Types in categorical linguistics (& elswhere) Peter Hines - - PowerPoint PPT Presentation

Types in categorical linguistics (& elswhere)

Peter Hines

Oxford – Oct. 2010 University of York

• N. V. M. S. Research

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Topic of the talk:

This talk will be about: Pure Category Theory.

... although it might have interesting interpretations in various settings. http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Possible interpretations:

This pure category theory can be interpreted as:

1

Logic & theoretical computing.

2

Categorical linguistics / semantics.

3

Categorical quantum mechanics.

Categorical linguistics will provide the fig-leaf for today’s category theory. http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Types in categorical models of meaning

Distributional semantics: We construct ‘meaning vectors’ for words Vectors can be compared, using an inner product. We derive a notion of ‘distance’ between two words. Bringing in categorical linguistics: Moving to a typed system allows us to compare sentences.

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

A (oversimplified) description

Words are assigned types, based on their rˆ

• le.

This extends to sentences; these are typed by their grammatical structure. The types are objects in a monoidal closed category. — closed categories have a reduction, or evaluation. All grammatical sentences reduce to the same type, S. By reducing sentences to the same type, they can be compared.

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

A reminder:

A monoidal closed category C has: A monoidal tensor ⊗ : C × C → C A unit object for the tensor I ⊗ X ∼ = X ∼ = X ⊗ I An internal hom [ → ] : Cop × C → C all satisfying various natural conditions. These can be used to define an evaluation arrow A ⊗ [A → B]

evalA,B

B

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

The kind of thing we wish to do ...

The aim: “Use typing and evaluation, to reduce all sentences to the same type”.

Noun Intransitive Verb

Type individual words

N [N → S]

Combine types using the tensor

N ⊗ [N → S]

Reduce, using the evaluation

S

Types are chosen so all (well-formed) sentences reduce to S

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

The kind of thing we wish to do ...

The aim: “Use typing and evaluation, to reduce all sentences to the same type”.

Noun Intransitive Verb

Type individual words

N [N → S]

Combine types using the tensor

N ⊗ [N → S]

Reduce, using the evaluation

S

Types are chosen so all (well-formed) sentences reduce to S

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

The kind of thing we wish to do ...

The aim: “Use typing and evaluation, to reduce all sentences to the same type”.

Noun Intransitive Verb

Type individual words

N [N → S]

Combine types using the tensor

N ⊗ [N → S]

Reduce, using the evaluation

S

Types are chosen so all (well-formed) sentences reduce to S

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

The kind of thing we wish to do ...

The aim: “Use typing and evaluation, to reduce all sentences to the same type”.

Noun Intransitive Verb

Type individual words

N [N → S]

Combine types using the tensor

N ⊗ [N → S]

Reduce, using the evaluation

S

Types are chosen so all (well-formed) sentences reduce to S

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

A few questions ...

How do we compare elements of the same type? How does comparison relate to

1

The monoidal tensor?

2

The internal hom.?

3

Evaluation?

What does the sentence type S look like? Does evaluation lose information ?

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

A few questions ...

How do we compare elements of the same type? How does comparison relate to

1

The monoidal tensor?

2

The internal hom.?

3

Evaluation?

What does the sentence type S look like? Does evaluation lose information ?

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

To avoid becoming too abstract(!)

We will use & compare two example sentences L1. Bobby loves Marilyn Monroe. L2. I like Fidel Castro and his beard. These are both lyrics from Bob Dylan songs.

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Defining elements of a certain type

An element of a type T is an arrow from the unit object to T. Let the ‘Noun Phrase’ type be N ∈ Ob(C). Then “Bobby” is an arrow I

Bobby

N Familiar examples: A member of a set is given by: a function f : {∗} → X. A state in quantum mechanics is given by: a linear map |ψ : ❈ → H.

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Defining elements of a certain type

An element of a type T is an arrow from the unit object to T. Let the ‘Noun Phrase’ type be N ∈ Ob(C). Then “Bobby” is an arrow I

Bobby

N Familiar examples: A member of a set is given by: a function f : {∗} → X. A state in quantum mechanics is given by: a linear map |ψ : ❈ → H.

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Defining elements of a certain type

An element of a type T is an arrow from the unit object to T. Let the ‘Noun Phrase’ type be N ∈ Ob(C). Then “Bobby” is an arrow I

Bobby

N Familiar examples: A member of a set is given by: a function f : {∗} → X. A state in quantum mechanics is given by: a linear map |ψ : ❈ → H.

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Comparing elements (I)

The precise form of categorical closure determines how we make comparisons. The grammar:

Lambek pregroups form a (non-symmetric) compact closed category.

The semantics

Distributional semantics uses Vector Spaces – another compact closed category. Tentative conclusion: let’s use a compact closed category!

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Comparing elements (I)

The precise form of categorical closure determines how we make comparisons. The grammar:

Lambek pregroups form a (non-symmetric) compact closed category.

The semantics

Distributional semantics uses Vector Spaces – another compact closed category. Tentative conclusion: let’s use a compact closed category!

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Comparing elements (I)

The precise form of categorical closure determines how we make comparisons. The grammar:

Lambek pregroups form a (non-symmetric) compact closed category.

The semantics

Distributional semantics uses Vector Spaces – another compact closed category. Tentative conclusion: let’s use a compact closed category!

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Comparing elements (II)

CCCs are symmetric monoidal categories with duals ( )† f : A → B

duality

⇐ ⇒ f † : B† → A† The dagger ( )† is a contravariant (order-reversing) functor. In a CCC, the internal hom is [A → B] = A† ⊗ B (This is a much simpler form that most closed categories).

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Comparing elements(III)

Our examples have self-dual objects: A† = A. Comparing elements of type N Elements Bobby and Fidel are compared using the composite: I

Fidel Bobby|Fidel

• N

Bobby†

• I

The generalised scalar product is Bobby|Fidel : I → I.

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Comparing elements(III)

Our examples have self-dual objects: A† = A. Comparing elements of type N Elements Bobby and Fidel are compared using the composite: I

Fidel Bobby|Fidel

• N

Bobby†

• I

The generalised scalar product is Bobby|Fidel : I → I.

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Generalised scalar products

Comparisons are of the form Bobby|Fidel : I → I In various categories, C(I, I) is:

Real numbers ❘, complex numbers ❈, the unit interval [0, 1], the set {T, F}, the natural numbers ◆, etc. In general: A comparison u|v gives a measure of the ‘similarity’ or ‘overlap’ of the elements u, v. For vector spaces, it is exactly the scalar product.

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Some further points:

The unit object I is not the sentence type S. For illustrative purposes, we will use C(I, I) = [0, 1]

• nothing in

common

• exactly

the same

• [0

1] Disclaimer: any actual values given are estimates (random guesses)

The comparison x|y : I → I exists for elements I

x,y

− → A of the same type .

– this holds for any type A ∈ Ob(C).

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Some further points:

The unit object I is not the sentence type S. For illustrative purposes, we will use C(I, I) = [0, 1]

• nothing in

common

• exactly

the same

• [0

1] Disclaimer: any actual values given are estimates (random guesses)

The comparison x|y : I → I exists for elements I

x,y

− → A of the same type .

– this holds for any type A ∈ Ob(C).

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Some further points:

The unit object I is not the sentence type S. For illustrative purposes, we will use C(I, I) = [0, 1]

• nothing in

common

• exactly

the same

• [0

1] Disclaimer: any actual values given are estimates (random guesses)

The comparison x|y : I → I exists for elements I

x,y

− → A of the same type .

– this holds for any type A ∈ Ob(C).

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Back to our sentences ...

L1. Bobby loves Marilyn Monroe. L2. I like Fidel Castro and his beard. Let’s instantiate a variable ... These are both Bob Dylan lyrics. We replace “I” by “Bob Dylan”.

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Back to our sentences ...

L1. Bobby loves Marilyn Monroe. L2. Bob Dylan likes Fidel Castro and his beard. Let’s instantiate a variable ... We replace “I” by “Bob Dylan”, ... and adjust the verb accordingly!

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

The first estimate ...

L1. Bobby loves Marilyn Monroe. L2. Bob Dylan likes Fidel Castro and his beard. Both “Bobby” and “Bob Dylan” are of type N — we can form their scalar product. As a reasonable estimate (random guess?) we put Bobby|Bob Dylan ≃ 0.98

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Putting things in context

From the context (i.e. Bob Dylan lyrics), we have assumed a close match between “I” and “Bobby”.

Unfortunately ... Historical / cultural context suggests that in L1. “Bobby” actually refers to “Robert Kennedy” However, this is not evident from the lyrics of either song.

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Making more comparisons

L1. Bobby loves Marilyn Monroe. L2. Bob Dylan likes Fidel Castro and his beard. These are both transitive verbs, so have type [N → [N → S]] As they have the same type, we may take their scalar product: likes|loves ≃ 0.75

(Another random guess - from Mehrnoosh)

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

One last comparison ...

L1. Bobby loves Marilyn Monroe. L2. Bob Dylan likes Fidel Castro and his beard.

How to compare “Marilyn Monroe” with “Fidel Castro and his beard”?

These are em not the same type: Marilyn Monroe has type N Fidel Castro and his beard has type N ⊗ C ⊗ N where C is the type for a binary connective such as: AND, OR, EXCLUSIVE OR, ...

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

One last comparison ...

L1. Bobby loves Marilyn Monroe. L2. Bob Dylan likes Fidel Castro and his beard.

How to compare “Marilyn Monroe” with “Fidel Castro and his beard”?

These are em not the same type: Marilyn Monroe has type N Fidel Castro and his beard has type N ⊗ C ⊗ N where C is the type for a binary connective such as: AND, OR, EXCLUSIVE OR, ...

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Typing connectives

We wish for Fidel Castro and his beard N ⊗ C ⊗ N to reduce to something of type N. For this to happen, the connective type C must be [N → [N → N]]

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

We may now make a comparison:

Applying an evaluation maps “Fidel Castro and his beard” into the type N. — this can then be compared to “Marilyn Monroe” We are happy to guess (hope?) Marilyn Monroe | Eval ◦ Castro and his beard = 0

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

A digression

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

A closer look at connectives

The ‘connective type’ C was chosen so that: N ⊗ C ⊗ N evaluates to N We wish for Noun Phrase and Noun Phrase to evaluate to another Noun Phrase. However, such connectives are used more generally.

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Other contexts for connectives

Bobby loves and obeys Marilyn Monroe (Verb phrases) Castro’s big and bushy beard (Adjectives) Bobby likes Marilyn and I like Fidel (Entire sentences)

The appropriate typing is: [X → [X → X]] where X varies, according to the context.

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Other contexts for connectives

Bobby loves and obeys Marilyn Monroe (Verb phrases) Castro’s big and bushy beard (Adjectives) Bobby likes Marilyn and I like Fidel (Entire sentences)

The appropriate typing is: [X → [X → X]] where X varies, according to the context.

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Other contexts for connectives

Bobby loves and obeys Marilyn Monroe (Verb phrases) Castro’s big and bushy beard (Adjectives) Bobby likes Marilyn and I like Fidel (Entire sentences)

The appropriate typing is: [X → [X → X]] where X varies, according to the context.

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Other contexts for connectives

Bobby loves and obeys Marilyn Monroe (Verb phrases) Castro’s big and bushy beard (Adjectives) Bobby likes Marilyn and I like Fidel (Entire sentences)

The appropriate typing is: [X → [X → X]] where X varies, according to the context.

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Other contexts for connectives

Bobby loves and obeys Marilyn Monroe (Verb phrases) Castro’s big and bushy beard (Adjectives) Bobby likes Marilyn and I like Fidel (Entire sentences)

The appropriate typing is: [X → [X → X]] where X varies, according to the context.

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Connectives and polymorphism

Our claim: To deal with connectives, we appear to need parameterised or polymorphic types. Abusing notation slightly, we write the type for “and” as ΛX . [X → [X → X]]

• r equivalently,

ΛX . [X ⊗ X → X]

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Connectives and polymorphism

Our claim: To deal with connectives, we appear to need parameterised or polymorphic types. Abusing notation slightly, we write the type for “and” as ΛX . [X → [X → X]]

• r equivalently,

ΛX . [X ⊗ X → X]

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

End of digression

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Back to comparing sentences

How does the scalar product | interact with the tensor ⊗ ? Some simple category theory: Given scalar products a|b : I → I x|y : I → I the interaction with the tensor is simply: a ⊗ x|b ⊗ y = a|b.x|y This is a general categorical identity.

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Can we now compare our sentences?

Using our (entirely fictitious) values:

Bobby loves Marilyn Monroe. Bob Dylan likes Fidel Castro and his beard. Bobby|Bob Dylan likes|loves Fidel & his beard|Marilyn Monroe 0.98 0.75 0.00

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Can we now compare our sentences?

Using our (entirely fictitious) values:

Bobby loves Marilyn Monroe. Bob Dylan likes Fidel Castro and his beard. Bobby|Bob Dylan likes|loves Fidel & his beard|Marilyn Monroe 0.98 0.75 0.00

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Can we compare L1 and L2 ?

We have two sentences, of type N ⊗ [N → [N → S]] ⊗ N We can take their inner product, to get L1|L2 = 0.98 × 0.75 × 0.00 = 0 Important: We have compared L1 and L2 as elements of type N ⊗ [N → [N → S]] ⊗ N. Do we get the same answer if we first ‘reduce’ them to terms of type S ??

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Can we compare L1 and L2 ?

We have two sentences, of type N ⊗ [N → [N → S]] ⊗ N We can take their inner product, to get L1|L2 = 0.98 × 0.75 × 0.00 = 0 Important: We have compared L1 and L2 as elements of type N ⊗ [N → [N → S]] ⊗ N. Do we get the same answer if we first ‘reduce’ them to terms of type S ??

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Evaluation and scalar products

Does evaluation preserve scalar products? I

x

• x′
• G

Eval

• I

y

• y′
• S

Is it true that x|y

?

= x′|y′

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Does evaluation preserve scalar products?

NO.

The simplest counterexamples come from quantum mechanics, where evaluation is (partial) measurement. Evaluation is an irreversible operation A ⊗ [A → B]

evalA,B

B Is this desirable, or undesirable, for categorical models of meaning?

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Does evaluation preserve scalar products?

NO.

The simplest counterexamples come from quantum mechanics, where evaluation is (partial) measurement. Evaluation is an irreversible operation A ⊗ [A → B]

evalA,B

B Is this desirable, or undesirable, for categorical models of meaning?

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Some motivation: Scruffy Cats

In distributional semantics: The element I

Cat

N provides information about cats in general ... An element I

Scruffy [N → N]

might tell us about the general concept of ‘scruffiness’. The tensor product I

Scruffy ⊗ Cat [N → N] ⊗ N

tells us all about ‘scruffiness’, along with everything about cats.

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Some motivation: Scruffy Cats

In distributional semantics: The element I

Cat

N provides information about cats in general ... An element I

Scruffy [N → N]

might tell us about the general concept of ‘scruffiness’. The tensor product I

Scruffy ⊗ Cat [N → N] ⊗ N

tells us all about ‘scruffiness’, along with everything about cats.

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Evaluation, and forgetfulness

The element I

Scruffy ⊗ Cat [N → N] ⊗ N

provides too much information! Composing with the evaluation map: I

Scruffy ⊗ Cat

• [N → N] ⊗ N

Eval

• N

defines a new element, that tells us about Scruffy Cats only.

It is vital that evaluation can forget information.

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Evaluation, and forgetfulness

The element I

Scruffy ⊗ Cat [N → N] ⊗ N

provides too much information! Composing with the evaluation map: I

Scruffy ⊗ Cat

• [N → N] ⊗ N

Eval

• N

defines a new element, that tells us about Scruffy Cats only.

It is vital that evaluation can forget information.

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

A more structural point of view

Taking a logical view of our type system: We work with compact closure This corresponds to a (degenerate) fragment of Linear Logic. This is resource-sensitive. (For example) the resource I

Scruffy [N → N]

is consumed in the evaluation ... and plays no further rˆ

• le.

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

How about a limited form of reversibility?

Let us compare Cat : I → N Dog : I → N do we get the same value when we compare Eval ◦ (Scruffy ⊗ Cat) : I → N, Eval ◦ (Scruffy ⊗ Dog) : I → N ?

In general, no!

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

How about a limited form of reversibility?

Let us compare Cat : I → N Dog : I → N do we get the same value when we compare Eval ◦ (Scruffy ⊗ Cat) : I → N, Eval ◦ (Scruffy ⊗ Dog) : I → N ?

In general, no!

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

In closed categories

Elements C(I, [X → Y]) are in 1:1 correspondence with Arrows C(X, Y) Most elements do not correspond to isomorphisms!

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

A special case: In HilbFD The element ❈

|Ψ H ⊗ K

∼ = [H → K]

maps to the arrow H

LΨ K

LΨ : H → K is unitary exactly when |Ψ is maximally entangled!

This is, of course, a very special condition.

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Must evaluation always lose information?

Sometimes, it is undesirable for ‘reduction’ to lose information! An example ...

Fidel Castro and his beard

The compound noun-phrase

N ⊗ [N ⊗ N → N] ⊗ N

The typing

N

After evaluation

The arrow named by I

and [N ⊗ N → N]

should not lose information about either

1

Fidel Castro,

2

Fidel Castro’s beard.

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Must evaluation always lose information?

Sometimes, it is undesirable for ‘reduction’ to lose information! An example ...

Fidel Castro and his beard

The compound noun-phrase

N ⊗ [N ⊗ N → N] ⊗ N

The typing

N

After evaluation

The arrow named by I

and [N ⊗ N → N]

should not lose information about either

1

Fidel Castro,

2

Fidel Castro’s beard.

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

A more serious example

The (polymorphic) connective type ΛX.[X ⊗ X → X] can be applied to the sentence type S Bobby likes Marilyn Monroe and I like Fidel Castro We do not wish the evaluation S ⊗ [S ⊗ S → S] ⊗ S

Eval

S

to lose information about either sub-sentence.

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Polymorphism and reversibility

The arrow S ⊗ S → S named by I

and

[S ⊗ S → S]

must be a monomorphism. This is closely related to models of polymorphic types.

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

building polymorphic types – a special case

We require an embedding: C(S ⊗ S, S ⊗ S) ֒ → C(S, S) ‘S contains a copy of S ⊗ S’ A special case We look at the special case where this is an isomorphism: S ⊗ S

∇

S

∆

• 1S
• 1S⊗S
• ∆ ◦ ∇ = 1S , ∇ ◦ ∆ = 1S⊗S

The two situations are (broadly speaking) interchangeable.

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

building polymorphic types – a special case

We require an embedding: C(S ⊗ S, S ⊗ S) ֒ → C(S, S) ‘S contains a copy of S ⊗ S’ A special case We look at the special case where this is an isomorphism: S ⊗ S

∇

S

∆

• 1S
• 1S⊗S
• ∆ ◦ ∇ = 1S , ∇ ◦ ∆ = 1S⊗S

The two situations are (broadly speaking) interchangeable.

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

A distinguished, closed, subcategory

Consider the subcategory of C generated by S ∈ Ob(C) , ( ⊗ ) We have the following isomorphisms: S ⊗ S ∼ = S [S → S] = S† ⊗ S = S ⊗ S ∼ = S We have a compact closed subcategory1 where all objects are isomorphic.

1without unit object http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

A distinguished, closed, subcategory

Consider the subcategory of C generated by S ∈ Ob(C) , ( ⊗ ) We have the following isomorphisms: S ⊗ S ∼ = S [S → S] = S† ⊗ S = S ⊗ S ∼ = S We have a compact closed subcategory1 where all objects are isomorphic.

1without unit object http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

A distinguished, closed, subcategory

Consider the subcategory of C generated by S ∈ Ob(C) , ( ⊗ ) We have the following isomorphisms: S ⊗ S ∼ = S [S → S] = S† ⊗ S = S ⊗ S ∼ = S We have a compact closed subcategory1 where all objects are isomorphic.

1without unit object http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

IMPORTANT!

In this subcategory, we cannot assume strict associativity A ⊗ (B ⊗ C) = (A ⊗ B) ⊗ C Associativity must be up to canonical isomorphism: tABC : A ⊗ (B ⊗ C) → (A ⊗ B) ⊗ C A classic result (J. Isbell / S. MacLane) Trying to combine: Strict associativity S ⊗ (S ⊗ S) = (S ⊗ S) ⊗ S self-similarity S ∼ = S ⊗ S forces S to collapse to the unit object. ‘Categories for the working mathematician’ uses this to justify “associativity up to isomorphism” instead of strict associativity.

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IMPORTANT!

In this subcategory, we cannot assume strict associativity A ⊗ (B ⊗ C) = (A ⊗ B) ⊗ C Associativity must be up to canonical isomorphism: tABC : A ⊗ (B ⊗ C) → (A ⊗ B) ⊗ C A classic result (J. Isbell / S. MacLane) Trying to combine: Strict associativity S ⊗ (S ⊗ S) = (S ⊗ S) ⊗ S self-similarity S ∼ = S ⊗ S forces S to collapse to the unit object. ‘Categories for the working mathematician’ uses this to justify “associativity up to isomorphism” instead of strict associativity.

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Another digression

(for logicians & hardcore category-theorists)

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Compact closed monoids

The identities S ∼ = S ⊗ S ∼ = [S → S] look like the defining equations of a C-monoid (a Cartesian closed monoid / model of untyped λ- calculus). This analogy can be taken seriously For any object X of this subcategory, C(X, X) is a compact closed monoid.

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The structure of C(S, S)

This has a monoidal tensor ⊗∆ : C(S, S) × C(S, S) → C(S, S) This is defined by convolution: S

∆

• f⊗∆g
• S ⊗ S

f⊗g

• S

S ⊗ S

∇

• This is:

Associative (up to isomorphism) Commutative (up to isomorphism) However, there is no unit object.

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The structure of C(S, S)

This has a monoidal tensor ⊗∆ : C(S, S) × C(S, S) → C(S, S) This is defined by convolution: S

∆

• f⊗∆g
• S ⊗ S

f⊗g

• S

S ⊗ S

∇

• This is:

Associative (up to isomorphism) Commutative (up to isomorphism) However, there is no unit object.

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Associativity of ⊗∆

There is an associativity isomorphism t∆ ∈ C(S, S) satisfying: t∆.(f ⊗∆ (g ⊗∆ h)) = ((f ⊗∆ g) ⊗∆ h).t∆ MacLane’s pentagon condition. It also satisfies: S

∆

• t∆
• S ⊗ S

1S⊗∆

S ⊗ (S ⊗ S)

tS,S,S

• S

S ⊗ S

∇

• (S ⊗ S) ⊗ S

∇⊗1S

• http://www.peterhines.net/downloads/talks/Types.pdf

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Associativity of ⊗∆

There is an associativity isomorphism t∆ ∈ C(S, S) satisfying: t∆.(f ⊗∆ (g ⊗∆ h)) = ((f ⊗∆ g) ⊗∆ h).t∆ MacLane’s pentagon condition. It also satisfies: S

∆

• t∆
• S ⊗ S

1S⊗∆

S ⊗ (S ⊗ S)

tS,S,S

• S

S ⊗ S

∇

• (S ⊗ S) ⊗ S

∇⊗1S

• http://www.peterhines.net/downloads/talks/Types.pdf

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Symmetry of ⊗∆

There is also a symmetry isomorphism σ∆ ∈ C(S, S) satisfying σ∆.(f ⊗∆ g) = (g ⊗∆ f).σ∆ MacLane’s hexagon condition. It also satisfies: S

∆

• σ∆
• S ⊗ S

σS,S

• S

S ⊗ S

∇

• http://www.peterhines.net/downloads/talks/Types.pdf

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Symmetry of ⊗∆

There is also a symmetry isomorphism σ∆ ∈ C(S, S) satisfying σ∆.(f ⊗∆ g) = (g ⊗∆ f).σ∆ MacLane’s hexagon condition. It also satisfies: S

∆

• σ∆
• S ⊗ S

σS,S

• S

S ⊗ S

∇

• http://www.peterhines.net/downloads/talks/Types.pdf

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End of digression

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Back on track ...

Building models of polymorphism depends on: A distinguished object S ∈ Ob(C). Distinguished isomorphisms:

∆ : S → S ⊗ S ∇ : S ⊗ S → S.

We also assume ∆ = ∇† = ∇−1 — this hold in most concrete examples! Question: do we have a † Frobenius algebra?

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Ceci n’est pas un Frobenius algebra

This fails at the first step: Units are a problem There are no natural candidates for the units ⊥ : I → S , ⊤ : S → I How about a † Frobenius algebra without units?

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What about associativity?

In a Frobenius algebra, we need associativity

S

• ∇

S

• S
• ∇

S S S

• ∇(∇ ⊗ 1S) : S ⊗ (S ⊗ S) → S

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What about associativity?

In a Frobenius algebra, we need associativity

S S

• S
• ∇

S

• ∇

S

• S
• (∇ ⊗ 1S)∇ : (S ⊗ S) ⊗ S → S

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(Strict) Associativity fails!

The (strict) associative condition for a Frobenius algebra fails ... for deeply unsatisfactory reasons! We do have associativity up to isomorphism.

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We have associativity, up to isomorphism

Adding in canonical isomorphisms: S

∆

• t∆
• S ⊗ S

1S⊗∆

S ⊗ (S ⊗ S)

tS,S,S

• S

∆

S ⊗ S

∆⊗1S

(S ⊗ S) ⊗ S

(Recall t∆ ∈ C(S, S), the associativity arrow for ⊗∆ )

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We have associativity, up to isomorphism

The same canonical isomorphisms make the dual diagram commute: S ⊗ (S ⊗ S)

1S⊗∇

• tS,S,S
• S ⊗ S

∇

S

t∆

• (S ⊗ S) ⊗ S

∇⊗1S

S ⊗ S

∇

S

We have associativity, and co-associativity, up to isomorphism.

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We have lax monoids / comonoids

Provided we don’t care about units: We have a (lax) monoid and comonoid at S.

We call these unitless monoids / comonoids, even though a monoid without a unit is a semigroup

How about the Frobenius condition?

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We have lax monoids / comonoids

Provided we don’t care about units: We have a (lax) monoid and comonoid at S.

We call these unitless monoids / comonoids, even though a monoid without a unit is a semigroup

How about the Frobenius condition?

http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

We have lax monoids / comonoids

Provided we don’t care about units: We have a (lax) monoid and comonoid at S.

We call these unitless monoids / comonoids, even though a monoid without a unit is a semigroup

How about the Frobenius condition?

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The Frobenius condition?

The Frobenius condition requires: The composite:

S S S

• ∆
• ∇

S S S

• http://www.peterhines.net/downloads/talks/Types.pdf

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The Frobenius condition?

The Frobenius condition requires: is equal to

S

• S
• ∆
• ∇
• S
• S

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As a commutative diagram

The Frobenius condition S ⊗ S

∇⊗1S

• ∇◦∆
• S ⊗ S ⊗ S

Strict equality!

• S ⊗ S

S ⊗ S ⊗ S

1S⊗∆

• We replace strict associativity by isomorphism:

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The Frobenius condition (up to iso.)

The following is satisfied: S ⊗ S

∇⊗1S

• ∇◦t−1

∆ ◦∆

• (S ⊗ S) ⊗ S

t−1

S,S,S

• S ⊗ S

S ⊗ (S ⊗ S)

1S⊗∆

• We have the Frobenius condition, up to canonical isomorphism.

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Anything else?

We have a unitless † Frobenius algebra (up to canonical iso.) — anything else ?? We have commutativity & co-commutativity e.g.

S ⊗ S

σS,S

• ∇

S

σ∆

• S ⊗ S

∇

S

Again, up to canonical isomorphism.

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Anything else?

We have a unitless † Frobenius algebra (up to canonical iso.) — anything else ?? We have commutativity & co-commutativity e.g.

S ⊗ S

σS,S

• ∇

S

σ∆

• S ⊗ S

∇

S

Again, up to canonical isomorphism.

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One final point

We also have the ‘classical structure’ condition: ∇ ◦ ∆ = 1S (This was our starting point!) Conclusion: the ‘polymorphism condition’, S ∼ = S ⊗ S, leads to a (lax, unitless) ‘classical structure’ – as used to specify orthonormal bases in categorical quantum mechanics.

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One final point

We also have the ‘classical structure’ condition: ∇ ◦ ∆ = 1S (This was our starting point!) Conclusion: the ‘polymorphism condition’, S ∼ = S ⊗ S, leads to a (lax, unitless) ‘classical structure’ – as used to specify orthonormal bases in categorical quantum mechanics.

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The real conclusion: “I had to say something, to strike them kind of weird, so I yelled ‘I like Fidel Castro, and his beard’. ”

– Bob Dylan, Motorpsycho Nightmare

A similar result can be obtained by talking about polymorphism.

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