Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed...
Transcript of Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed...
![Page 1: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/1.jpg)
Equality in the Dependently Typed LambdaCalculus:
An Introduction to Homotopy Type Theoryor: Connecting Topology and Logic with Category Theory
Nicolai Kraus
School of Computer ScienceUniversity of Nottingham, UK
21.10.2011
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 1 / 24
(1/24) COMPUTING 2011 – 2011-10-21
![Page 2: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/2.jpg)
Basics
Typed λ Calculus
Natural DeductionCurry-Howard∼= Type Theory
A→ B AB
Γ ` f : A→ B Γ ` u : AΓ ` f u : B
BA→ B
Γ, x : A ` t : BΓ ` λx.t : A→ B
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 2 / 24
(2/24) COMPUTING 2011 – 2011-10-21
![Page 3: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/3.jpg)
Basics
Dependently Typed λ Calculus
Types may depend on terms:
Vec A n
are Lists over A with length n.
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 3 / 24
(3/24) COMPUTING 2011 – 2011-10-21
![Page 4: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/4.jpg)
Basics
Dependently Typed λ Calculus
Natural DeductionCurry-Howard∼= Type Theory special case
∃x∈AB Σ(x:A).B A× B
∀x∈AB Π(x:A).B A→ B
Usage, e.g. Agda & Epigram:proof assistants, formal verification, proof-carrying code
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 4 / 24
(4/24) COMPUTING 2011 – 2011-10-21
![Page 5: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/5.jpg)
The J Eliminator
Problems...
Typechecking requires Computation.
Equality is no longer decidable in general.
We want decidable typechecking.
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 5 / 24
(5/24) COMPUTING 2011 – 2011-10-21
![Page 6: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/6.jpg)
The J Eliminator
Problems...
Typechecking requires Computation.
Equality is no longer decidable in general.
We want decidable typechecking.
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 5 / 24
(5/24) COMPUTING 2011 – 2011-10-21
![Page 7: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/7.jpg)
The J Eliminator
Problems...
Typechecking requires Computation.
Equality is no longer decidable in general.
We want decidable typechecking.
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 5 / 24
(5/24) COMPUTING 2011 – 2011-10-21
![Page 8: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/8.jpg)
The J Eliminator
...and Answers
Two kinds of Equality!
Definitional Equality“Real” decidable equality such as (λa.b)x =β b[x/a]
Propositional EqualityEquality needing a proof
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 6 / 24
(6/24) COMPUTING 2011 – 2011-10-21
![Page 9: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/9.jpg)
The J Eliminator
...and Answers
Two kinds of Equality!
Definitional Equality“Real” decidable equality such as (λa.b)x =β b[x/a]
Propositional EqualityEquality needing a proof
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 6 / 24
(6/24) COMPUTING 2011 – 2011-10-21
![Page 10: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/10.jpg)
The J Eliminator
...and Answers
Two kinds of Equality!
Definitional Equality“Real” decidable equality such as (λa.b)x =β b[x/a]
Propositional EqualityEquality needing a proof
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 6 / 24
(6/24) COMPUTING 2011 – 2011-10-21
![Page 11: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/11.jpg)
The J Eliminator
Propositional Equality
Γ ` x, y : AΓ ` IdA x y : type Form
Γ ` x : AΓ ` reflx : IdA x x
Intro
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 7 / 24
(7/24) COMPUTING 2011 – 2011-10-21
![Page 12: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/12.jpg)
The J Eliminator
Propositional Equality
Γ `A: typeΓ, x, y : A, p : IdA x y `M(x,y,p): typeΓ, r : A `m : M(r, r, reflr)Γ `a, b : AΓ `q : IdA a b
Γ ` J M m a b q : M(a,b,q)Elim (J)
. . .J M m a a refl a = ma
Comp
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 8 / 24
(8/24) COMPUTING 2011 – 2011-10-21
![Page 13: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/13.jpg)
The J Eliminator
Subst from J
P : A→ Set and a, b : A.q : IdA a bp : P a
Can we get something of type P b?
I.e. is (P : A→ Set)→ (a, b : A)→ IdA a b → P a→ P b
inhabited?
Sure! Using J with
M = λ x y p . P x → Py
m = λ x.x
Call it subst.
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 9 / 24
(9/24) COMPUTING 2011 – 2011-10-21
![Page 14: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/14.jpg)
The J Eliminator
Subst from J
P : A→ Set and a, b : A.q : IdA a bp : P a
Can we get something of type P b?
I.e. is (P : A→ Set)→ (a, b : A)→ IdA a b → P a→ P b
inhabited?
Sure! Using J with
M = λ x y p . P x → Py
m = λ x.x
Call it subst.
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 9 / 24
(9/24) COMPUTING 2011 – 2011-10-21
![Page 15: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/15.jpg)
The J Eliminator
Subst from J
P : A→ Set and a, b : A.q : IdA a bp : P a
Can we get something of type P b?
I.e. is (P : A→ Set)→ (a, b : A)→ IdA a b → P a→ P b
inhabited?
Sure! Using J with
M = λ x y p . P x → Py
m = λ x.x
Call it subst.
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 9 / 24
(9/24) COMPUTING 2011 – 2011-10-21
![Page 16: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/16.jpg)
The J Eliminator
Subst from J
P : A→ Set and a, b : A.q : IdA a bp : P a
Can we get something of type P b?
I.e. is (P : A→ Set)→ (a, b : A)→ IdA a b → P a→ P b
inhabited?
Sure! Using J with
M = λ x y p . P x → Py
m = λ x.x
Call it subst.
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 9 / 24
(9/24) COMPUTING 2011 – 2011-10-21
![Page 17: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/17.jpg)
The J Eliminator
Subst from J
P : A→ Set and a, b : A.q : IdA a bp : P a
Can we get something of type P b?
I.e. is (P : A→ Set)→ (a, b : A)→ IdA a b → P a→ P b
inhabited?
Sure! Using J with
M = λ x y p . P x → Py
m = λ x.x
Call it subst.
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 9 / 24
(9/24) COMPUTING 2011 – 2011-10-21
![Page 18: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/18.jpg)
The J Eliminator
Subst from J
P : A→ Set and a, b : A.q : IdA a bp : P a
Can we get something of type P b?
I.e. is (P : A→ Set)→ (a, b : A)→ IdA a b → P a→ P b
inhabited?
Sure! Using J with
M = λ x y p . P x → Py
m = λ x.x
Call it subst.
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 9 / 24
(9/24) COMPUTING 2011 – 2011-10-21
![Page 19: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/19.jpg)
UIP
Uniqueness of Identity Proofs
How many inhabitants can IdA a b have in general?
For some time, it was assumed that there is at most one (UIP),i.e. given p, q : IdA a b, the type Id p q is inhabited.
Hofmann-Streicher groupoid model: not derivable from J.
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 10 / 24
(10/24) COMPUTING 2011 – 2011-10-21
![Page 20: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/20.jpg)
UIP
Uniqueness of Identity Proofs
How many inhabitants can IdA a b have in general?
For some time, it was assumed that there is at most one (UIP),i.e. given p, q : IdA a b, the type Id p q is inhabited.
Hofmann-Streicher groupoid model: not derivable from J.
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 10 / 24
(10/24) COMPUTING 2011 – 2011-10-21
![Page 21: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/21.jpg)
UIP
Uniqueness of Identity Proofs
How many inhabitants can IdA a b have in general?
For some time, it was assumed that there is at most one (UIP),i.e. given p, q : IdA a b, the type Id p q is inhabited.
Hofmann-Streicher groupoid model: not derivable from J.
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 10 / 24
(10/24) COMPUTING 2011 – 2011-10-21
![Page 22: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/22.jpg)
UIP
Uniqueness of Identity Proofs - Refuted
as 99
p**
q
44 b
m
��
u
v
��
c
dw1 99 w2ee e
n
YY fv1 99
a, b, d, e : A c, f : B
s : IdA a a p, q : IdA a b u, v : IdA b e . . .
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 11 / 24
(11/24) COMPUTING 2011 – 2011-10-21
![Page 23: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/23.jpg)
UIP
UIP is weird anyway
BOOL = {true, false}
isomorphisms:
id : BOOL→ BOOL
¬ : BOOL→ BOOL
So, identity equals negation?!
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 12 / 24
(12/24) COMPUTING 2011 – 2011-10-21
![Page 24: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/24.jpg)
UIP
UIP is weird anyway
BOOL = {true, false}
isomorphisms:
id : BOOL→ BOOL
¬ : BOOL→ BOOL
So, identity equals negation?!
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 12 / 24
(12/24) COMPUTING 2011 – 2011-10-21
![Page 25: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/25.jpg)
Extensionality
Extensionality
Given:
f : A→ B
g : A→ B
p : Π(x : A).IdB (f x) (gx)
Can we construct something of type IdA→B f g (Leibniz)? No!
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 13 / 24
(13/24) COMPUTING 2011 – 2011-10-21
![Page 26: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/26.jpg)
Extensionality
Extensionality
Given:
f : A→ B
g : A→ B
p : Π(x : A).IdB (f x) (gx)
Can we construct something of type IdA→B f g (Leibniz)? No!
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 13 / 24
(13/24) COMPUTING 2011 – 2011-10-21
![Page 27: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/27.jpg)
Extensionality
Extensionality
Given:
f : A→ B
g : A→ B
p : Π(x : A).IdB (f x) (gx)
Can we construct something of type IdA→B f g (Leibniz)? No!
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 13 / 24
(13/24) COMPUTING 2011 – 2011-10-21
![Page 28: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/28.jpg)
Extensionality
Extensionality
Idea: Adding extensionality as additional axiom.
But then, assume p is a (nontrivial) equality proof using thisaxiom.
Consequence:
subst (λh → N) p 0
Non-canonical natural numbers!
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 14 / 24
(14/24) COMPUTING 2011 – 2011-10-21
![Page 29: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/29.jpg)
Extensionality
Extensionality
Idea: Adding extensionality as additional axiom.
But then, assume p is a (nontrivial) equality proof using thisaxiom.
Consequence:
subst (λh → N) p 0
Non-canonical natural numbers!
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 14 / 24
(14/24) COMPUTING 2011 – 2011-10-21
![Page 30: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/30.jpg)
Extensionality
Extensionality
Idea: Adding extensionality as additional axiom.
But then, assume p is a (nontrivial) equality proof using thisaxiom.
Consequence:
subst (λh → N) p 0
Non-canonical natural numbers!
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 14 / 24
(14/24) COMPUTING 2011 – 2011-10-21
![Page 31: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/31.jpg)
Extensionality
Extensionality
Idea: Adding extensionality as additional axiom.
But then, assume p is a (nontrivial) equality proof using thisaxiom.
Consequence:
subst (λh → N) p 0
Non-canonical natural numbers!
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 14 / 24
(14/24) COMPUTING 2011 – 2011-10-21
![Page 32: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/32.jpg)
Univalence and weak omega groupoids
Vladimir Voevodsky
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 15 / 24
(15/24) COMPUTING 2011 – 2011-10-21
![Page 33: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/33.jpg)
Univalence and weak omega groupoids
Voevodsky’s suggestion
Do not use UIP...because it is weird and has undesirable consequences!
Do not use the Extensionality Axiom!... because of the same reason!
Use Univalence instead!... because it is better - as we will see in a moment!
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 16 / 24
(16/24) COMPUTING 2011 – 2011-10-21
![Page 34: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/34.jpg)
Univalence and weak omega groupoids
Voevodsky’s suggestion
Do not use UIP...because it is weird and has undesirable consequences!
Do not use the Extensionality Axiom!... because of the same reason!
Use Univalence instead!... because it is better - as we will see in a moment!
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 16 / 24
(16/24) COMPUTING 2011 – 2011-10-21
![Page 35: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/35.jpg)
Univalence and weak omega groupoids
Voevodsky’s suggestion
Do not use UIP...because it is weird and has undesirable consequences!
Do not use the Extensionality Axiom!... because of the same reason!
Use Univalence instead!... because it is better - as we will see in a moment!
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 16 / 24
(16/24) COMPUTING 2011 – 2011-10-21
![Page 36: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/36.jpg)
Univalence and weak omega groupoids
Lumsdaine’s and v.d.Berg’s result
a
p
&&
p′
xxb
Weak ω groupoid
for example:
a := b := x
p := p′ := reflx
H := H′ := reflreflx
reflreflreflx. . .
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 17 / 24
(17/24) COMPUTING 2011 – 2011-10-21
![Page 37: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/37.jpg)
Univalence and weak omega groupoids
Lumsdaine’s and v.d.Berg’s result
a
p
&&
p′
xxb
H�#
Weak ω groupoid
for example:
a := b := x
p := p′ := reflx
H := H′ := reflreflx
reflreflreflx. . .
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 17 / 24
(17/24) COMPUTING 2011 – 2011-10-21
![Page 38: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/38.jpg)
Univalence and weak omega groupoids
Lumsdaine’s and v.d.Berg’s result
a
p
&&
p′
xxb
H�#
H′
;C
Weak ω groupoid
for example:
a := b := x
p := p′ := reflx
H := H′ := reflreflx
reflreflreflx. . .
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 17 / 24
(17/24) COMPUTING 2011 – 2011-10-21
![Page 39: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/39.jpg)
Univalence and weak omega groupoids
Lumsdaine’s and v.d.Berg’s result
a
p
&&
p′
xxb
H�#
H′
;C
JT
Weak ω groupoid
for example:
a := b := x
p := p′ := reflx
H := H′ := reflreflx
reflreflreflx. . .
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 17 / 24
(17/24) COMPUTING 2011 – 2011-10-21
![Page 40: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/40.jpg)
Homotopy Theoretic Model of Intensional Type Theory
A very well-known structure. . .
. . . in Topology!
(source: Wikipedia)
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 18 / 24
(18/24) COMPUTING 2011 – 2011-10-21
![Page 41: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/41.jpg)
Homotopy Theoretic Model of Intensional Type Theory
A disc
a (nondepen-dent!) type -we call it X
a topologicalspace - we call itX
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 19 / 24
(19/24) COMPUTING 2011 – 2011-10-21
![Page 42: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/42.jpg)
Homotopy Theoretic Model of Intensional Type Theory
A disc
two terms two points
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 19 / 24
(19/24) COMPUTING 2011 – 2011-10-21
![Page 43: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/43.jpg)
Homotopy Theoretic Model of Intensional Type Theory
A disc
? a path
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 19 / 24
(19/24) COMPUTING 2011 – 2011-10-21
![Page 44: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/44.jpg)
Homotopy Theoretic Model of Intensional Type Theory
A disc
a, b : X
p : Id a b
a, b ∈ X
p : [0, 1]→ X
p(0) = a
p(1) = b
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 19 / 24
(19/24) COMPUTING 2011 – 2011-10-21
![Page 45: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/45.jpg)
Homotopy Theoretic Model of Intensional Type Theory
A disc
p−1 : Id b ap−1 : [0, 1]→ X
p−1(t) = p(1− t)
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 19 / 24
(19/24) COMPUTING 2011 – 2011-10-21
![Page 46: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/46.jpg)
Homotopy Theoretic Model of Intensional Type Theory
A disc
p : a ≡ bq : Id b c
a, b ∈ X
p : [0, 1]→ X
p(0) = a
p(1) = b
q : [0, 1]→ X
q(0) = b
q(1) = c
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 19 / 24
(19/24) COMPUTING 2011 – 2011-10-21
![Page 47: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/47.jpg)
Homotopy Theoretic Model of Intensional Type Theory
A disc
q ◦ p : Id a c
q ◦ p :
[0, 1]→ X
x 7→{p(2x), x < 0.5
q(2x − 1), else
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 19 / 24
(19/24) COMPUTING 2011 – 2011-10-21
![Page 48: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/48.jpg)
Homotopy Theoretic Model of Intensional Type Theory
Another set
Id a c notinhabited
notpath-connected
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 19 / 24
(19/24) COMPUTING 2011 – 2011-10-21
![Page 49: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/49.jpg)
Homotopy Theoretic Model of Intensional Type Theory
A disc
p, p′ : Id a bp, p′ : [0, 1]→ X
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 19 / 24
(19/24) COMPUTING 2011 – 2011-10-21
![Page 50: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/50.jpg)
Homotopy Theoretic Model of Intensional Type Theory
A disc
H : Id p p′
H : [0, 1]2 → X
H(0, ·) = p
H(1, ·) = p′
H(t, 0) = a
H(t, 1) = b
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 19 / 24
(19/24) COMPUTING 2011 – 2011-10-21
![Page 51: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/51.jpg)
Homotopy Theoretic Model of Intensional Type Theory
A disc
H : Id p p′
H : [0, 1]2 → X
H(0, ·) = p
H(1, ·) = p′
H(t, 0) = a
H(t, 1) = b
p : [0, 1]1 → X
a : [0, 1]0 → X
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 19 / 24
(19/24) COMPUTING 2011 – 2011-10-21
![Page 52: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/52.jpg)
Homotopy Theoretic Model of Intensional Type Theory
A ring
H : Id p p′
H : [0, 1]2 → X
H(0, ·) = p
H(1, ·) = p′
H(t, 0) = a
H(t, 1) = b
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 19 / 24
(19/24) COMPUTING 2011 – 2011-10-21
![Page 53: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/53.jpg)
Homotopy Theoretic Model of Intensional Type Theory
A disc
H : Id p p′
H : [0, 1]2 → X
H(0, ·) = p
H(1, ·) = p′
H(t, 0) = a
H(t, 1) = b
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 19 / 24
(19/24) COMPUTING 2011 – 2011-10-21
![Page 54: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/54.jpg)
Homotopy Theoretic Model of Intensional Type Theory
A disc
H′ : Id p p′
H′ : [0, 1]2 → X
H′(0, ·) = p
H′(1, ·) = p′
H′(t, 0) = a
H′(t, 1) = b
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 19 / 24
(19/24) COMPUTING 2011 – 2011-10-21
![Page 55: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/55.jpg)
Homotopy Theoretic Model of Intensional Type Theory
A disc
K : IdH′H
K : [0, 1]3 → X
K(0, ·, ·) = H′
. . .
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 19 / 24
(19/24) COMPUTING 2011 – 2011-10-21
![Page 56: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/56.jpg)
Homotopy Theoretic Model of Intensional Type Theory
Putting it together
a
p
&&
p′
xxb
H�#
H′
;C
JT
K
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 20 / 24
(20/24) COMPUTING 2011 – 2011-10-21
![Page 57: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/57.jpg)
Univalence again
Voevodsky again
Univalence AxiomThe (canonical) mapping from equalities to weak equivalences is a
weak equivalence.
No need for UIP
Extensionality
Only canonical members of Na “completely natural axiom” so that everything works as inhomotopical intuition
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 21 / 24
(21/24) COMPUTING 2011 – 2011-10-21
![Page 58: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/58.jpg)
Univalence again
Voevodsky again
Univalence AxiomThe (canonical) mapping from equalities to weak equivalences is a
weak equivalence.
No need for UIP
Extensionality
Only canonical members of Na “completely natural axiom” so that everything works as inhomotopical intuition
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 21 / 24
(21/24) COMPUTING 2011 – 2011-10-21
![Page 59: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/59.jpg)
Univalence again
Voevodsky again
Univalence AxiomThe (canonical) mapping from equalities to weak equivalences is a
weak equivalence.
No need for UIP
Extensionality
Only canonical members of Na “completely natural axiom” so that everything works as inhomotopical intuition
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 21 / 24
(21/24) COMPUTING 2011 – 2011-10-21
![Page 60: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/60.jpg)
Univalence again
Voevodsky again
Univalence AxiomThe (canonical) mapping from equalities to weak equivalences is a
weak equivalence.
No need for UIP
Extensionality
Only canonical members of Na “completely natural axiom” so that everything works as inhomotopical intuition
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 21 / 24
(21/24) COMPUTING 2011 – 2011-10-21
![Page 61: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/61.jpg)
Univalence again
Voevodsky again
Univalence AxiomThe (canonical) mapping from equalities to weak equivalences is a
weak equivalence.
No need for UIP
Extensionality
Only canonical members of Na “completely natural axiom” so that everything works as inhomotopical intuition
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 21 / 24
(21/24) COMPUTING 2011 – 2011-10-21
![Page 62: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/62.jpg)
Summary, Researchers and Acknowledgements
Summary
Hopes:Homotopic Models:
new results and intuition in both type and homotopy theorybetter understanding of the connection between logic andtopology
Univalence:avoiding a couple of problems in a natural way
UIPExtensionalityCanonicity of natural numbers
better foundation than Set Theory for (constructive)mathematicsat the same time, natively supported by proof assistants
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 22 / 24
(22/24) COMPUTING 2011 – 2011-10-21
![Page 63: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/63.jpg)
Summary, Researchers and Acknowledgements
Summary
Hopes:Homotopic Models:
new results and intuition in both type and homotopy theorybetter understanding of the connection between logic andtopology
Univalence:avoiding a couple of problems in a natural way
UIPExtensionalityCanonicity of natural numbers
better foundation than Set Theory for (constructive)mathematicsat the same time, natively supported by proof assistants
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 22 / 24
(22/24) COMPUTING 2011 – 2011-10-21
![Page 64: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/64.jpg)
Summary, Researchers and Acknowledgements
Summary
Hopes:Homotopic Models:
new results and intuition in both type and homotopy theorybetter understanding of the connection between logic andtopology
Univalence:avoiding a couple of problems in a natural way
UIPExtensionalityCanonicity of natural numbers
better foundation than Set Theory for (constructive)mathematicsat the same time, natively supported by proof assistants
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 22 / 24
(22/24) COMPUTING 2011 – 2011-10-21
![Page 65: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/65.jpg)
Summary, Researchers and Acknowledgements
Summary
Hopes:Homotopic Models:
new results and intuition in both type and homotopy theorybetter understanding of the connection between logic andtopology
Univalence:avoiding a couple of problems in a natural way
UIPExtensionalityCanonicity of natural numbers
better foundation than Set Theory for (constructive)mathematicsat the same time, natively supported by proof assistants
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 22 / 24
(22/24) COMPUTING 2011 – 2011-10-21
![Page 66: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/66.jpg)
Summary, Researchers and Acknowledgements
Summary
Hopes:Homotopic Models:
new results and intuition in both type and homotopy theorybetter understanding of the connection between logic andtopology
Univalence:avoiding a couple of problems in a natural way
UIPExtensionalityCanonicity of natural numbers
better foundation than Set Theory for (constructive)mathematicsat the same time, natively supported by proof assistants
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 22 / 24
(22/24) COMPUTING 2011 – 2011-10-21
![Page 67: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/67.jpg)
Summary, Researchers and Acknowledgements
Summary
Hopes:Homotopic Models:
new results and intuition in both type and homotopy theorybetter understanding of the connection between logic andtopology
Univalence:avoiding a couple of problems in a natural way
UIPExtensionalityCanonicity of natural numbers
better foundation than Set Theory for (constructive)mathematicsat the same time, natively supported by proof assistants
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 22 / 24
(22/24) COMPUTING 2011 – 2011-10-21
![Page 68: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/68.jpg)
Summary, Researchers and Acknowledgements
(Other) People I want to mention
Thorsten AltenkirchPeter ArndtSteve AwodeyThierry CoquandNicola GambinoRichard GarnerChris KapulkinDan LicataMike ShulmanThomas StreicherMichael Warren. . . and many more
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 23 / 24
(23/24) COMPUTING 2011 – 2011-10-21
![Page 69: Equality in the Dependently Typed Lambda Calculus: An ... · Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: ConnectingTopologyandLogicwithCategoryTheory](https://reader034.fdocuments.net/reader034/viewer/2022050414/5f8b1be9b7742270c441a299/html5/thumbnails/69.jpg)
Summary, Researchers and Acknowledgements
Even more people I want to Thank
You.
Nicolai Kraus ( School of Computer Science University of Nottingham, UK )Equality in the Dependently Typed Lambda Calculus: An Introduction to Homotopy Type Theory or: Connecting Topology and Logic with Category Theory2011-10-21 24 / 24
(24/24) COMPUTING 2011 – 2011-10-21