Redirected from "weak equivalence type".
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 of types, definitional isomorphism
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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.
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 equivalence types
There are various different rules one can use for equivalence types, depending upon what notion of equivalence one wishes to use:
- One-To-One correspondences
- Half-adjoint equivalences
- Biinvertible functions
- Functions with contractible fibers
One-To-One correspondence types
Let denote the isContr modality which says whether the type is a contractible type, and let
be the uniqueness quantifier over the type family . A binary correspondence between types and is simply a binary type family . A binary correspondence is one-to-one if
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for all there is a unique such that , and
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for all there is a unique such that .
Written out in the language of dependent type theory, one has
In the presence of some form of function extensionality, the type is guaranteed to be a mere proposition.
The rules for equivalence types then state that equivalences, the elements of equivalence types, are (codes for) one-to-one correspondences (in the same way that functions, the elements of function types, are (codes for) families of elements):
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:
Half-adjoint equivalence types
A half-adjoint equivalence between types and is a record consisting of the following fields:
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a function
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a function
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a homotopy
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a homotopy
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a homotopy expressing the coherence law for equivalences, where is the function application of to the identification .
Thus, the rules for half-adjoint equivalence types state that half-adjoint equivalence types are record types with the above fields:
Formation rules for half-adjoint equivalence types:
Introduction rules for half-adjoint equivalence types:
Elimination rules for half-adjoint equivalence types:
Computation rules for half-adjoint equivalence types:
Uniqueness rules for half-adjoint equivalence types:
Bi-invertible function types
A bi-invertible function between types and is a record consisting of the following fields:
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a function
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a function
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a function
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a homotopy
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a homotopy
Thus, the rules for bi-invertible function types state that bi-invertible function types are record types with the above fields:
Formation rules for bi-invertible function types:
Introduction rules for bi-invertible function types:
Elimination rules for bi-invertible function types:
Computation rules for bi-invertible function types:
Uniqueness rules for bi-invertible function 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: