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

� � 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 exactly common the same [ 0 1 ] Disclaimer: any actual values given are estimates (random guesses) x , y The comparison � x | y � : I → I exists for elements I − → 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 exactly common the same [ 0 1 ] Disclaimer: any actual values given are estimates (random guesses) x , y The comparison � x | y � : I → I exists for elements I − → 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 exactly common the same [ 0 1 ] Disclaimer: any actual values given are estimates (random guesses) x , y The comparison � x | y � : I → I exists for elements I − → 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 Λ X . [ X → [ X → X ]] or equivalently, Λ X . [ 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 Λ X . [ X → [ X → X ]] or equivalently, Λ X . [ 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 � L 1 | L 2 � = 0 . 98 × 0 . 75 × 0 . 00 = 0 Important: We have compared L 1 and L 2 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 � L 1 | L 2 � = 0 . 98 × 0 . 75 × 0 . 00 = 0 Important: We have compared L 1 and L 2 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? y x I G I � � � ���������������� � � � � � � � Eval � x ′ � 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 eval A , B � B A ⊗ [ A → 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 eval A , B � B A ⊗ [ A → 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: Cat � N The element I provides information about cats in general ... Scruffy � [ N → N ] An element I might tell us about the general concept of ‘scruffiness’. The tensor product Scruffy ⊗ Cat � [ N → N ] ⊗ N I 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: Cat � N The element I provides information about cats in general ... Scruffy � [ N → N ] An element I might tell us about the general concept of ‘scruffiness’. The tensor product Scruffy ⊗ Cat � [ N → N ] ⊗ N I 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 Scruffy ⊗ Cat � [ N → N ] ⊗ N I provides too much information! Composing with the evaluation map: Scruffy ⊗ Cat � [ N → N ] ⊗ N I 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 Scruffy ⊗ Cat � [ N → N ] ⊗ N I provides too much information! Composing with the evaluation map: Scruffy ⊗ Cat � [ N → N ] ⊗ N I 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 Scruffy � [ N → N ] I is consumed in the evaluation ... and plays no further rˆ ole. 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 Hilb FD The element | Ψ � � H ⊗ K ∼ = [ H → K ] ❈ maps to the arrow L Ψ � K H L Ψ : 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 and � [ N ⊗ N → N ] The arrow named by I should not lose information about either Fidel Castro, 1 Fidel Castro’s beard. 2 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 and � [ N ⊗ N → N ] The arrow named by I should not lose information about either Fidel Castro, 1 Fidel Castro’s beard. 2 http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

A more serious example The (polymorphic) connective type Λ X . [ 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 Eval � S S ⊗ [ S ⊗ S → S ] ⊗ 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 and � [ S ⊗ S → S ] I 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 1 S ⊗ S S ⊗ S 1 S ∆ ∆ ◦ ∇ = 1 S , ∇ ◦ ∆ = 1 S ⊗ 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 1 S ⊗ S S ⊗ S 1 S ∆ ∆ ◦ ∇ = 1 S , ∇ ◦ ∆ = 1 S ⊗ 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 subcategory 1 where all objects are isomorphic. 1 without 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 subcategory 1 where all objects are isomorphic. 1 without 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 subcategory 1 where all objects are isomorphic. 1 without 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: t ABC : 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. 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: t ABC : 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. http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

Another digression (for logicians & hardcore category-theorists) http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

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

� � � � 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 S ⊗ S f ⊗ ∆ g f ⊗ g S S ⊗ S ∇ This is: Associative (up to isomorphism) Commutative (up to isomorphism) However, there is no unit object. http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

� � � � 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 S ⊗ S f ⊗ ∆ g f ⊗ g S S ⊗ S ∇ This is: Associative (up to isomorphism) Commutative (up to isomorphism) However, there is no unit object. http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

� � � � � 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: 1 S ⊗ ∆ ∆ � S ⊗ ( S ⊗ S ) S S ⊗ S t S , S , S t ∆ S ⊗ S ( S ⊗ S ) ⊗ S S ∇ ∇⊗ 1 S http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

� � � � � 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: 1 S ⊗ ∆ ∆ � S ⊗ ( S ⊗ S ) S S ⊗ S t S , S , S t ∆ S ⊗ S ( S ⊗ S ) ⊗ S S ∇ ∇⊗ 1 S http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

� � � � 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 ⊗ S S σ S , S σ ∆ S ⊗ S S ∇ http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

� � � � 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 ⊗ S S σ S , S σ ∆ S ⊗ S S ∇ 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 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? http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

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

What about associativity? In a Frobenius algebra, we need associativity S � � � � � � � ���� ���� � S ∇ � � � � � � � � � � � � � � ���� ���� � � S ∇ S � � � � � � � � S S ∇ ( ∇ ⊗ 1 S ) : S ⊗ ( S ⊗ S ) → S http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

What about associativity? In a Frobenius algebra, we need associativity S S � � � � � � � ���� ���� � S S ∇ � � � � � � � � � � � � � ���� ���� � � � ∇ S � � � � � � � � S ( ∇ ⊗ 1 S ) ∇ : ( S ⊗ S ) ⊗ S → S http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

(Strict) Associativity fails! The (strict) associative condition for a Frobenius algebra fails ... for deeply unsatisfactory reasons! We do have associativity up to isomorphism. http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

� � � We have associativity, up to isomorphism Adding in canonical isomorphisms: 1 S ⊗ ∆ ∆ � S ⊗ ( S ⊗ S ) S ⊗ S S t S , S , S t ∆ � S ⊗ S � ( S ⊗ S ) ⊗ S S ∆ ∆ ⊗ 1 S (Recall t ∆ ∈ C ( S , S ) , the associativity arrow for ⊗ ∆ ) http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

� � � We have associativity, up to isomorphism The same canonical isomorphisms make the dual diagram commute: 1 S ⊗∇ ∇ � S S ⊗ ( S ⊗ S ) S ⊗ S t S , S , S t ∆ � S ⊗ S � S ( S ⊗ S ) ⊗ S ∇⊗ 1 S ∇ We have associativity, and co-associativity, up to isomorphism . 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? 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? 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? http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

The Frobenius condition? The Frobenius condition requires: The composite: S S � � � � � � ���� ���� � � S ∆ � � � � � � � ���� ���� � ∇ S � � � � � � � � S S http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

The Frobenius condition? The Frobenius condition requires: is equal to S S � � � � � � � � � � � � � ���� ���� ���� ���� � � � ∆ ∇ � � � � � � � � � � � � � � � � S S http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

� � � � As a commutative diagram The Frobenius condition ∇⊗ 1 S S ⊗ S S ⊗ S ⊗ S Strict equality ! ∇◦ ∆ S ⊗ S S ⊗ S ⊗ S 1 S ⊗ ∆ We replace strict associativity by isomorphism: http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

� � � � The Frobenius condition (up to iso.) The following is satisfied: ∇⊗ 1 S S ⊗ S ( S ⊗ S ) ⊗ S t − 1 ∇◦ t − 1 ∆ ◦ ∆ S , S , S S ⊗ ( S ⊗ S ) S ⊗ S 1 S ⊗ ∆ We have the Frobenius condition, up to canonical isomorphism. http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

� � Anything else? We have a unitless † Frobenius algebra (up to canonical iso.) — anything else ?? We have commutativity & co-commutativity e.g. ∇ � S S ⊗ S σ S , S σ ∆ � S S ⊗ S ∇ Again, up to canonical isomorphism. http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

� � Anything else? We have a unitless † Frobenius algebra (up to canonical iso.) — anything else ?? We have commutativity & co-commutativity e.g. ∇ � S S ⊗ S σ S , S σ ∆ � S S ⊗ S ∇ Again, up to canonical isomorphism. http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

One final point We also have the ‘classical structure’ condition: ∇ ◦ ∆ = 1 S (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. http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types

One final point We also have the ‘classical structure’ condition: ∇ ◦ ∆ = 1 S (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. http://www.peterhines.net/downloads/talks/Types.pdf Fun & games with all types of types