nLab identity function

Definition

Given a set $S$, the identity function on $S$ is the function $id_S\colon S \to S$ that maps any element $x$ of $S$ to itself:

$$id_S = (x \mapsto x) = \lambda x.\; x ;$$

or equivalently,

$$id_S(x) = x .$$

Thus, the identity function is the function for which every element $x$ of $S$ is a fixed point.

The identity functions are the identity morphisms in the category Set of sets.

More generally, in any concrete category, the identity morphism of each object is given by the identity function on its underlying set.

Definition

In (dependent) type theory, the identity function on a type $T$ may be declared as

(1)$$\mathrm{id}_T \;\coloneqq\; \big(\lambda(t \colon T).t \big) \;\colon\; T \to T \,.$$

 Properties

Definition

In dependent type theory, the type of identity functions on a type $A$ is the type

$$\sum_{f:A \to A} \prod_{x:A} f(x) =_A x$$

of functions with a witness that every element $x:A$ is a fixed point of the function.

In general, an identity function is an element of this type. However, any element of the type of identity functions is unique up to identification:

Theorem

For all types $A$, the type

$$\sum_{f:A \to A} \prod_{x:A} f(x) =_A x$$

is a proposition.

Proof

Suppose we have $i:A \to A$ with $I:\prod_{x:A} i(x) =_A x$ and $j:A \to A$ with $J:\prod_{x:A} j(x) =_A x$. Then we have a homotopy

$$\lambda x:A.I(x) \bullet J(x)^{-1}:\prod_{x:A} i(x) =_A j(x)$$

and by function extensionality, an identification

$$\mathrm{homToId}(i, j, \lambda x:A.I(x) \bullet J(x)^{-1}):i =_{A \to A} j$$

and by dependent function application on this identification, a dependent identification

$$\mathrm{apd}(i, j, \mathrm{homToId}(i, j, \lambda x:A.I(x) \bullet J(x)^{-1}, I, J):I =_{f:A \to A.\prod_{x:A} f(x) =_A j(x)}^{i, j, \lambda x:A.I(x) \bullet J(x)^{-1}} J$$

Then by the extensionality principle for dependent pair types, we have

$$\mathrm{pairIdsToId}(i, j, I, J)(\mathrm{homToId}(i, j, \lambda x:A.I(x) \bullet J(x)^{-1}), \mathrm{apd}(i, j, \mathrm{homToId}(i, j, \lambda x:A.I(x) \bullet J(x)^{-1}, I, J))$$

in the identity type

$$(i, I) =_{\sum_{f:A \to A} \prod_{x:A} f(x) =_A x} (j, J)$$

By induction on dependent pair types, this implies that for all elements

$$F:\sum_{f:A \to A} \prod_{x:A} f(x) =_A x \; \mathrm{and} \; G:\sum_{f:A \to A} \prod_{x:A} f(x) =_A x \; ,$$

$$F =_{\sum_{f:A \to A} \prod_{x:A} f(x) =_A x} G$$

Thus, the type

$$\sum_{f:A \to A} \prod_{x:A} f(x) =_A x$$

is a proposition.

Theorem

For all types $A$, the type

$$\sum_{f:A \to A} \prod_{x:A} f(x) =_A x$$

is contractible.

Proof

Since the type

$$\sum_{f:A \to A} \prod_{x:A} f(x) =_A x$$

is a proposition, it suffices to construct an element of the above type. Given a type $A$, by the weakening rules and the generic variable rule in dependent type theory, one gets the identity family of elements $x:A \vdash x:A$. Then by lambda abstraction, one gets a function $\lambda x:A.x:A \to A$, and by the typal computation rule for weak function types one gets the dependent function $\beta_{A \to A}^{x:A.x}:\prod_{x:A} (\lambda x:A.x)(x) =_A x$. For strict function types, we have $\beta_{A \to A}^{x:A.x} \equiv \mathrm{refl}_A$. Thus, one has the identity function

$$(\lambda x:A.x, \beta_{A \to A}^{x:A.x}):\sum_{f:A \to A} \prod_{x:A} f(x) =_A x$$

Since the type

$$\sum_{f:A \to A} \prod_{x:A} f(x) =_A x$$

is a proposition and has element $(\lambda x:A.x, \beta_{A \to A}^{x:A.x})$, the type is contractible.

Thus, it makes sense to refer to the canonical identity function $\lambda x:A.x$ as the identity function on type $A$.

Now, we show that the identity function is an equivalence of types:

Theorem

Any identity function is a (non-coherent) involution.

Proof

Given a function $i:A \to A$ and a dependent function $I:\prod_{x:A} i(x) =_A x$, by function application on $i$, we have the family of identifications

$$x:A \vdash \mathrm{ap}(i, i(x), x, I(x)):i(i(x)) =_A i(x)$$

Thus by concatenation of identifications and lambda abstraction, we have the dependent function

$$\lambda x:A.\mathrm{ap}(i, i(x), x, I(x)) \bullet I(x):\prod_{x:A} i(i(x)) =_A x$$

which indicates that the identity function $i:A \to A$ is a non-coherent involution.

By definition of involution, it follows that:

Corollary

The inverse function of the identity function is the identity function itself.

Theorem

Any identity function is bi-invertible.

Proof

Given a function $i:A \to A$ and a dependent function $I:\prod_{x:A} i(x) =_A x$, we have an element

$$((i, \lambda x:A.\mathrm{ap}(i, i(x), x, I(x)) \bullet I(x)), (i, \lambda x:A.\mathrm{ap}(i, i(x), x, I(x)) \bullet I(x)))$$

of type

$$\left(\sum_{j:A \to A} \prod_{x:A} i(j(x)) =_A x\right) \times \left(\sum_{k:A \to A} \prod_{x:A} k(i(x)) =_A x\right)$$

indicating that $i:A \to A$ is bi-invertible.

By definition, it follows that:

Corollary

Any identity function on a type $A$ is an equivalence of types.

Hence, in dependent type theory, the identity function is also called the identity equivalence.

There are other ways to prove that the identity function is an equivalence of types. One of them is given below:

Theorem

Suppose that function types are defined using judgmental computation rules. Then for all terms $x \colon T$, the fiber of the identity function $\lambda(t \colon T).t \,\colon\, T \to T$ (1) at $x$ is a contractible type.

Proof

The fiber type in question is the dependent type

$$\sum_{y \colon T} \big( \lambda(t:T).t \big)(y) =_T x \,.$$

By the judgmental computation rule of function types, the term $\big(\lambda(t:T).t\big)(x)$ reduces down to $x$, and the fiber type becomes

$$\sum_{y \colon T} y \;=_T\; x$$

This is always a contractible type:

• the center of contraction is given by the dependent pair $(x, \mathrm{refl}_T(x))$,

• for any $z \colon \sum_{y \colon T} y \,=_T\, x$, there is an identification

$$z \;=_{\sum_{y:T} y =_T x}\; \big( x, \mathrm{refl}_T(x) \big) \,.$$

By induction on dependent sum types, it suffices to construct an identification

$$(w, p) \;=_{\sum_{y:T} y =_T x}\; \big( x, \mathrm{refl}_T(x) \big)$$

for any $w \colon T$ and $p \;\colon\; w =_T x$, and by induction on identity types, it suffices to construct an identification

$$\big(x, \mathrm{refl}_T(x)\big) \;=_{\sum_{y:T} y =_T x}\; \big(x, \mathrm{refl}_T(x)\big)$$

but that’s just reflexivity on $(x, \mathrm{refl}_T(x))$

$$\mathrm{refl}_{\sum_{y:T} y =_T x} \big( (x, \mathrm{refl}_T(x)) \big) \;\colon\; \big(x, \mathrm{refl}_T(x)\big) \;=_{\sum_{y:T} y =_T x}\; \big(x, \mathrm{refl}_T(x)\big) \,.$$

Thus, for all $x \colon T$, the fiber type of the identity function $\lambda(t:T).t:T \to T$ at $x$ is contractible.

Related concepts

• neutral element

• identity morphism

• identity functor

• identity natural transformation

• identity type

• transport

• fixed point