nLab book HoTT

Idea

Book HoTT is the dependent type theory which appears in the HoTT book. It is notable in that, unlike most other dependent type theories which have been formally written in natural deduction, it does not have a separate type judgment. Instead, it only has term judgments, and an infinite sequence of Russell universes indexed by a natural numbers primitive. Types are represented by terms of a Russell universe.

However, when formally defining the natural numbers primitive, in order to ensure that the dependent type theory does not have a separate type judgment, one has to define the natural numbers primitive in a separate layer, meaning that formal book HoTT is a layered type theory. There are many different ways to define the layer containing the natural numbers primitive:

• One could define the first-order theory of a category with finite products and a parameterized natural numbers object, and use the hom-set $\mathrm{Hom}(1, \mathbb{N})$ to index the Russell universes.
• One could define the simply typed first order theory of ZFC, and use the von Neumann natural numbers or the Zermelo natural numbers to index the Russell universes.
• One could define the simply typed first order theory of Peano arithmetic, and use the natural numbers in Peano arithmetic to index the Russell universes.
• One could define a separate dependent type theory with a natural numbers type, such as the extensional type theory in two-level type theory, and use the natural numbers type $\mathbb{N}$ to index the Russell universes.

Formal presentation

The presentation of formal book HoTT we have chosen is a two-layered type theory. The first level consists of a basic objective type theory consisting only of type judgments, term judgments, identity types, and a natural numbers type. The second level is the homotopy type theory, and contains term judgments and judgmental equality of terms, Russell universes, and all the other type formers in homotopy type theory. To distinguish between the two layers, we shall call the types in the first layer “metatypes”, as the first layer behaves as a metatheory or external theory.

First layer

Judgments and contexts

We begin with the formal rules of the first layer. The first layer consists of three judgments: metatype judgments $A \; \mathrm{metatype}$, where we judge $A$ to be a metatype, metatyping judgments, where we judge $a$ to be an element of $A$, $a \in A$, and metacontext judgments, where we judge $\Xi$ to be a metacontext, $\Xi \; \mathrm{metactx}$. Metacontexts are lists of metatyping judgments $a \in A$, $b \in B$, $c \in C$, et cetera, and are formalized by the rules for the empty metacontext and extending the metacontext by a metatyping judgment

$$\frac{}{() \; \mathrm{metactx}} \qquad \frac{\Xi \; \mathrm{metactx} \quad \Xi \vdash A \; \mathrm{metatype}}{(\Xi, a \in A) \; \mathrm{metactx}}$$

Structural rules

The three standard structural rules, the variable rule, the weakening rule, and the substitution rule, are also included in the theory. Let $\mathcal{J}$ be any arbitrary judgment. Then we have the following rules:

• The variable rule:

$$\frac{\Xi, a \in A, \Omega \; \mathrm{metactx}}{\vdash \Xi, a \in A, \Omega \vdash a \in A}$$

• The weakening rule:

$$\frac{\Xi, \Omega \vdash \mathcal{J} \quad \Xi \vdash A \; \mathrm{metatype}}{\Xi, a \in A, \Omega \vdash \mathcal{J}}$$

• The substitution rule:

$$\frac{\Xi \vdash a \in A \quad \Xi, b \in A, \Omega \vdash \mathcal{J}}{\Xi, \Omega[a/b] \vdash \mathcal{J}[a/b]}$$

Identity metatypes

In addition, there are identity metatypes: the natural deduction rules for identity metatypes are as follows

$$\frac{\Xi \vdash A \; \mathrm{metatype}}{\Xi, a \in A, b \in A \vdash a =_A b \; \mathrm{metatype}} \qquad \frac{\Xi \vdash A \; \mathrm{metatype}}{\Xi, a \in A \vdash \mathrm{refl}_A(a) \in a =_A a}$$

$$\frac{\Xi, x \in A, y \in A, p \in x =_A y \vdash C \; \mathrm{metatype} \quad \Xi, z \in A \vdash t \in C[z/x, z/y, \mathrm{refl}_A(z)/p] \quad \Xi \vdash a \in A \quad \Xi \vdash b \in A \quad \Xi \vdash q \in a =_A b}{\Xi \vdash J(x.y.p.C, z.t, x, y, p) \in C[a/x, b/y, q/p]}$$

$$\frac{\Xi, x \in A, y \in A, p \in x =_A y \vdash C \; \mathrm{metatype} \quad \Xi, z \in A \vdash t:C[z/x, z/y, \mathrm{refl}_A(z)/p] \quad \Xi \vdash a:A}{\Xi \vdash \beta_{=_A}(a) \in J(x.y.p.C, z.t, a, a, \mathrm{refl}_A(a)) =_{C[a/x, a/y, \mathrm{refl}_A(a)/p]} t[a/z]}$$

Natural numbers primitive

Finally, we have the natural numbers primitive, given by the following natural deduction rules:

$$\frac{\Xi \; \mathrm{metactx}}{\Xi \vdash \mathcal{N} \; \mathrm{metatype}} \qquad \frac{\Xi \; \mathrm{metactx}}{\Xi \vdash 0_{\mathcal{N}} \in \mathcal{N}} \qquad \frac{\Xi \vdash n \in \mathcal{N}}{\Xi \vdash s_\mathcal{N}(n) \in \mathcal{N}}$$

$$\frac{\Xi, x \in \mathcal{N} \vdash C \; \mathrm{metatype} \quad \Xi \vdash c_{0_\mathcal{N}} \in C[0_\mathcal{N}/x] \quad \Xi, x \in \mathcal{N}, c \in C \vdash c_{s_\mathcal{N}} \in C[s_\mathcal{N}(x)/x] \quad \Xi \vdash n \in \mathbb{N}}{\Gamma \vdash \mathrm{ind}_\mathcal{N}^C(n, c_{0_\mathcal{N}}, c_{s_\mathcal{N}}) \in C[n/x]}$$

$$\frac{\Xi, x \in \mathcal{N} \vdash C \; \mathrm{metatype} \quad \Xi \vdash c_{0_\mathcal{N}} \in C[0_\mathcal{N}/x] \quad \Xi, x \in \mathcal{N}, c \in C \vdash c_{s_\mathcal{N}} \in C[s_\mathcal{N}(x)/x]}{\Xi \vdash \beta_\mathcal{N}^{0_\mathcal{N}} \in \mathrm{ind}_\mathcal{N}^C(0_\mathcal{N}, c_{0_\mathcal{N}}, c_{s_\mathcal{N}}) =_{C[0_\mathcal{N}/x]} c_{0_\mathcal{N}}}$$

$$\frac{\Xi, x \in \mathcal{N} \vdash C \; \mathrm{metatype} \quad \Xi \vdash c_{0_\mathcal{N}} \in C[0_\mathcal{N}/x] \quad \Xi, x \in \mathcal{N}, c \in C \vdash c_{s_\mathcal{N}} \in C[s_\mathcal{N}(x)/x]}{\Gamma \vdash \beta_\mathcal{N}^{s_\mathcal{N}(n)} \in \mathrm{ind}_\mathcal{N}^C(s_\mathcal{N}(n), c_{0_\mathcal{N}}, c_{s_\mathcal{N}}) =_{C[s_\mathcal{N}(n)/x]} c_{s_\mathcal{N}}(n, \mathrm{ind}_\mathcal{N}^C(n, c_{0_\mathcal{N}}, c_{s_\mathcal{N}}))}$$

Second layer

Judgments, contexts, and Russell universes

Now, we introduce the second layer, which consists of a dependent type theory with two judgments, the typing judgment $a:A$, which says that $a$ is a term of the type $A$, and judgmental equality $a \equiv a':A$, which says that $a$ and $a'$ are judgmentally equal terms of the type $A$. Instead of type judgments, we introduce a special kind of type called a Russell universe, whose terms are the types themselves. We also assume cumulativity for the Russell universes. Russell universes are formalized with the following rules:

$$\frac{\Xi \vdash i \in \mathcal{N}}{\Xi \vdash U_i:U_{s_\mathcal{N}(i)}} \quad \frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vdash A:U_i}{\Xi \vdash A:U_{s_\mathcal{N}(i)}}$$

Contexts are defined as a metacontext with a list of typing judgments, with the metacontext always preceding the list of typing judgments:

$$\frac{\Xi \; \mathrm{metactx}}{\Xi \vert () \; \mathrm{ctx}} \qquad \frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma \vdash A:U_i}{\Xi \vert (\Gamma, a:A) \; \mathrm{ctx}}$$

The general rules for Russell universes then follows:

$$\frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma \; \mathrm{ctx}}{\Xi \vert \Gamma \vdash U_i:U_{s_\mathcal{N}(i)}} \quad \frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma \vdash A:U_i}{\Xi \vert \Gamma \vdash A:U_{s_\mathcal{N}(i)}}$$

In addition, if there is a term $a:A$ of a type $A$, then the type $A$ is a term of some Russell universe $U_i$.

$$\frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma \vdash a:A}{\Xi \vert \Gamma \vdash A:U_{i}}$$

Structural rules

There are three structural rules in dependent type theory, the variable rule, the weakening rule, and the substitution rule.

The variable rule states that we may derive a typing judgment if the typing judgment is in the context already:

$$\frac{\Xi \vert \Gamma, a:A, \Delta \; \mathrm{ctx}}{\Xi \vert \Gamma, a:A, \Delta \vdash a:A}$$

Let $\mathcal{J}$ be any arbitrary judgment. Then we have the following rules:

The weakening rule:

$$\frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma \vdash A:U_i \quad \Xi \vert \Gamma, \Delta \vdash \mathcal{J}}{\Xi \vert \Gamma, a:A, \Delta \vdash \mathcal{J}}$$

The substitution rule:

$$\frac{\Xi \vert \Gamma \vdash a:A \quad \Xi \vert \Gamma, b:A, \Delta \vdash \mathcal{J}}{\Xi \vert \Gamma, \Delta[a/b] \vdash \mathcal{J}[a/b]}$$

The weakening and substitution rules are admissible rules: they do not need to be explicitly included in the type theory as they could be proven by induction on the structure of all possible derivations.

Structural rules for judgmental equality

Judgmental equality has its own structural rules: introduction rules for judgmentally equal terms, reflexivity, symmetry, transitivity, the principle of substitution, and the variable conversion rule.

• Introduction rules for judgmentally equal terms

  $$\frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma \vdash A:U_i \quad \Xi \vert \Gamma \vdash a \equiv b:A}{\Xi \vert \Gamma \vdash a:A}$$
  $$\frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma \vdash A:U_i \quad \Xi \vert \Gamma \vdash a \equiv b:A}{\Xi \vert \Gamma \vdash b:A}$$

• Reflexivity of judgmental equality

  $$\frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma \vdash A:U_i \quad \Xi \vert \Gamma \vdash a:A}{\Xi \vert \Gamma \vdash a \equiv a:A}$$

• Symmetry of judgmental equality

  $$\frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma \vdash A:U_i \quad \Xi \vert \Gamma \vdash a \equiv b:A}{\Xi \vert \Gamma \vdash b \equiv a:A}$$

• Transitivity of judgmental equality

  $$\frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma \vdash A:U_i \quad \Xi \vert \Gamma \vdash a \equiv b:A \quad \Xi \vert \Gamma \vdash b \equiv c:A}{\Xi \vert \Gamma \vdash a \equiv c:A}$$

• Principle of substitution:

  $$\frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma \vdash A:U_i \quad \Xi \vert \Gamma \vdash a \equiv b : A \quad \Xi \vert \Gamma, x:A, \Delta \vdash c:B}{\Xi \vert \Gamma, \Delta[b/x] \vdash c[a/x] \equiv c[b/x]: B[b/x]}$$

• Variable conversion rule:

  $$\frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma \vdash A \equiv B:U_i \quad \Xi \vert \Gamma, x:A, \Delta \vdash \mathcal{J}}{\Xi \vert \Gamma, x:B, \Delta \vdash \mathcal{J}}$$

Dependent product types

• Formation rules for dependent product types:

$$\frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma \vdash A:U_i \quad \Xi \vert \Gamma, x:A \vdash B:U_i}{\Xi \vert \Gamma \vdash \prod_{x:A} B(x):U_i}$$

• Introduction rules for dependent product types:

$$\frac{\Xi \vert \Gamma, x:A \vdash b:B}{\Xi \vert \Gamma \vdash \lambda(x:A).b(x):\prod_{x:A} B(x)}$$

• Elimination rules for dependent product types:

$$\frac{\Xi \vert \Gamma \vdash f:\prod_{x:A} B(x) \quad \Xi \vert \Gamma \vdash a:A}{\Xi \vert \Gamma \vdash f[a/x]:B[a/x]}$$

• Computation rules for dependent product types:

$$\frac{\Xi \vert \Gamma, x:A \vdash b:B \quad \Xi \vert \Gamma \vdash a:A}{\Xi \vert \Gamma \vdash \lambda(x:A).b(x)[a/x] \equiv b[a/x]:B[a/x]}$$

• Uniqueness rules for dependent product types:

$$\frac{\Xi \vert \Gamma \vdash f:\prod_{x:A} B(x)}{\Xi \vert \Gamma \vdash f \equiv \lambda(x).f(x):\prod_{x:A} B(x)}$$

Dependent sum types

• Formation rules for dependent sum types:

$$\frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma \vdash A:U_i \quad \Gamma, x:A \vdash B:U_i}{\Xi \vert \Gamma \vdash \sum_{x:A} B(x):U_i}$$

• Introduction rules for dependent sum types:

$$\frac{\Xi \vert \Gamma, x:A \vdash b:B \quad \Xi \vert \Gamma \vdash a:A \quad \Xi \vert \Gamma \vdash b:B[a/x]}{\Xi \vert \Gamma \vdash (a, b):\sum_{x:A} B(x)}$$

• Elimination rules for dependent sum types:

$$\frac{\Xi \vert \Gamma \vdash z:\sum_{x:A} B(x)}{\Xi \vert \Gamma \vdash \pi_1(z):A} \qquad \frac{\Xi \vert \Gamma \vdash z:\sum_{x:A} B(x)}{\Xi \vert \Gamma \vdash \pi_2(z):B[\pi_1(z)/x]}$$

• Computation rules for dependent sum types:

$$\frac{\Xi \vert \Gamma, x:A \vdash b:B \quad \Xi \vert \Gamma \vdash a:A}{\Xi \vert \Gamma \vdash \pi_1(a, b) \equiv a:A} \qquad \frac{\Xi \vert \Gamma, x:A \vdash b:B \quad \Xi \vert \Gamma \vdash a:A}{\Xi \vert \Gamma \vdash \pi_2(a, b) \equiv b:B[\pi_1(a, b)/x]}$$

• Uniqueness rules for dependent sum types:

$$\frac{\Xi \vert \Gamma \vdash z:\sum_{x:A} B(x)}{\Xi \vert \Gamma \vdash z \equiv (\pi_1(z), \pi_2(z)):\sum_{x:A} B(x)}$$

Sum types

• Formation rules for sum types:

$$\frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma \vdash A:U_i \quad \Xi \vert \Gamma \vdash B:U_i}{\Xi \vert \Gamma \vdash A + B:U_i}$$

• Introduction rules for sum types:

$$\frac{\Xi \vert \Gamma \vdash a:A}{\Xi \vert \Gamma \vdash \mathrm{inl}(a):A + B} \qquad \frac{\Xi \vert \Gamma \vdash b:B}{\Xi \vert \Gamma \vdash \mathrm{inr}(b):A + B}$$

• Elimination rules for sum types:

$$\frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma, z:A + B \vdash C:U_i \quad \Xi \vert \Gamma, x:A \vdash c:C[\mathrm{inl}(x)/z] \quad \Xi \vert \Gamma, y:B \vdash d:C[\mathrm{inr}(y)/z] \quad \Xi \vert \Gamma \vdash e:A + B}{\Xi \vert \Gamma \vdash \mathrm{ind}_{A + B}(z.C, x.c, y.d, e):C[e/z]}$$

• Computation rules for sum types:

$$\frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma, z:A + B \vdash C:U_i \quad \Xi \vert \Gamma, x:A \vdash c:C[\mathrm{inl}(x)/z] \quad \Xi \vert \Gamma, y:B \vdash d:C[\mathrm{inr}(y)/z] \quad \Xi \vert \Gamma \vdash a:A}{\Xi \vert \Gamma \vdash \mathrm{ind}_{A + B}(z.C, x.c, y.d, \mathrm{inl}(a)) \equiv c[a/x]:C[\mathrm{inl}(a)/z]}$$

$$\frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma, z:A + B \vdash C:U_i \quad \Xi \vert \Gamma, x:A \vdash c:C[\mathrm{inl}(x)/z] \quad \Xi \vert \Gamma, y:B \vdash d:C[\mathrm{inr}(y)/z] \quad \Xi \vert \Gamma \vdash b:A}{\Xi \vert \Gamma \vdash \mathrm{ind}_{A + B}(z.C, x.c, y.d, \mathrm{inr}(b)) \equiv d[b/x]:C[\mathrm{inr}(b)/z]}$$

Empty type

• Formation rules for the empty type:

$$\frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma \; \mathrm{ctx}}{\Xi \vert \Gamma \vdash \mathbb{0}:U_i}$$

• Elimination rules for the empty type:

$$\frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma, x:\mathbb{0} \vdash C:U_i \quad \Gamma \vdash p:\mathbb{0}}{\Gamma \vdash \mathrm{ind}_\mathbb{0}(x.C, p):C[p/x]}$$

Unit type

• Formation rules for the unit type:

$$\frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma \; \mathrm{ctx}}{\Xi \vert \Gamma \vdash \mathbb{1} \; \mathrm{type}}$$

• Introduction rules for the unit type:

$$\frac{\Xi \vert \Gamma \; \mathrm{ctx}}{\Xi \vert \Gamma \vdash *:\mathbb{1}}$$

• Elimination rules for the unit type:

$$\frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma, x:\mathbb{1} \vdash C:U_i \quad \Gamma, y:\mathbb{1} \vdash c[y/x]:C[y/x] \quad \Gamma \vdash p:\mathbb{1}}{\Gamma \vdash \mathrm{ind}_\mathbb{1}(x.C, y.c, p):C[p/x]}$$

• Computation rules for the unit type:

$$\frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma, x:\mathbb{1} \vdash C:U_i \quad \Gamma, y:\mathbb{1} \vdash c[y/x]:C[y/x]}{\Gamma \vdash \mathrm{ind}_\mathbb{1}(x.C, y.c, *) \equiv c[*/x]:C[*/x]}$$

Natural numbers

• Formation rules for the natural numbers:

  $$\frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma \; \mathrm{ctx}}{\Xi \vert \Gamma \vdash \mathbb{N}:U_i}$$

• Introduction rules for the natural numbers:

  $$\frac{\Xi \vert \Gamma \; \mathrm{ctx}}{\Xi \vert \Gamma \vdash 0:\mathbb{N}} \qquad \frac{\Xi \vert \Gamma \vdash n:\mathbb{N}}{\Xi \vert \Gamma \vdash s(n):\mathbb{N}}$$

• Elimination rules for the natural numbers:

  $$\frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma, x:\mathbb{N} \vdash C:U_i \quad \Xi \vert \Gamma \vdash c_0:C[0/x] \quad \Xi \vert \Gamma, x:\mathbb{N}, y:C \vdash c_s:C[s(x)/x] \quad \Xi \vert \Gamma \vdash n:\mathbb{N}}{\Xi \vert \Gamma \vdash \mathrm{ind}_\mathbb{N}(x.C, c_0, x.y.c_s, n):C[n/x]}$$

• Computation rules for the natural numbers:

  $$\frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma, x:\mathbb{N} \vdash C:U_i \quad \Xi \vert \Gamma \vdash c_0:C[0/x] \quad \Xi \vert \Gamma, x:\mathbb{N}, y:C \vdash c_s:C[s(x)/x]}{\Xi \vert \Gamma \vdash \mathrm{ind}_\mathbb{N}(x.C, c_0, x.y.c_s, 0) \equiv c_0:C[0/x]}$$

$$\frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma, x:\mathbb{N} \vdash C:U_i \quad \Xi \vert \Gamma \vdash c_0:C[0/x] \quad \Xi \vert \Gamma, x:\mathbb{N}, y:C \vdash c_s:C[s(x)/x]}{\Xi \vert \Gamma \vdash \mathrm{ind}_\mathbb{N}(x.C, c_0, x.y.c_s, s(n) \equiv c_s(n, \mathrm{ind}_\mathbb{N}(x.C, c_0, x.y.c_s, n)):C[s(n)/x]}$$

Identity types

There is another version of equality in book HoTT, called typal equality. Typal equality is represented by the identity type.

• Formation rule for identity types:

  $$\frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma \vdash A:U_i \quad \Xi \vert \Gamma \vdash a:A \quad \Xi \vert \Gamma \vdash b:A}{\Xi \vert \Gamma \vdash a =_A b:U_i}$$

• Introduction rule for identity types:

  $$\frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma \vdash A:U_i \quad \Xi \vert \Gamma \vdash a:A}{\Xi \vert \Gamma \vdash \mathrm{refl}_A(a) : a =_A a}$$

• Elimination rule for identity types:

  $$\frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma, x:A, y:A, p:a =_A b \vdash C:U_i \quad \Xi \vert \Gamma, z:A \vdash t:C[z/a, z/b, \mathrm{refl}_A(z)/p] \quad \Xi \vert \Gamma \vdash a:A \quad \Xi \vert \Gamma \vdash b:A \quad \Xi \vert \Gamma \vdash q:a =_A b}{\Xi \vert \Gamma \vdash J(x,y,p.C, z.t, a, b, q):C[a, b, q/x, y, p]}$$

• Computation rules for identity types:

  $$\frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma, x:A, y:A, p:a =_A b \vdash C:U_i \quad \Xi \vert \Gamma, z:A \vdash t:C[z/a, z/b, \mathrm{refl}_A(z)/p] \quad \Xi \vert \Gamma \vdash a:A}{\Xi \vert \Gamma \vdash J(x.y.p.C, z.t, a, a, \mathrm{refl}(a)) \equiv t:C[a, a, \mathrm{refl}_A(a)/x, y, p]}$$

Univalence

Given types $A:U_i$ and $B:U_i$, a function $f:A \to B$ is an equivalence of types if the fiber of $f$ at each element of $B$ has exactly one element. The property of the type $A$ having exactly one element is represented by the isContr modality which states that the type $A$ is contractible. The fiber of $f$ at an element $b:B$ is given by the type

$$\sum_{b:B} f(a) =_B b$$

We formally define the property of $R$ being a one-to-one correspondence or an equivalence of types as the type:

$$\mathrm{isEquiv}_{A, B}(f) \coloneqq \prod_{b:B} \mathrm{isContr}\left(\sum_{a:A} f(a) =_B b\right)$$

We define the type of equivalences from $A$ to $B$ as

$$A \simeq B \coloneqq \sum_{f:A \to B} \mathrm{isEquiv}_{A, B}(f)$$

There is a function

$$\mathrm{idtoequiv}_{A, B}:(A =_{U_i} B) \to (A \simeq B)$$

which is inductively defined on reflexivity to be the identity function on $A$

$$\mathrm{idtoequiv}_{A, A}(\mathrm{refl}_{U_i}(A)) \equiv id_A$$

The univalence axiom then states that $\mathrm{idtoequiv}_{A, B}$ is an equivalence for all external natural numbers $i \in \mathcal{N}$ and types $A:U_i$ and $B:U_i$:

$$\frac{\Xi \vdash i \in \mathcal{N} \quad \Xi \vert \Gamma \vdash A:U_i \quad \Xi \vert \Gamma \vdash B:U_i}{\Xi \vert \Gamma \vdash \mathrm{ua}_{U_i}(A, B):\mathrm{isEquiv}_{A =_{U_i} B, A \simeq B}(\mathrm{idtoequiv}_{A, B})}$$

See also

• dependent type theory

• homotopy type theory

 References

• Homotopy Type Theory: Univalent Foundations of Mathematics, The Univalent Foundations Project, Institute for Advanced Study, 2013. (web, pdf)