Reynolds Parametricity Patricia Johann Appalachian State University - - PowerPoint PPT Presentation

Reynolds’ Parametricity

Patricia Johann Appalachian State University

cs.appstate.edu/

∼johannp

Based on joint work with Neil Ghani, Fredrik Nordvall Forsberg, Federico Orsanigo, and Tim Revell

OPLSS 2016

Course Outline

Topic: Reynolds’ theory of parametric polymorphism for System F Goals: - extract the fibrational essence of Reynolds’ theory

• generalize Reynolds’ construction to very general models
• Lecture 1: Reynolds’ theory of parametricity for System F
• Lecture 2: Introduction to fibrations
• Lecture 3: A bifibrational view of parametricity
• Lecture 4: Bifibrational parametric models for System F

Course Outline

Topic: Reynolds’ theory of parametric polymorphism for System F Goals: - extract the fibrational essence of Reynolds’ theory

• generalize Reynolds’ construction to very general models
• Lecture 1: Reynolds’ theory of parametricity for System F
• Lecture 2: Introduction to fibrations
• Lecture 3: A bifibrational view of parametricity
• Lecture 4: Bifibrational parametric models for System F

Where Were We?

• Last time we recalled Reynolds’ standard relational parametricity

Where Were We?

• Last time we recalled Reynolds’ standard relational parametricity
• This is the main inspiration for the bifibrational model of parametricity

for System F we will develop

Where Were We?

• Last time we recalled Reynolds’ standard relational parametricity
• This is the main inspiration for the bifibrational model of parametricity

for System F we will develop

• View Reynolds’ construction and results through the lens of the rela-

tions (bi)fibration on Set

Where Were We?

• Last time we recalled Reynolds’ standard relational parametricity
• This is the main inspiration for the bifibrational model of parametricity

for System F we will develop

• View Reynolds’ construction and results through the lens of the rela-

tions (bi)fibration on Set

• Generalize Reynolds’ constructions to bifibrational models of System

F for which we can prove (bifibrational versions of) the IEL and Ab- straction Theorem

Where Were We?

• Last time we recalled Reynolds’ standard relational parametricity
• This is the main inspiration for the bifibrational model of parametricity

for System F we will develop

• View Reynolds’ construction and results through the lens of the rela-

tions (bi)fibration on Set

• Generalize Reynolds’ constructions to bifibrational models of System

F for which we can prove (bifibrational versions of) the IEL and Ab- straction Theorem

• Reynolds’ construction is (ignoring size issues) such a model

Motivation: Indexed Families of Sets

• A fibration captures a family (EB)B∈B of categories EB indexed over
• bjects of a(nother) category B

Motivation: Indexed Families of Sets

• A fibration captures a family (EB)B∈B of categories EB indexed over
• bjects of a(nother) category B
• A fibration is a functor U : E → B

– B is the base category of U – E is the total category of U

Motivation: Indexed Families of Sets

• A fibration captures a family (EB)B∈B of categories EB indexed over
• bjects of a(nother) category B
• A fibration is a functor U : E → B

– B is the base category of U – E is the total category of U Intuitively, E =

B∈B EB

Motivation: Indexed Families of Sets

• A fibration captures a family (EB)B∈B of categories EB indexed over
• bjects of a(nother) category B
• A fibration is a functor U : E → B

– B is the base category of U – E is the total category of U Intuitively, E =

B∈B EB

• U must have some additional properties for describing indexing

Motivation: Indexed Families of Sets

• A fibration captures a family (EB)B∈B of categories EB indexed over
• bjects of a(nother) category B
• A fibration is a functor U : E → B

– B is the base category of U – E is the total category of U Intuitively, E =

B∈B EB

• U must have some additional properties for describing indexing
• We are interested in indexing because Reynolds’ interpretations are

type-indexed

Display Maps

• Simple case: Indexing for sets

– B is a set I of indices, – E is X =

i∈I Xi, where (Xi)i∈I is a (wlog, disjoint) family of sets

– U : X → I maps each x ∈ X to the index i ∈ I such that x ∈ Xi

Display Maps

• Simple case: Indexing for sets

– B is a set I of indices, – E is X =

i∈I Xi, where (Xi)i∈I is a (wlog, disjoint) family of sets

– U : X → I maps each x ∈ X to the index i ∈ I such that x ∈ Xi

• U is called the display map for (Xi)i∈I

Display Maps

• Simple case: Indexing for sets

– B is a set I of indices, – E is X =

i∈I Xi, where (Xi)i∈I is a (wlog, disjoint) family of sets

– U : X → I maps each x ∈ X to the index i ∈ I such that x ∈ Xi

• U is called the display map for (Xi)i∈I
• It is customary to draw it vertically, like this:

X

U

• I

Display Maps

• Simple case: Indexing for sets

– B is a set I of indices, – E is X =

i∈I Xi, where (Xi)i∈I is a (wlog, disjoint) family of sets

– U : X → I maps each x ∈ X to the index i ∈ I such that x ∈ Xi

• U is called the display map for (Xi)i∈I
• It is customary to draw it vertically, like this:

X

U

• I
• The set

Xi = U −1(i) = {x ∈ X| Ux = i} is called the fibre of X over i

Categories from Indexed Families - Example I

• The slice category Set/I

Categories from Indexed Families - Example I

• The slice category Set/I

– An object in Set/I is a function U : X → I in Set

Categories from Indexed Families - Example I

• The slice category Set/I

– An object in Set/I is a function U : X → I in Set – A morphism from U ′ : X′ → I and U : X → I in Set/I is a function g : X′ → X in Set such that U ◦ g = U ′ X′

g

• U′
• X

U

• I

Categories from Indexed Families - Example I

• The slice category Set/I

– An object in Set/I is a function U : X → I in Set – A morphism from U ′ : X′ → I and U : X → I in Set/I is a function g : X′ → X in Set such that U ◦ g = U ′ X′

g

• U′
• X

U

• I
• We can view g as a family of functions (gi)i∈I, where gi : X′

i → Xi

Categories from Indexed Families - Example I

• The slice category Set/I

– An object in Set/I is a function U : X → I in Set – A morphism from U ′ : X′ → I and U : X → I in Set/I is a function g : X′ → X in Set such that U ◦ g = U ′ X′

g

• U′
• X

U

• I
• We can view g as a family of functions (gi)i∈I, where gi : X′

i → Xi

• Identities and composition are inherited from Set

Categories from Indexed Families - Example II

• The arrow category Set→

Categories from Indexed Families - Example II

• The arrow category Set→

– An object of Set→ is a function U : X → I in Set for some index set I

Categories from Indexed Families - Example II

• The arrow category Set→

– An object of Set→ is a function U : X → I in Set for some index set I – A morphism from U ′ : Y → J to U : X → I in Set→ is a pair (g : Y → X, f : J → I) of functions in Set such that U ◦ g = f ◦ U ′ Y

g

• U′
• X

U

• J

f

I

Categories from Indexed Families - Example II

• The arrow category Set→

– An object of Set→ is a function U : X → I in Set for some index set I – A morphism from U ′ : Y → J to U : X → I in Set→ is a pair (g : Y → X, f : J → I) of functions in Set such that U ◦ g = f ◦ U ′ Y

g

• U′
• X

U

• J

f

I

• We can view g as a family of functions (gj)j∈J, where gj : Yj → Xf(j)

(since g(y) ∈ U −1(f(j)) for any y ∈ Yj = U ′−1(j) )

Categories from Indexed Families - Example II

• The arrow category Set→

– An object of Set→ is a function U : X → I in Set for some index set I – A morphism from U ′ : Y → J to U : X → I in Set→ is a pair (g : Y → X, f : J → I) of functions in Set such that U ◦ g = f ◦ U ′ Y

g

• U′
• X

U

• J

f

I

• We can view g as a family of functions (gj)j∈J, where gj : Yj → Xf(j)

(since g(y) ∈ U −1(f(j)) for any y ∈ Yj = U ′−1(j) )

• Identities and composition are componentwise inherited from Set.

Categories from Indexed Families - Example II

• The arrow category Set→

– An object of Set→ is a function U : X → I in Set for some index set I – A morphism from U ′ : Y → J to U : X → I in Set→ is a pair (g : Y → X, f : J → I) of functions in Set such that U ◦ g = f ◦ U ′ Y

g

• U′
• X

U

• J

f

I

• We can view g as a family of functions (gj)j∈J, where gj : Yj → Xf(j)

(since g(y) ∈ U −1(f(j)) for any y ∈ Yj = U ′−1(j) )

• Identities and composition are componentwise inherited from Set.
• Set→ induces a codomain functor cod : Set→ → Set mapping

U : X → I to I and (g, f) to f

Substitution

• Consider U : X → I for X = (Xi)i∈I for some index set I

Substitution

• Consider U : X → I for X = (Xi)i∈I for some index set I
• Substitution along f : J → I turns the family (Xi)i∈I into a family

(Yj)j∈J such that Yj = Xf(j)

Substitution

• Consider U : X → I for X = (Xi)i∈I for some index set I
• Substitution along f : J → I turns the family (Xi)i∈I into a family

(Yj)j∈J such that Yj = Xf(j)

• (Yj)j∈J is obtained by pullback of U along f

Y

g

• U′
• X

U

• J

f

I

Substitution

• Consider U : X → I for X = (Xi)i∈I for some index set I
• Substitution along f : J → I turns the family (Xi)i∈I into a family

(Yj)j∈J such that Yj = Xf(j)

• (Yj)j∈J is obtained by pullback of U along f

Y

g

• U′
• X

U

• J

f

I

• Y = {(j, x) ∈ J ×X | U(x) = f(j)} with projection functions g and U ′

Substitution

• Consider U : X → I for X = (Xi)i∈I for some index set I
• Substitution along f : J → I turns the family (Xi)i∈I into a family

(Yj)j∈J such that Yj = Xf(j)

• (Yj)j∈J is obtained by pullback of U along f

Y

g

• U′
• X

U

• J

f

I

• Y = {(j, x) ∈ J ×X | U(x) = f(j)} with projection functions g and U ′
• U ′ : Y → J gives a new family of sets (Yj)j∈J whose fibres are

Yj = U ′−1(j) = {x ∈ X | U(x) = f(j)} = U −1(f(j)) = Xf(j)

Substitution

• Consider U : X → I for X = (Xi)i∈I for some index set I
• Substitution along f : J → I turns the family (Xi)i∈I into a family

(Yj)j∈J such that Yj = Xf(j)

• (Yj)j∈J is obtained by pullback of U along f

Y

g

• U′
• X

U

• J

f

I

• Y = {(j, x) ∈ J ×X | U(x) = f(j)} with projection functions g and U ′
• U ′ : Y → J gives a new family of sets (Yj)j∈J whose fibres are

Yj = U ′−1(j) = {x ∈ X | U(x) = f(j)} = U −1(f(j)) = Xf(j)

• We usually write f ∗(U) for the display map U ′

Substitution - Example 1

• Let f be an element f : {∗} → I

Substitution - Example 1

• Let f be an element f : {∗} → I
• Then f picks out an element i of I (i.e., f(∗) = i)

Substitution - Example 1

• Let f be an element f : {∗} → I
• Then f picks out an element i of I (i.e., f(∗) = i)
• Y∗ = U ′−1(∗) = {x ∈ X | U(x) = i} = U −1(i) = Xi

Substitution - Example 1

• Let f be an element f : {∗} → I
• Then f picks out an element i of I (i.e., f(∗) = i)
• Y∗ = U ′−1(∗) = {x ∈ X | U(x) = i} = U −1(i) = Xi
• Thus Y =

j∈{∗} Yj = Y∗ = Xi

Substitution - Example 1

• Let f be an element f : {∗} → I
• Then f picks out an element i of I (i.e., f(∗) = i)
• Y∗ = U ′−1(∗) = {x ∈ X | U(x) = i} = U −1(i) = Xi
• Thus Y =

j∈{∗} Yj = Y∗ = Xi

• So substituting along a particular element of I selects the fibre of X
• ver that element

Substitution - Example 2

• Let f be a non-indexed set f : J → {∗}

Substitution - Example 2

• Let f be a non-indexed set f : J → {∗}
• Then, for every j ∈ J,

Yj = U ′−1(j) = {x ∈ X | U(x) = ∗} = U −1(∗) = X∗ = X

Substitution - Example 2

• Let f be a non-indexed set f : J → {∗}
• Then, for every j ∈ J,

Yj = U ′−1(j) = {x ∈ X | U(x) = ∗} = U −1(∗) = X∗ = X

• So Y =

j∈J Yj = J × X (since the Yj are disjoint)

Substitution - Example 3

• Let f be a projection f : I × J → I

Substitution - Example 3

• Let f be a projection f : I × J → I
• Then, for every pair (i, j),

Y(i,j) = U ′−1(i, j) = {x ∈ X | U(x) = i} = U −1(i) = Xi

Substitution - Example 3

• Let f be a projection f : I × J → I
• Then, for every pair (i, j),

Y(i,j) = U ′−1(i, j) = {x ∈ X | U(x) = i} = U −1(i) = Xi

• So Y =

(i,j)∈I×J Y(i,j) = (i,j)∈I×J Xi = Xi × J

Substitution - Example 3

• Let f be a projection f : I × J → I
• Then, for every pair (i, j),

Y(i,j) = U ′−1(i, j) = {x ∈ X | U(x) = i} = U −1(i) = Xi

• So Y =

(i,j)∈I×J Y(i,j) = (i,j)∈I×J Xi = Xi × J

• There is a “dummy” index j in the family f ∗(U) that plays no role

Substitution - Example 3

• Let f be a projection f : I × J → I
• Then, for every pair (i, j),

Y(i,j) = U ′−1(i, j) = {x ∈ X | U(x) = i} = U −1(i) = Xi

• So Y =

(i,j)∈I×J Y(i,j) = (i,j)∈I×J Xi = Xi × J

• There is a “dummy” index j in the family f ∗(U) that plays no role
• Logically speaking, substitution along a projection is weakening

Substitution - Example 4

• Let f be a diagonal map f : I → I × I

Substitution - Example 4

• Let f be a diagonal map f : I → I × I
• Then, for every i ∈ I,

Yi = U ′−1(i) = {x ∈ X | U(x) = (i, i)} = U −1(i, i) = X(i,i)

Substitution - Example 4

• Let f be a diagonal map f : I → I × I
• Then, for every i ∈ I,

Yi = U ′−1(i) = {x ∈ X | U(x) = (i, i)} = U −1(i, i) = X(i,i)

• So Y =

i∈I Yi = (i,i)∈I×I X(i,i)

Substitution - Example 4

• Let f be a diagonal map f : I → I × I
• Then, for every i ∈ I,

Yi = U ′−1(i) = {x ∈ X | U(x) = (i, i)} = U −1(i, i) = X(i,i)

• So Y =

i∈I Yi = (i,i)∈I×I X(i,i)

• In other words, Y is restriction of

(i,i′)∈I×I X(i,i′) to the diagonal i = i′

Substitution - Example 4

• Let f be a diagonal map f : I → I × I
• Then, for every i ∈ I,

Yi = U ′−1(i) = {x ∈ X | U(x) = (i, i)} = U −1(i, i) = X(i,i)

• So Y =

i∈I Yi = (i,i)∈I×I X(i,i)

• In other words, Y is restriction of

(i,i′)∈I×I X(i,i′) to the diagonal i = i′

• Logically speaking, substitution along a diagonal is contraction

Best Substitution Morphisms - Part I

• The pair (g, f) in the pullback diagram

Y

g

• f ∗(U)
• X

U

• J

f

I

is a morphism from f ∗(U) to U in the arrow category Set→

Best Substitution Morphisms - Part I

• The pair (g, f) in the pullback diagram

Y

g

• f ∗(U)
• X

U

• J

f

I

is a morphism from f ∗(U) to U in the arrow category Set→

• We call (g, f) a substitution morphism from f ∗(U) to U

Best Substitution Morphisms - Part II

• (g, f) is such that if

– U ′′ : Z → K is any object in Set→ – (g′, f ′) : U ′′ → U is a morphism in Set→ – f ′ : K → I factors through f : J → I via v : K → J (i.e., f ′ = f ◦v) Z

g′

• U′′
• Y

g

• f ∗(U)
• X

U

• K

f ′

• v

J

f

I

Best Substitution Morphisms - Part II

• (g, f) is such that if

– U ′′ : Z → K is any object in Set→ – (g′, f ′) : U ′′ → U is a morphism in Set→ – f ′ : K → I factors through f : J → I via v : K → J (i.e., f ′ = f ◦v) then there exists a unique h : Z → Y in Set→ such that – cod(h, v) = v for cod : Set→ → Set – g ◦ h = g′ Z

g′

• h
• U′′
• Y

g

• f ∗(U)
• X

U

• K

f ′

• v

J

f

I

Best Substitution Morphisms - Part II

• (g, f) is such that if

– U ′′ : Z → K is any object in Set→ – (g′, f ′) : U ′′ → U is a morphism in Set→ – f ′ : K → I factors through f : J → I via v : K → J (i.e., f ′ = f ◦v) then there exists a unique h : Z → Y in Set→ such that – cod(h, v) = v for cod : Set→ → Set – g ◦ h = g′ Z

g′

• h
• U′′
• Y

g

• f ∗(U)
• X

U

• K

f ′

• v

J

f

I

• That is, (g, f) is the best substitution morphism from f ∗(U) to U

Best Substitution Morphisms - Part II

• (g, f) is such that if

– U ′′ : Z → K is any object in Set→ – (g′, f ′) : U ′′ → U is a morphism in Set→ – f ′ : K → I factors through f : J → I via v : K → J (i.e., f ′ = f ◦v) then there exists a unique h : Z → Y in Set→ such that – cod(h, v) = v for cod : Set→ → Set – g ◦ h = g′ Z

g′

• h
• U′′
• Y

g

• f ∗(U)
• X

U

• K

f ′

• v

J

f

I

• That is, (g, f) is the best substitution morphism from f ∗(U) to U
• The existence of such best substitution morphisms is what makes cod :

Set→ → Set a fibration

Cartesian Morphisms

• Let U : E → B be a functor

Cartesian Morphisms

• Let U : E → B be a functor
• A morphism g : Q → P in E is cartesian over f : X → Y in B if

Cartesian Morphisms

• Let U : E → B be a functor
• A morphism g : Q → P in E is cartesian over f : X → Y in B if

– Ug = f

Cartesian Morphisms

• Let U : E → B be a functor
• A morphism g : Q → P in E is cartesian over f : X → Y in B if

– Ug = f – for every g′ : Q′ → P in E with Ug′ = f ◦ v for some v : UQ′ → X, there exists a unique h : Q′ → Q with Uh = v and g′ = g ◦ h

Cartesian Morphisms

• Let U : E → B be a functor
• A morphism g : Q → P in E is cartesian over f : X → Y in B if

– Ug = f – for every g′ : Q′ → P in E with Ug′ = f ◦ v for some v : UQ′ → X, there exists a unique h : Q′ → Q with Uh = v and g′ = g ◦ h E

U

• Q′

h

• g′
• Q

g

P

UQ′

v

• Ug′
• B

X

f

Y

Opcartesian Morphisms

• Let U : E → B be a functor

Opcartesian Morphisms

• Let U : E → B be a functor
• A morphism g : P → Q in E is opcartesian over f : X → Y in B if

Opcartesian Morphisms

• Let U : E → B be a functor
• A morphism g : P → Q in E is opcartesian over f : X → Y in B if

– Ug = f

Opcartesian Morphisms

• Let U : E → B be a functor
• A morphism g : P → Q in E is opcartesian over f : X → Y in B if

– Ug = f – for every g′ : P → Q′ in E with Ug′ = v ◦ f for some v : Y → UQ′, there exists a unique h : Q → Q′ with Uh = v and g′ = h ◦ g

Opcartesian Morphisms

• Let U : E → B be a functor
• A morphism g : P → Q in E is opcartesian over f : X → Y in B if

– Ug = f – for every g′ : P → Q′ in E with Ug′ = v ◦ f for some v : Y → UQ′, there exists a unique h : Q → Q′ with Uh = v and g′ = h ◦ g E

U

• Q′

h

• g′
• Q′

Q

g

P

P

g′

• g

Q

h

• UQ′

v

• Ug′
• UQ′

B X

f

Y

X

Ug′

• f

Y

v

Observations and Notation

• Let P in E and f : X → Y with UP = Y

Observations and Notation

• Let P in E and f : X → Y with UP = Y
• (Op)cartesian morphisms over f wrt P are unique up to isomorphism

Observations and Notation

• Let P in E and f : X → Y with UP = Y
• (Op)cartesian morphisms over f wrt P are unique up to isomorphism
• f §

P is the cartesian morphism over f with codomain P

Observations and Notation

• Let P in E and f : X → Y with UP = Y
• (Op)cartesian morphisms over f wrt P are unique up to isomorphism
• f §

P is the cartesian morphism over f with codomain P

• f P

§ is the opcartesian morphism over f with domain P

Observations and Notation

• Let P in E and f : X → Y with UP = Y
• (Op)cartesian morphisms over f wrt P are unique up to isomorphism
• f §

P is the cartesian morphism over f with codomain P

• f P

§ is the opcartesian morphism over f with domain P

• f ∗P is the domain of f §

P

Observations and Notation

• Let P in E and f : X → Y with UP = Y
• (Op)cartesian morphisms over f wrt P are unique up to isomorphism
• f §

P is the cartesian morphism over f with codomain P

• f P

§ is the opcartesian morphism over f with domain P

• f ∗P is the domain of f §

P

• ΣfP is the codomain of f P

§

Fibrations and Opfibrations

• U : E → B is a fibration if for every object P of E and every f : X →

UP in B, there is a cartesian morphism f §

P : Q → P in E over f

Fibrations and Opfibrations

• U : E → B is a fibration if for every object P of E and every f : X →

UP in B, there is a cartesian morphism f §

P : Q → P in E over f

• U : E → B is an opfibration if for every object P of E and every

f : UP → Y in B, there is an opcartesian morphism f P

§ : P → Q in E

• ver f

Fibrations and Opfibrations

• U : E → B is a fibration if for every object P of E and every f : X →

UP in B, there is a cartesian morphism f §

P : Q → P in E over f

• U : E → B is an opfibration if for every object P of E and every

f : UP → Y in B, there is an opcartesian morphism f P

§ : P → Q in E

• ver f
• U : E → B is a bifibration if it is both a fibration and an opfibration

Fibrations and Opfibrations

• U : E → B is a fibration if for every object P of E and every f : X →

UP in B, there is a cartesian morphism f §

P : Q → P in E over f

• U : E → B is an opfibration if for every object P of E and every

f : UP → Y in B, there is an opcartesian morphism f P

§ : P → Q in E

• ver f
• U : E → B is a bifibration if it is both a fibration and an opfibration
• If U : E → B is a fibration, opfibration, or bifibration, then an object

P in E is over its image UP and similarly for morphisms

Fibrations and Opfibrations

• U : E → B is a fibration if for every object P of E and every f : X →

UP in B, there is a cartesian morphism f §

P : Q → P in E over f

• U : E → B is an opfibration if for every object P of E and every

f : UP → Y in B, there is an opcartesian morphism f P

§ : P → Q in E

• ver f
• U : E → B is a bifibration if it is both a fibration and an opfibration
• If U : E → B is a fibration, opfibration, or bifibration, then an object

P in E is over its image UP and similarly for morphisms

• A morphism is vertical if it is over id

Fibrations and Opfibrations

• U : E → B is a fibration if for every object P of E and every f : X →

UP in B, there is a cartesian morphism f §

P : Q → P in E over f

• U : E → B is an opfibration if for every object P of E and every

f : UP → Y in B, there is an opcartesian morphism f P

§ : P → Q in E

• ver f
• U : E → B is a bifibration if it is both a fibration and an opfibration
• If U : E → B is a fibration, opfibration, or bifibration, then an object

P in E is over its image UP and similarly for morphisms

• A morphism is vertical if it is over id
• The fibre EX over an object X in B is the subcategory of E of objects
• ver X and morphisms over id X

Indexing and Reindexing Functors

• The function mapping each object P of E to f ∗P extends to the rein-

dexing functor f ∗ : EY → EX along f mapping each k : P → P ′ in EY to the (unique) morphism f ∗k such that k ◦ f §

P = f § P ′ ◦ f ∗k

U E f ∗ B f X Y

Indexing and Reindexing Functors

• The function mapping each object P of E to f ∗P extends to the rein-

dexing functor f ∗ : EY → EX along f mapping each k : P → P ′ in EY to the (unique) morphism f ∗k such that k ◦ f §

P = f § P ′ ◦ f ∗k

• The function mapping each object P of E to ΣfP extends to the oprein-

dexing functor Σf : EX → EY along f mapping each k : P → P ′ in EX to the (unique) morphism Σfk such that Σfk ◦ f P

§ = f P ′ §

• k

U E f ∗ Σf B f X Y

New Fibrations from Old

• |C| is the discrete category of C

New Fibrations from Old

• |C| is the discrete category of C
• The discrete functor |U| : |E| → |B| is induced by the restriction of

U : E → B to |E|

New Fibrations from Old

• |C| is the discrete category of C
• The discrete functor |U| : |E| → |B| is induced by the restriction of

U : E → B to |E|

• Cn is the n-fold product of C (in Cat)

New Fibrations from Old

• |C| is the discrete category of C
• The discrete functor |U| : |E| → |B| is induced by the restriction of

U : E → B to |E|

• Cn is the n-fold product of C (in Cat)
• The n-fold product of U : E → B, denoted U n : En → Bn, is given by

U n(X1, ..., Xn) = (UX1, ..., UXn) and U n(f1, ..., fn) = (Uf1, ..., Ufn)

New Fibrations from Old

• |C| is the discrete category of C
• The discrete functor |U| : |E| → |B| is induced by the restriction of

U : E → B to |E|

• Cn is the n-fold product of C (in Cat)
• The n-fold product of U : E → B, denoted U n : En → Bn, is given by

U n(X1, ..., Xn) = (UX1, ..., UXn) and U n(f1, ..., fn) = (Uf1, ..., Ufn)

• Lemma
• 1. If U : E → B is a functor, then |U| : |E| → |B| is a bifibration,

called the discrete fibration for U

• 2. If U is a (bi)fibration then so is U n : En → Bn for any n ∈ Nat

Fibred Functors

• Let U : E → B and U ′ : E′ → B′ be fibrations

Fibred Functors

• Let U : E → B and U ′ : E′ → B′ be fibrations
• A fibred functor F : U ′ → U comprises two functors

Fo : B′ → B and Fr : E′ → E such that

Fibred Functors

• Let U : E → B and U ′ : E′ → B′ be fibrations
• A fibred functor F : U ′ → U comprises two functors

Fo : B′ → B and Fr : E′ → E such that – U ◦ Fr = Fo ◦ U ′ E′

Fr

• U′
• E

U

• B′

Fo

B

Fibred Functors

• Let U : E → B and U ′ : E′ → B′ be fibrations
• A fibred functor F : U ′ → U comprises two functors

Fo : B′ → B and Fr : E′ → E such that – U ◦ Fr = Fo ◦ U ′ E′

Fr

• U′
• E

U

• B′

Fo

B

– cartesian morphisms are preserved, i.e., if f in E′ is cartesian over g in B′ then Frf in E is cartesian over Fog in B

Fibred Natural Transformations

• Let F, F ′ : U ′ → U be fibred functors

Fibred Natural Transformations

• Let F, F ′ : U ′ → U be fibred functors
• A fibred natural transformation η : F ′ → F comprises two natural

transformations ηo : F ′

• → Fo

and ηr : F ′

r → Fr

Fibred Natural Transformations

• Let F, F ′ : U ′ → U be fibred functors
• A fibred natural transformation η : F ′ → F comprises two natural

transformations ηo : F ′

• → Fo

and ηr : F ′

r → Fr

such that U ◦ ηr = ηo ◦ U ′

Fibred Natural Transformations

• Let F, F ′ : U ′ → U be fibred functors
• A fibred natural transformation η : F ′ → F comprises two natural

transformations ηo : F ′

• → Fo

and ηr : F ′

r → Fr

such that U ◦ ηr = ηo ◦ U ′ F ′

rX U′

• ηrX
• F ′

rf

• FrX

U

• Frf
• F ′

rY ηrY

FrY

F ′

• U ′X

ηoUX

• F ′
• U′f
• FoUX

FoUf

• F ′
• U ′Y

ηoUY

FoUY

Coming Up

• View Reynolds’ construction and results through the lens of the rela-

tions (bi)fibration on Set

Coming Up

• View Reynolds’ construction and results through the lens of the rela-

tions (bi)fibration on Set

• Generalize Reynolds’ constructions to (bi)fibrational models of Sys-

tem F for which we can prove (fibrational versions of) the IEL and Abstraction Theorem Reynolds’ construction is (ignoring size issues) an instance

Coming Up

• View Reynolds’ construction and results through the lens of the rela-

tions (bi)fibration on Set

• Generalize Reynolds’ constructions to (bi)fibrational models of Sys-

tem F for which we can prove (fibrational versions of) the IEL and Abstraction Theorem Reynolds’ construction is (ignoring size issues) an instance

• Reynolds’ construction is (ignoring size issues) such a model

References

• Categorical Logic and Type Theory. B. Jacobs. Elsevier, 1999.
• Bifibrational functorial semantics for parametric polymorphism.

N. Ghani, P. Johann, F. Nordvall Forsberg, F. Orsanigo, and T. Revell. MFPS’15.