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An Introduction to Homotopy Type Theory Morteza Moniri Department of Mathematics Shahid Beheshti University Mathematical Logic and its Applications September 4, 2019 Arak University of Technology Morteza Moniri Homotopy Type Theory 1/21

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An Introduction to Homotopy Type Theory

Morteza Moniri

Department of Mathematics Shahid Beheshti University

Mathematical Logic and its Applications September 4, 2019 Arak University of Technology

Morteza Moniri Homotopy Type Theory 1/21

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Outline

  • HoTT is a new branch of mathematics that combines

aspects of homotopy theory and type theory.

Morteza Moniri Homotopy Type Theory 2/21

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Outline

  • HoTT is a new branch of mathematics that combines

aspects of homotopy theory and type theory.

  • The univalence axiom of Voevodsky based on homotopic

interpretation of type theory implies that isomorphic structures can be identified.

Morteza Moniri Homotopy Type Theory 2/21

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Outline

  • HoTT is a new branch of mathematics that combines

aspects of homotopy theory and type theory.

  • The univalence axiom of Voevodsky based on homotopic

interpretation of type theory implies that isomorphic structures can be identified.

  • This suggests a new foundation of mathematics and

convenient machine implementation.

Morteza Moniri Homotopy Type Theory 2/21

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Outline

  • HoTT is a new branch of mathematics that combines

aspects of homotopy theory and type theory.

  • The univalence axiom of Voevodsky based on homotopic

interpretation of type theory implies that isomorphic structures can be identified.

  • This suggests a new foundation of mathematics and

convenient machine implementation.

  • A special year on univalent foundation of Mathematics was

held in 2012-2013 at the Institute for Advanced Study, school of mathematics.

Morteza Moniri Homotopy Type Theory 2/21

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Outline

  • HoTT is a new branch of mathematics that combines

aspects of homotopy theory and type theory.

  • The univalence axiom of Voevodsky based on homotopic

interpretation of type theory implies that isomorphic structures can be identified.

  • This suggests a new foundation of mathematics and

convenient machine implementation.

  • A special year on univalent foundation of Mathematics was

held in 2012-2013 at the Institute for Advanced Study, school of mathematics.

  • Recently, philosopher of science, James Ladyman, argues

that HoTT can be considered as an autonomous foundation

  • f mathematics, independent of Homotopy Theory.

Morteza Moniri Homotopy Type Theory 2/21

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Introduction

  • The notion of type theory begins with Bertrand Russel’s

efforts to resolve certain paradoxes in set theory.

Morteza Moniri Homotopy Type Theory 3/21

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Introduction

  • The notion of type theory begins with Bertrand Russel’s

efforts to resolve certain paradoxes in set theory.

  • His investigation is the foundation of mathematics, led him

to consider the type of individuals, the type of first order functions (predicates), ...

Morteza Moniri Homotopy Type Theory 3/21

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Introduction

  • The notion of type theory begins with Bertrand Russel’s

efforts to resolve certain paradoxes in set theory.

  • His investigation is the foundation of mathematics, led him

to consider the type of individuals, the type of first order functions (predicates), ...

  • If θ(x) is a predicate, what is the place from which x may

be taken s.t. θ(x) is a sensible assertion? For example, if θ(x) stands for " x is either true or false", then x should be a proposition.

Morteza Moniri Homotopy Type Theory 3/21

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Introduction

  • The notion of type theory begins with Bertrand Russel’s

efforts to resolve certain paradoxes in set theory.

  • His investigation is the foundation of mathematics, led him

to consider the type of individuals, the type of first order functions (predicates), ...

  • If θ(x) is a predicate, what is the place from which x may

be taken s.t. θ(x) is a sensible assertion? For example, if θ(x) stands for " x is either true or false", then x should be a proposition.

  • The basic notion in modern type theory is that each object

is assigned a type, and this type is something to which the

  • bject is explicitly linked.

Morteza Moniri Homotopy Type Theory 3/21

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Curry-Howard Isomorphism

Curry-Howard (1934): Corresponding between Computations in type theory and Natural Deduction proofs f : A → B x : A f(x) : B

Morteza Moniri Homotopy Type Theory 4/21

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Proposition as types

HoTT is based on constructive intentional type theory (Mortin-Löf 1975-1982-1984).

Morteza Moniri Homotopy Type Theory 5/21

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Proposition as types

HoTT is based on constructive intentional type theory (Mortin-Löf 1975-1982-1984). Propositions as types:

Morteza Moniri Homotopy Type Theory 5/21

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Proposition as types

HoTT is based on constructive intentional type theory (Mortin-Löf 1975-1982-1984). Propositions as types:

  • Each proposition is the type (set) of its proofs (these are

the only types).

Morteza Moniri Homotopy Type Theory 5/21

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Proposition as types

HoTT is based on constructive intentional type theory (Mortin-Löf 1975-1982-1984). Propositions as types:

  • Each proposition is the type (set) of its proofs (these are

the only types).

  • Propositions are syntactic objects, like 2 + 3 = 5. On the
  • ther hand, "2 + 2 = 4 is true" is a judgement.

Morteza Moniri Homotopy Type Theory 5/21

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Proposition as types

HoTT is based on constructive intentional type theory (Mortin-Löf 1975-1982-1984). Propositions as types:

  • Each proposition is the type (set) of its proofs (these are

the only types).

  • Propositions are syntactic objects, like 2 + 3 = 5. On the
  • ther hand, "2 + 2 = 4 is true" is a judgement.
  • A proposition is true if we have a token of the

corresponding type.

Morteza Moniri Homotopy Type Theory 5/21

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Logical operators

  • Correspondence between logical operators and types:

Morteza Moniri Homotopy Type Theory 6/21

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Logical operators

  • Correspondence between logical operators and types:

A → B = BA A ∧ B = A × B A ∨ B = A + B ¬A = ∅A ⊥= ∅ ⊤ = {0}

Morteza Moniri Homotopy Type Theory 6/21

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Logical operators

  • Correspondence between logical operators and types:

A → B = BA A ∧ B = A × B A ∨ B = A + B ¬A = ∅A ⊥= ∅ ⊤ = {0} Identity Type: For a, b of type U, we have the type IdU(a, b).

Morteza Moniri Homotopy Type Theory 6/21

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Dependent types

  • Dependent types:

Morteza Moniri Homotopy Type Theory 7/21

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Dependent types

  • Dependent types:

A proof of ∀x : A ϕ(x) is a function with domain A s.t. for x : A we have f(x) : ϕ(x). ∀x : A ϕ(x) = ∏

x∈A

ϕ(x) ∃x : A ϕ(x)

  • a pair (x,y) s.t. x:A and y:ϕ(x)

= ⨿

x:A

ϕ(x)

  • coproduct or disjoint union

Morteza Moniri Homotopy Type Theory 7/21

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Types in HoTT

  • The basic way to construct types in HoTT is essentially

similar to constructing types in MLTT.

Morteza Moniri Homotopy Type Theory 8/21

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Types in HoTT

  • The basic way to construct types in HoTT is essentially

similar to constructing types in MLTT.

  • "exp1 = exp2" means expressions exp1 and exp2 name the

same tokens or types.

Morteza Moniri Homotopy Type Theory 8/21

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Types in HoTT

  • The basic way to construct types in HoTT is essentially

similar to constructing types in MLTT.

  • "exp1 = exp2" means expressions exp1 and exp2 name the

same tokens or types.

  • "exp1 := exp2" means "by definition, expression exp1

names the same tokens or types as exp2".

Morteza Moniri Homotopy Type Theory 8/21

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Identity Type

About the Identity Type:

Morteza Moniri Homotopy Type Theory 9/21

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Identity Type

About the Identity Type:

  • a ∈ U

b ∈ U } IdU(a, b) is a type

Morteza Moniri Homotopy Type Theory 9/21

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Identity Type

About the Identity Type:

  • a ∈ U

b ∈ U } IdU(a, b) is a type

  • refla : IdU(a, a) (reflexivity of identity)
  • We have

a = b ⇒ IdU(a, b) ̸= ∅

Morteza Moniri Homotopy Type Theory 9/21

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Identity Type

About the Identity Type:

  • a ∈ U

b ∈ U } IdU(a, b) is a type

  • refla : IdU(a, a) (reflexivity of identity)
  • We have

a = b ⇒ IdU(a, b) ̸= ∅ a = b ̸⇐ IdU(a, b) ̸= ∅ This failure is definition of intentionality for a type theory

Morteza Moniri Homotopy Type Theory 9/21

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Path Induction

For any type U and property P that can be asserted of identifications between tokens of U, if we can prove that P holds of all trivial self-identifications refla for all a ∈ U, then it holds of all identifications in U.

Morteza Moniri Homotopy Type Theory 10/21

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Univalence Axiom

Univalence Axiom (Voevodsky): (A ≃ B) ≃ IdU(A, B).

Morteza Moniri Homotopy Type Theory 11/21

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Univalence Axiom

Univalence Axiom (Voevodsky): (A ≃ B) ≃ IdU(A, B).

  • A ≃ B is the type of equivalences between A and B, i.e.

types of functions f : A → B for which there exists a quasi-invense.

Morteza Moniri Homotopy Type Theory 11/21

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Isomorphisms are Identities

  • Steve Awodey argues that the Univalence Axiom implies

that Isomorphic Structures are Identical.

Morteza Moniri Homotopy Type Theory 12/21

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Isomorphisms are Identities

  • Steve Awodey argues that the Univalence Axiom implies

that Isomorphic Structures are Identical.

  • why is UA plausible?

Morteza Moniri Homotopy Type Theory 12/21

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Isomorphisms are Identities

  • Steve Awodey argues that the Univalence Axiom implies

that Isomorphic Structures are Identical.

  • why is UA plausible?
  • Two structures A and B are isomorphic if for each structural

property P(x), if P(A) then P(B), i.e. A ∼ = B P(A) P(B) Now Consider P(x) := IdU(A, X). We have A ∼ = B IdU(A, A) IdU(A, B) That means isomorphic objects are identical.

Morteza Moniri Homotopy Type Theory 12/21

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Consistency of HoTT

Voevodsky: HoTT and Univalent Axiom are consistent relative to ZFC (it has a model in the category of Kan complexes).

Morteza Moniri Homotopy Type Theory 13/21

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Numbers in HoTT

  • The type of natural numbers N is generated by 0 : N and

succ : N → N.

Morteza Moniri Homotopy Type Theory 14/21

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Numbers in HoTT

  • The type of natural numbers N is generated by 0 : N and

succ : N → N.

  • The type of Z, Q and R can be constructed using type

constructors of HoTT similar to the usual constructions of such numbers.

Morteza Moniri Homotopy Type Theory 14/21

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Homotopy Theory

The study of topological spaces and functions between them up to continuous distortion. That is if there is a continuous deformation that transforms one topological space into another or one continuous function into another, then in homotopy theory they are regarded as equivalent.

Morteza Moniri Homotopy Type Theory 15/21

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Homotopy Theory

H1: Let A, B be topological spaces. Two continuous functions f, g : A → B are homotopic if there exists a continuous function h : [0, 1] × A → B s.t. h(0, x) = f(x) and h(1, x) = g(x). We denote this by f ∼ g.

Morteza Moniri Homotopy Type Theory 16/21

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Homotopy Theory

H1: Let A, B be topological spaces. Two continuous functions f, g : A → B are homotopic if there exists a continuous function h : [0, 1] × A → B s.t. h(0, x) = f(x) and h(1, x) = g(x). We denote this by f ∼ g. H2: Let x, y ∈ X. A path between x, y is a continuous function γ : [0, 1] → X s.t. γ(0) = x, γ(1) = y.

Morteza Moniri Homotopy Type Theory 16/21

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Homotopy Theory

H1: Let A, B be topological spaces. Two continuous functions f, g : A → B are homotopic if there exists a continuous function h : [0, 1] × A → B s.t. h(0, x) = f(x) and h(1, x) = g(x). We denote this by f ∼ g. H2: Let x, y ∈ X. A path between x, y is a continuous function γ : [0, 1] → X s.t. γ(0) = x, γ(1) = y. H3: Two spaces x, y are homotopy equivalent if there exist continuous maps f : X → Y, g : Y → X s.t. gof ∼ IX, fog ∼ IY .

Morteza Moniri Homotopy Type Theory 16/21

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Homotopy Theory

H1: Let A, B be topological spaces. Two continuous functions f, g : A → B are homotopic if there exists a continuous function h : [0, 1] × A → B s.t. h(0, x) = f(x) and h(1, x) = g(x). We denote this by f ∼ g. H2: Let x, y ∈ X. A path between x, y is a continuous function γ : [0, 1] → X s.t. γ(0) = x, γ(1) = y. H3: Two spaces x, y are homotopy equivalent if there exist continuous maps f : X → Y, g : Y → X s.t. gof ∼ IX, fog ∼ IY . H4: Example: (i) Any two paths between any two points in Euclidean plane are homotopic.

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Homotopy Theory

H1: Let A, B be topological spaces. Two continuous functions f, g : A → B are homotopic if there exists a continuous function h : [0, 1] × A → B s.t. h(0, x) = f(x) and h(1, x) = g(x). We denote this by f ∼ g. H2: Let x, y ∈ X. A path between x, y is a continuous function γ : [0, 1] → X s.t. γ(0) = x, γ(1) = y. H3: Two spaces x, y are homotopy equivalent if there exist continuous maps f : X → Y, g : Y → X s.t. gof ∼ IX, fog ∼ IY . H4: Example: (i) Any two paths between any two points in Euclidean plane are homotopic. (ii) A disk and a single point are homotopy equivalent. But, a circle and its root are not.

Morteza Moniri Homotopy Type Theory 16/21

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Homotopy interpretation of HoTT

The Homotopy interpretation of HoTT (Awodey-Warren 2009)

Morteza Moniri Homotopy Type Theory 17/21

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Homotopy interpretation of HoTT

The Homotopy interpretation of HoTT (Awodey-Warren 2009)

  • Types as spaces (as understood in Homotopy Theory)

Morteza Moniri Homotopy Type Theory 17/21

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Homotopy interpretation of HoTT

The Homotopy interpretation of HoTT (Awodey-Warren 2009)

  • Types as spaces (as understood in Homotopy Theory)
  • Tokens as points in corresponding types

Morteza Moniri Homotopy Type Theory 17/21

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Homotopy interpretation of HoTT

The Homotopy interpretation of HoTT (Awodey-Warren 2009)

  • Types as spaces (as understood in Homotopy Theory)
  • Tokens as points in corresponding types
  • Each x ∈ X can be regarded as the constant function

kx : {0} → X.

Morteza Moniri Homotopy Type Theory 17/21

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Homotopy interpretation of HoTT

The Homotopy interpretation of HoTT (Awodey-Warren 2009)

  • Types as spaces (as understood in Homotopy Theory)
  • Tokens as points in corresponding types
  • Each x ∈ X can be regarded as the constant function

kx : {0} → X.

  • Identifications between tokens as paths between

corresponding points

Morteza Moniri Homotopy Type Theory 17/21

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Homotopy interpretation of HoTT

The Homotopy interpretation of HoTT (Awodey-Warren 2009)

  • Types as spaces (as understood in Homotopy Theory)
  • Tokens as points in corresponding types
  • Each x ∈ X can be regarded as the constant function

kx : {0} → X.

  • Identifications between tokens as paths between

corresponding points

  • the constant path fx : [0, 1] → X is considered as the

self-identification of x to itself.

Morteza Moniri Homotopy Type Theory 17/21

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Homotopy interpretation of HoTT

The Homotopy interpretation of HoTT (Awodey-Warren 2009)

  • Types as spaces (as understood in Homotopy Theory)
  • Tokens as points in corresponding types
  • Each x ∈ X can be regarded as the constant function

kx : {0} → X.

  • Identifications between tokens as paths between

corresponding points

  • the constant path fx : [0, 1] → X is considered as the

self-identification of x to itself.

  • Interpretation of Path induction: Any path between x

and y is homotopic to a constant path at x.

Morteza Moniri Homotopy Type Theory 17/21

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Ladyman: HoTT as an autonomous foundation for mathematics 1- Framework: Formal view

Morteza Moniri Homotopy Type Theory 18/21

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Ladyman: HoTT as an autonomous foundation for mathematics 1- Framework: Formal view 2- Semantics: A type is a mathematical concept and its tokens are instances of the concept (e.g. Number-2) Intentionality (e.g. even divisor of 9 ̸= even divisor of 11). When we work formally in HoTT, we construct expressions according to formula rules. They are names of tokens and types.

Morteza Moniri Homotopy Type Theory 18/21

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Ladyman: HoTT as an autonomous foundation for mathematics 1- Framework: Formal view 2- Semantics: A type is a mathematical concept and its tokens are instances of the concept (e.g. Number-2) Intentionality (e.g. even divisor of 9 ̸= even divisor of 11). When we work formally in HoTT, we construct expressions according to formula rules. They are names of tokens and types. 3- Metaphysics: The only metaphysical commitment required is to the existence of concepts.

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Ladyman: HoTT as an autonomous foundation for mathematics 4- Epistemology: The truth of a proposition is demonstrated by exhibiting a certificate, and a proof is a step-by-step construction of a certificate to the conclusion from certificates to the premises.

Morteza Moniri Homotopy Type Theory 19/21

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Ladyman: HoTT as an autonomous foundation for mathematics 4- Epistemology: The truth of a proposition is demonstrated by exhibiting a certificate, and a proof is a step-by-step construction of a certificate to the conclusion from certificates to the premises. 5- Methodology: We begin by formulating types corresponding to the kinds of mathematical entities under

  • discussion. Then from these we form types expressing the

proposition to be proved and the premises to be assumed.

Morteza Moniri Homotopy Type Theory 19/21

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References

Awodey, Steve; Structuralism, invariance, and univalence.

  • Philos. Math. (3) 22 (2014), no. 1, 1 - 11

Martin-Löf, Per; On the meanings of the logical constants and the justifications of the logical laws. Nordic J. Philos. Logic 1 (1996), no. 1, 11 - 60 Ladyman, James; Presnell, Stuart; Does homotopy type theory provide a foundation for mathematics? British J.

  • Phil. Sci. 69 (2018), no. 2, 377 - 420.

Homotopy Type Theory: Univalent Foundations of

  • Mathematics. The Univalent Foundations Program, Institute

for Advanced Study (IAS), Princeton, NJ, 2013.

Morteza Moniri Homotopy Type Theory 20/21

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Thanks for your attention

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