nLab
identity type

Context

Equality and Equivalence

Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idemponent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels

semantics

Induction

Contents

everything is identical with itself (WdL §863)

no two things are like each other (WdL §903)

Idea

In intensional type theory under the propositions as types paradigm, an identity type (or equality type) is the incarnation of equality. That is, for any type AA and any terms x,y:Ax,y:A, the type Id A(x,y)Id_A(x,y) is “the type of proofs that x=yx=y” or “the type of reasons why x=yx=y”.

To contrast with “computational” or definitional equality, sometimes inhabitation of an identity type is sometimes called propositional equality. The identity type Id A(x,y)Id_A(x,y) is sometimes written Eq A(x,y)Eq_A(x,y) or just (x=y)(x=y), but in this article we reserve the latter for definitional equality.

In extensional type theory, such as that modeled in the internal logic of a 1-category, equality is an h-proposition, and hence each Id A(x,y)Id_A(x,y) is a subsingleton. However, in the internal type theory of higher categories, such as the internal logic of an (∞,1)-topos, identity types represent path objects and are highly nontrivial. One speaks of homotopy type theory. In these cases, one may write for instance Path A(x,y)Path_A(x,y) instead of Id A(x,y)Id_A(x,y).

Definition

The definition of identity types was originally given in explicit form by Martin-Löf, in terms of introduction and elimination rules. Later, it was realized that this was a special case of the general notion of inductive type. We will discuss both formulations.

With introduction and elimination rules

The rules for forming identity types and terms are as follows (expressed in sequent calculus). First the rule that defines the identity type itself, as a dependent type, in some context Γ\Gamma.

type formation

ΓA:TypeΓ,x:A,y:AId A(x,y):Type\frac{\Gamma \vdash A:Type} {\Gamma, x:A, y:A \vdash Id_A(x,y):Type}

Now the basic “introduction” rule, which says that everything is equal to itself in a canonical way.

term introduction

ΓA:TypeΓ,x:Ar(x):Id A(x,x)\frac{\Gamma \vdash A:Type} {\Gamma, x:A \vdash r(x) : Id_A(x,x)}

To a category theorist, it might be more natural to call this 1 X1_X. The traditional notation r(x)r(x) indicates that this is a canonical proof of the reflexivity of equality.

Then we have the “elimination” rule, which is easily the most subtle and powerful.

term elimination

Γ,x:A,y:A,p:Id A(x,y),Δ(x,y,p)C(x,y,p):TypeΓ,x:A,Δ(x,x,r(x))t:C(x,x,r(x))Γ,x:A,y:A,p:Id A(x,y),Δ(x,y,p)J(t;x,y,p):C(x,y,p)\frac{\Gamma, x:A, y:A, p:Id_A(x,y), \Delta(x,y,p) \vdash C(x,y,p):Type \qquad \Gamma, x:A, \Delta(x,x,r(x)) \vdash t : C(x,x,r(x))} {\Gamma, x:A, y:A, p:Id_A(x,y), \Delta(x,y,p) \vdash J(t;x,y,p) : C(x,y,p)}

Ignore the presence of the additional context Δ\Delta for now; it is unnecessary if we also have dependent product types. The elimination rule then says that if:

  1. for any x,y:Ax,y:A and any reason p:Id A(x,y)p:Id_A(x,y) why they are the same, we have a type C(x,y,p)C(x,y,p), and
  2. if xx and yy are actually identical and p:Id A(x,x)p:Id_A(x,x) is the reflexivity proof r(x)r(x), then we have a specified term t:C(x,x,r(x))t:C(x,x,r(x)),

then we can construct a canonically defined term J(t;x,y,p):C(x,y,p)J(t;x,y,p):C(x,y,p) for any xx, yy, and p:Id A(x,y)p:Id_A(x,y), by “transporting” the term tt along the proof of equality pp. In homotopical or categorical models, this can be viewed as a “path-lifting” property, i.e. that the display maps are some sort of fibration. This can be made precise with the identity type weak factorization system?.

A particular case is when CC is a term representing a proposition according to the propositions-as-types philosophy. In this case, the elimination rule says that in order to prove a statement is true about all x,y,px,y,p, it suffices to prove it in the special case for x,x,r(x)x,x,r(x).

Finally, we have the “computation” or β-reduction rule. This says that if we substitute along a reflexivity proof, nothing happens.

computation rule

Γ,x:A,y:A,p:Id A(x,y),Δ(x,y,p)C(x,y,p):TypeΓ,x:A,Δ(x,x,r(x))t:C(x,x,r(x))Γ,x:A,Δ(x,x,r(x))J(t;x,x,r(x))=t\frac{\Gamma, x:A, y:A, p:Id_A(x,y), \Delta(x,y,p) \vdash C(x,y,p):Type \qquad \Gamma, x:A, \Delta(x,x,r(x)) \vdash t : C(x,x,r(x))} {\Gamma, x:A, \Delta(x,x,r(x)) \vdash J(t;x,x,r(x)) = t}

Note that the equality == in the conclusion of the computation rule is definitional equality, not an instance of the identity/equality type itself.

These rules may seem a little ad-hoc, but they are actually a particular case of the general notion of inductive type.

In terms of inductive types

Using inductive types the notion of identity types is encoded in a single line. In Coq notation we can say

Inductive id {A} : A -> A -> Type := idpath : forall x, id x x.   

In other words, the identity type of AA is inductively generated by reflexivity, in the same way that the natural numbers are inductively generated by zero and successor. From this, the above introduction, elimination, and computation rules are all derived automatically.

This is the approach to identity types taken by practical work in homotopy type theory, which is usually implemented in Coq or Agda. See, for instance, Paths.v

An essentially equivalent way to give the definition, due to Paulin-Mohring, is

Inductive id {A} (x:A) : A -> Type := idpath : id x x.   

The difference here is that now xx is a parameter of the inductive definition rather than an index. In other words, the first definition says “for each type AA, we have an type Id AId_A dependent on A×AA\times A, inductively defined by a constructor idpathidpath which takes an element x:Ax\colon A as input and yields output in Id A(x,x)Id_A(x,x)” while the second definition says “for each type AA and each element x:Ax\colon A, we have a type Id A(x)Id_A(x) dependent on AA, inductively defined by a constructor idpathidpath which takes no input and yields output in Id A(x)(x)Id_A(x)(x).” The two formulations can be proven equivalent, but sometimes one is more convenient than the other.

Extensionality and η\eta-conversion

Almost all types in type theory can be given both β-reduction and η-reduction rules. β\beta-reduction specifies what happens when we apply an eliminator to a term obtained by a constructor; η\eta-reduction specifies the reverse. Above we have formulated only the β\beta-reduction rule for identity types; the η\eta-conversion rule would be the following:

Γ,x:A,y:A,p:Id A(x,y),Δ(x,y,p)C(x,y,p):TypeΓ,x:A,y:A,p:Id A(x,y),Δ(x,x,r(x))t:C(x,y,p)Γ,x:A,y:A,p:Id A(x,y),Δ(x,y,p)J(t[y/x,r(x)/p];x,y,p)=t\frac{\Gamma, x:A, y:A, p:Id_A(x,y), \Delta(x,y,p) \vdash C(x,y,p):Type \qquad \Gamma, x:A, y:A, p:Id_A(x,y), \Delta(x,x,r(x)) \vdash t : C(x,y,p)} {\Gamma, x:A, y:A, p:Id_A(x,y), \Delta(x,y,p) \vdash J(t[y/x, r(x)/p];x,y,p) = t}

This says that if CC is a type which we can use the eliminator JJ to construct a term of, but we already have a term tt of that type, then if we restrict tt to reflexivity inputs and then apply JJ to construct a term of type CC, the result is the same as the term tt we started with. As in the β\beta-reduction rule, the == in the conclusion refers to definitional equality.

This η\eta-conversion rule has some very strong consequences. For instance, suppose x:Ax\colon A, y:Ay\colon A, and p:Id A(x,y)p\colon Id_A(x,y), and let CAC \coloneqq A. Then with t=xt=x, the η\eta-conversion rule tells us that x=J(x[y/x,r(x)/p];x,y,p)x = J(x[y/x,r(x)/p];x,y,p). And with t=yt=y, the η\eta-conversion rule tells us that y=J(y[y/x,r(x)/p];x,y,p)y = J(y[y/x,r(x)/p];x,y,p). But substituting yy for xx (and r(x)r(x) for pp) in the term yy simply yields the term xx, which is the same as the result of substituting yy for xx and r(x)r(x) for pp in the term xx. Thus, we have

x=J(x;x,y,p)=yx = J(x;x,y,p) = y

In other words, if Id A(x,y)Id_A(x,y) is inhabited (that is, xx and yy are propositionally equal) then in fact xx and yy are definitionally equal. Moreover, by a similar argument we can show that

p=J(p[y/x,r(x)/p];x,y,p)=J(r(x)[y/x,r(x)/p];x,y,p)=r(x).p = J(p[y/x, r(x)/p];x,y,p) = J(r(x)[y/x,r(x)/p];x,y,p) = r(x).

(Here we are eliminating into the type C(x,y,p)Id A(x,y)C(x,y,p) \coloneqq Id_A(x,y). The term r(x)r(x) may be regarded as belonging to this type, because we have already shown that xx and yy are definitionally equal.)

Thus, the definitional η\eta-conversion rule for identity types implies that the type theory is extensional in a very strong sense. (This was observed already in (Streicher).) For this reason, in homotopy type theory we do not assume the η\eta-conversion rule for identity types.

This sort of extensionality in type theory is also problematic for non-homotopical reasons: since type-checking in dependent type theory depends on definitional equality, but the above rule implies that definitional equality depends on inhabitation of identity types, this makes definitional equality and hence type-checking undecidable in the formal computational sense. Thus, η\eta-conversion for identity types is often omitted (as in Coq).

On the other hand, it is possible to prove a propositional version of η\eta-conversion using only the identity types as defined above without definitional η\eta-conversion. In other words, given the hypotheses of the above η\eta-conversion rule, we can construct a term belonging to the type

Id C(x,y,p)(J(t[y/x,r(x)/p];x,y,p),t). Id_{C(x,y,p)}(J(t[y/x, r(x)/p];x,y,p), t).

This has none of the bad consequences of definitional η\eta-conversion, and in particular does not imply that the type theory is extensional. The argument that p:Id A(x,y)p\colon Id_A(x,y) implies x=yx=y becomes the tautologous statement that if p:Id A(x,y)p\colon Id_A(x,y) then p:Id A(x,y)p\colon Id_A(x,y), while the subsequent argument that p=r(x)p= r(x) fails because xx and yy are no longer definitionally equal, so r(x)r(x) does not have type Id A(x,y)Id_A(x,y). We can transport it along pp to obtain a term of this type, but then we obtain only that pp is equal to the transport of r(x)r(x) along pp, which is a perfectly intensional/homotopical statement.

Categorical semantics

We discuss the categorical semantics for identity types in the extensional case, and identity types in the categorical semantics of homotopy type theory in the intensional case.

In categorical models of extensional type theory, generally every morphism of the category is allowed to represent a dependent type, and the extensional identity types are represented by diagonal maps AA×AA\to A\times A.

By contrast, in models of intensional type theory, there is only a particular class of display maps or fibrations which are allowed to represent dependent types, and intensional identity types are represented by path objects PAA×AP A \to A \times A.

Both of these cases apply in particular to models in the category of contexts of the type theory itself, i.e. the term model?.

Prerequisites

By the standard construction of mapping path spaces out of path space objects, the existence of identity types allows one to construct a weak factorization system.

Conversely, since any weak factorization system gives rise to path objects by factorization of diagonal maps, one may hope to construct a model of type theory with identity types in a category equipped with a WFS (L,R)(L,R). There are four obstacles in the way of such a construction.

  1. In order to handle the additional context Δ\Delta in the explicit definition above, it turns out to be necessary to assume that LL-maps are preserved by pullback along RR-maps between RR-objects. (Such a condition is also necessary in order to interpret type-theoretic dependent products in a locally cartesian closed category.)

  2. This enables us to define identity types with their elimination and computation rules “locally”, i.e. for each type individually. However, every construction in type theory is stable under substitution. This means that if y:YA(y):Typey\colon Y\vdash A(y)\colon Type is a dependent type and f:XYf\colon X\to Y is a morphism, then the identity type x:XId A(f(x))(,):Typex\colon X \vdash Id_{A(f(x))}(-,-)\colon Type is the same whether we first construct Id A(y)Id_{A(y)} and then substitute f(x)f(x) for yy, or first substitute f(x)f(x) for yy to obtain A(f(x))A(f(x)) and then construct its identity type. In order for this to hold up to isomorphism, we need to require that the WFS have stable path objects — a choice of path object data in each slice category which is preserved by pullback. In (Warren) it is shown that any simplicial model category in which the cofibrations are the monomorphisms can be equipped with stable path objects, while (Garner-van den Berg) it is shown that the presence of internal path-categories also suffices.

  3. The eliminator term JJ of identity types in type theory is also preserved by substitution. This imposes an additional coherence requirement which is tricky to obtain categorically. See the references by Warren and Garner-van den Berg for methods that ensure this, such as by invoking an algebraic weak factorization system. It can also be handled a la Voevodsky by using a (possibly univalent) universe.

  4. Finally, substitution in type theory is strictly functorial/associative, whereas it is modeled categorically by pullback which is generally not strictly so. This is a general issue with the categorical interpretation of dependent type theory, not something specific to identity types. It can be resolved by passing to a split fibration which is equivalent to the codomain fibration, or by making use of a universe. See categorical model of dependent types.

Interpretation in a type-theoretic model category

Assume then that a category 𝒞\mathcal{C} with suitable WFSs has been chosen, for instance a type-theoretic model category. Then

  • The interpretation of a type A:Type \vdash A : Type is as a fibrant object [A:Type][\vdash A : Type] which we will just write “AA” for short.

  • type formation

    The identity type a,b:AId A(a,b):Typea, b : A \vdash Id_A(a,b) : Type is interpreted as the path space object fibration

    A I A×A \array{ A^I \\ \downarrow \\ A \times A }
  • term introduction

    By definition of path space object, there exists a lift σ\sigma in

    A I σ A (id,id) A×A. \array{ && A^I \\ & {}^{\mathllap{\sigma}}\nearrow& \downarrow \\ A &\stackrel{(id,id)}{\to}& A \times A } \,.

    By the universal property of the pullback this is equivalently a section of the pullback of the path space object along the diagonal morphism (id,id):AA×A(id,id) : A \to A \times A.

    (id,id) *A I A I σ A = A (id,id) A×A. \array{ && (id,id)^* A^I &\to& A^I \\ &{}^{\mathllap{\sigma}}\nearrow& \downarrow & & \downarrow \\ A &=& A &\stackrel{(id,id)}{\to}& A \times A } \,.

    Since (id,id) *A I(id, id)^* A^I is the interpretation of the substitution
    a:AId A(a,a):Typea : A \vdash Id_A (a,a) : Type in this sense σ\sigma is now the interpretation of a term a:Ar A:Id A(a,a)a : A \vdash r_A : Id_A (a,a).

  • term elemination

    A type depending on an identity type

    a,b:A,p:Id A(a,b)C(a,b,p) a, b : A, p : Id_A(a,b) \vdash C(a,b,p)

    is interpreted as a fibration

    C A I. \array{ C \\ \downarrow \\ A^I } \,.

    The substitution C(a,a,r a)C(a,a,r_a) is interpreted by the pullback

    (id,id) *C C A (id,id) A×A. \array{ (id,id)^* C &\to& C \\ \downarrow && \downarrow \\ A &\stackrel{(id,id)}{\to}& A \times A } \,.

    Therefore a term t:C(a,a,r a)t : C(a,a,r_a) is interpreted as a section of this pullback

    (id,id) *C C t A = A (id,id) A×A. \array{ && (id,id)^* C &\to& C \\ &{}^{\mathllap{t}}\nearrow& \downarrow && \downarrow \\ A &=& A &\stackrel{(id,id)}{\to}& A \times A } \,.

    By the universal property of the pullback, this is equivalently a morphism tt in

    C t A (id,id) A×A. \array{ && C \\ & {}^{\mathllap{t}}\nearrow & \downarrow \\ A &\stackrel{(id,id)}{\to}& A \times A } \,.

    The elimination rule says that given such tt, there exists a compatible section of CId AC \to Id_A. If we redraw the previous diagram as a square, then this section is a lift in that diagram

    A C r A I = A I. \array{ A &\to& C \\ {}^{\mathllap{r}}\downarrow &\nearrow& \downarrow \\ A^I &=& A^I } \,.

    In particular, if CC itself is the pullback of a fibration DBD \to B along a morphism A IBA^I \to B, then rr has the left lifting property also against that fibration

    A C D r A I = A I B. \array{ A &\to& C &\to& D \\ {}^{\mathllap{r}}\downarrow &\nearrow& \downarrow && \downarrow \\ A^I &=& A^I &\to& B } \,.

    So the term elimination rule says that the interpretaton AA IA \to A^I of a:Ar(a):Id A(a,a)a : A \vdash r(a) : Id_A (a,a) has the left lifting property against all fibrations, hence that AA IA \to A^I is to be interpreted as an acyclic cofibration.

Weak ω\omega-groupoids

Some of the first work noticing the homotopical / higher-categorical interpretation of identity types (see below) focused on the fact that the tower of iterated identity types of a type has the structure of an internal algebraic ω-groupoid.

In retrospect, this is roughly an algebraic version of the standard fact that every object of a model category (or more generally a category of fibrant objects or a category with a weak factorization system) admits a simplicial resolution which is an internal Kan complex, i.e. a nonalgebraic \infty-groupoid. Note, however, that the first technical condition above (stability of LL-maps under pullback along RR-maps) seems to be necessary for the algebraic version of the result to go through.

References

Explicit definition

A survey is in chapter 1 of

  • Michael Warren, Homotopy theoretic aspects of constructive type theory, PhD thesis (2008) (pdf)

Extensionality and intensionality isses are studied in

By inductive types

Weak factorization systems

Types as weak ω\omega-groupoids

Revised on November 30, 2014 19:22:36 by David Corfield (46.208.236.195)