T ¬ Num TP 1 √ON .... ⋀ E→T Cat ⋁ N-√DOG T→T ....

Figure 7: Syntax and Semantics connected by interpretation Functor. RSyn as a Category has more structures than partial orders.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Conclusion

Category Theory has been called “abstract nonsense”, but it provides a very sensible metalanguage to describe the ontological organization of the Syntactic Category System (SCS).

I A functional hierarchy is a poset Category.

I A f-category inventory is a tiny Category of posets.

I All possible f-category inventories form a huge poset.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

Conclusion

An important Categorical relation epi-Adjunction

I indirectly formalizes cross-functional-hierarchy parallelism (via universal spine)

I directly formalizes cross-granularity-level inheritance (via Adjunction composition) (More details are in my dissertation, Song 2019).

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

BP BA

a a a

BV B0 a BN a a

BW BZ

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

ReferencesI

Fong, B. & D. Spivak Seven Sketches in Compositionality: An Invitation to Applied Category Theory. MIT, 2018. Wiltschko, M. The Universal Structure of Categories: Towards a Formal Typology. CUP, 2014. Biberauer, T. & I. Roberts Rethinking formal hierarchies: A proposed uniﬁcation. COPiL7, 2015.

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system Introduction Methodology Category Theory Syntactic Category System Conclusion

ReferencesII

Erne,´ M. Adjunctions and Galois connections: Origins, history and development. Galois Connections and Applications. Springer Science+Business Media, B.V., 2004. Song, C. Formal ﬂexibility of syntactic categories. PhD dissertation, 2019 (in progress).

Song 2019 TAL, Cambridge More than abstract nonsense: A Category-theoretic sketch of the syntactic category system