Context
Type theory
Equality and Equivalence
equivalence
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equality (definitional, propositional, computational, judgemental, extensional, intensional, decidable)
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identity type, equivalence in homotopy type theory
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isomorphism, weak equivalence, homotopy equivalence, weak homotopy equivalence, equivalence in an (∞,1)-category
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natural equivalence, natural isomorphism
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gauge equivalence
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Examples.
principle of equivalence
equation
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fiber product, pullback
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homotopy pullback
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Examples.
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linear equation, differential equation, ordinary differential equation, critical locus
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Euler-Lagrange equation, Einstein equation, wave equation
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Schrödinger equation, Knizhnik-Zamolodchikov equation, Maurer-Cartan equation, quantum master equation, Euler-Arnold equation, Fuchsian equation, Fokker-Planck equation, Lax equation
Contents
Idea
In dependent type theory, the equivalence type is to types what the identity type is to terms: it represents the collection of “equalities” between types (equality of types being given by the notion of equivalence in type theory), in the same way that the identity type represents the collection of equalities between terms (equality of terms being given by the notion of identity/identification/path).
Definition
In dependent type theory, the equivalence type between two types and is the type whose terms are equivalences between and . Like any other notion of type in dependent type theory, there are two different notions of equivalence types in type theory: strict and weak equivalence types. Strict equivalence types use judgmental equality in the conversion rules, while weak equivalence types use identity types in the conversion rules. Weak equivalence types could also be defined analytically from other type formers in the type theory.
Strict equivalence types
Given types and , one could define the type of strict equivalences betwen and . These are given by the following rules:
Formation rule for strict equivalence types:
Introduction rule for strict equivalence types:
Elimination rules for strict equivalence types:
Computation rules for strict equivalence types:
Uniqueness rules for strict equivalence types:
Weak equivalence types
As a dependent sum type of the isEquiv type family
Given a notion of the isEquiv type family on the function type , the equivalence type is defined by
Locally small equivalence types
Given a type universe and a notion of a -small isEquiv type family for some type , the locally -small equivalence type is defined by
could be the type of -small spans, the type of -small multivalued partial functions, or the type of -small correspondences.
Rules for weak equivalence types
In the same way that isEquiv could be defined in a way such that given the function and a witness , satisfies the universal property of a wrapped copy of , one could define weak equivalence types such that given an equivalence , satsifes the universal property of a wrapped copy of :
Formation rules for equivalence types:
Introduction rules for equivalence types:
Elimination rules for equivalence types:
Computation rules for equivalence types:
Uniqueness rules for equivalence types:
Coinductive definition
Given two types and and two functions and , and are inverses of each other if for every element and , there is an equivalence of types between and :
This leads to a coinductive definition of the type of equivalences
Formation rule for equivalence types:
Introduction rule for equivalence types:
Elimination rules for equivalence types:
Computation rules for equivalence types:
Properties
Relation to interval types
Given types and and an equivalence , one could define the dependent type indexed by the interval type as , , and .
One-to-one correspondences
Given types and and an equivalence , one could define a correspondence as the dependent identity type
where is defined as in the previous section. By the properties of dependent identity types, the correspondence is always a one-to-one correspondence.
Quasi-inverse functions with contractible fibers
By the rules for function types, given an equivalence , one could derive functions and . One could show that these functions are quasi-inverse functions of each other: for all and and equivalences , there are identities
where is the inverse identity of . By the introduction rule for dependent product types, there are homotopies
which indicate that and are quasi-inverse functions of each other.
By the rules for dependent sum types and dependent product types, one could show that the above functions each have contractible fibers, making both of them coherent inverse functions of each other.
Heterogeneous identity types
Given the definition of the equivalence type as the type of encodings for one-to-one correspondences, the heterogeneous identity type is defined by the rule
Identity equivalences, inverse equivalences, and composition of equivalences
The identity equivalence on a type is defined as an equivalence such that for all elements ,
Given an equivalence , the inverse equivalence of is an equivalence such that for all elements and ,
Given equivalences and , the composite of and is an equivalence such that for all elements and ,
Relation to universes and univalence
Given a Russell universe , there are two ways to say that types and are equal: by the identity type , and the equivalence type . The univalence axiom says that these two types and are the same, which is represented by an equivalence between the two types
For Tarski universes , one instead says that is the same as , represented as
Action on equivalences
We introduce a modal operator to the type theory, which we assume in general not to be idempotent or monadic; this is given by the formation rule
preserves equivalences: given types and , there is a function , called the action on equivalences for .
Categorical semantics
The categorical semantics of an equivalence type is an object of isomorphisms.
See also
References
For the definition of the equivalence type as a dependent sum type, see: