nLab axiom of set truncation

Idea

In dependent type theory, an axiom of set truncation is a axiom or unjustified rule which implies the uniqueness of identity proofs which states that every type is an h-set. Adding an axiom of set truncation to a dependent type theory results in a set-level type theory.

Examples

TODO: show that each of the axioms below implies the uniqueness of identity proofs.

Uniqueness of identity proofs

For identification types

In dependent type theory, uniqueness of identity proofs for identification types is given by the following rule:

$$\frac{\Gamma \vdash A \; type \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash p:\mathrm{Id}_A(a, b) \quad \Gamma \vdash q:\mathrm{Id}_A(a, b)}{\Gamma \vdash \mathrm{UIP}_A(a, b, p, q):\mathrm{Id}_{\mathrm{Id}_A(a, b)}(p, q)}$$

For heterogeneous identification types

There is uniqueness of identity proofs for heterogeneous identification types:

$$\frac{ \begin{array}{l} \Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \\ \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash p:\mathrm{Id}_A(a, b) \quad \Gamma \vdash y:B \quad \Gamma \vdash z:B \\ \Gamma \vdash q:\mathrm{hId}_{B}(a, b, p, y, z) \quad \Gamma \vdash r:\mathrm{hId}_{B}(a, b, p, y, z) \end{array} }{\Gamma \vdash \mathrm{UIP}_{B}(a, b, p, y, z, q, r):\mathrm{Id}_{\mathrm{hId}_{B}(a, b, p, y, z)}(q, r)}$$

Uniqueness of identity proofs for identification types follows from uniqueness of identity proofs for heterogeneous identification types by taking canonical element $\mathrm{pt}:\mathbb{1}$ of the unit type $\mathbb{1}$, identification $\mathrm{refl}_{\mathbb{1}}(\mathrm{pt}):\mathrm{Id}_{\mathbb{1}}(\mathrm{pt}, \mathrm{pt})$, and elements $a:A$ and $b:A$. Since the following two types are equivalent:

$$\mathrm{Id}_{A}(a, b) \simeq \mathrm{hId}_{A}(\mathrm{pt}, \mathrm{pt}, \mathrm{refl}_{\mathbb{1}}(\mathrm{pt}), a, b)$$

and because $\mathrm{hId}_{A}(\mathrm{pt}, \mathrm{pt}, \mathrm{refl}_{\mathbb{1}}(\mathrm{pt}), a, b)$ is always a mere proposition by uniqueness of identity proofs for heterogeneous identification types, it follows that $\mathrm{Id}_{A}(a, b)$ is also always a mere proposition.

Similarly, there is also uniqueness of identity proofs for dependent heterogeneous identification types:

$$\frac{ \begin{array}{l} \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \\ \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash p:\mathrm{Id}_A(a, b) \quad \Gamma \vdash y:B(a) \quad \Gamma \vdash z:B(b) \\ \Gamma \vdash q:\mathrm{hId}_{x:A.B(x)}(a, b, p, y, z) \quad \Gamma \vdash r:\mathrm{hId}_{x:A.B(x)}(a, b, p, y, z) \end{array} }{\Gamma \vdash \mathrm{UIP}_{x:A.B(x)}(a, b, p, y, z, q, r):\mathrm{Id}_{\mathrm{hId}_{x:A.B(x)}(a, b, p, y, z)}(q, r)}$$

Uniqueness of identity proofs for identification types follows from uniqueness of identity proofs for dependent heterogeneous identification types by defining the type $A$ to be the dependent type $C(\mathrm{pt})$ for type family $x:\mathbb{1} \vdash C(x)$ indexed by the unit type $\mathbb{1}$, and taking the canonical element $\mathrm{pt}:\mathbb{1}$, identification $\mathrm{refl}_{\mathbb{1}}(\mathrm{pt}):\mathrm{Id}_{\mathbb{1}}(\mathrm{pt}, \mathrm{pt})$, and elements $a:C(\mathrm{pt})$ and $b:C(\mathrm{pt})$. Since the two types are equivalent:

$$\mathrm{Id}_{A}(a, b) \equiv \mathrm{Id}_{C(\mathrm{pt})}(a, b) \simeq \mathrm{hId}_{x:\mathbb{1}.C(x)}(\mathrm{pt}, \mathrm{pt}, \mathrm{refl}_{\mathbb{1}}(\mathrm{pt}), a, b)$$

and because $\mathrm{hId}_{x:\mathbb{1}.C(x)}(\mathrm{pt}, \mathrm{pt}, \mathrm{refl}_{\mathbb{1}}(\mathrm{pt}), a, b)$ is always a mere proposition by uniqueness of identity proofs for dependent heterogeneous identification types, it follows that $\mathrm{Id}_{C(\mathrm{pt})}(a, b)$ and thus by definition $\mathrm{Id}_{A}(a, b)$ is also always a mere proposition.

Axiom K

For identification types

In dependent type theory, axiom K is given by the following rule:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma\vdash a:A \quad \Gamma \vdash p:\mathrm{Id}_A(a, a)}{\Gamma\vdash K_A(a, p):\mathrm{Id}_{\mathrm{Id}_A(a, a)}(p, refl_A(a))}$$

For heterogeneous identification types

There is an axiom K for heterogeneous identification types:

$$\frac{ \begin{array}{l} \Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \\ \Gamma \vdash f:A \to B \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash p:\mathrm{Id}_A(a, b) \\ \Gamma \vdash q:\mathrm{hId}_{B}(a, b, p, f(a), f(b)) \end{array} }{\Gamma \vdash K_{B}(f, a, b, p, q):\mathrm{Id}_{\mathrm{hId}_{B}(a, b, p, f(a), f(b))}(q, \mathrm{ap}_{B}(f, a, b, p))}$$

Axiom K for identification types follows from axiom K for heterogeneous identification types by taking canonical element $\mathrm{pt}:\mathbb{1}$ of the unit type $\mathbb{1}$, identification $\mathrm{refl}_{\mathbb{1}}(\mathrm{pt}):\mathrm{Id}_{\mathbb{1}}(\mathrm{pt}, \mathrm{pt})$, and elements $a:A$ and $b:A$. Since the following two types are equivalent:

$$\mathrm{Id}_{A}(a, b) \simeq \mathrm{hId}_{A}(\mathrm{pt}, \mathrm{pt}, \mathrm{refl}_{\mathbb{1}}(\mathrm{pt}), a, b)$$

and because $\mathrm{hId}_{A}(\mathrm{pt}, \mathrm{pt}, \mathrm{refl}_{\mathbb{1}}(\mathrm{pt}), a, a)$ is always a contractible type with center of contraction $\mathrm{ap}_{A}(f, \mathrm{pt}, \mathrm{pt}, \mathrm{refl}_{\mathbb{1}}(\mathrm{pt}))$ for element $a:A$, and function $f:\mathbb{1} \to A$, by axiom K for heterogeneous identification types, it follows that $\mathrm{Id}_{A}(a, a)$ is also always a contractible type, with center of contraction $\mathrm{refl}_A(a)$.

Similarly, there is also an axiom K for dependent heterogeneous identification types:

$$\frac{ \begin{array}{l} \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \\ \Gamma \vdash f:\prod_{x:A} B(x) \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash p:\mathrm{Id}_A(a, b) \\ \Gamma \vdash q:\mathrm{hId}_{x:A.B(x)}(a, b, p, f(a), f(b)) \end{array} }{\Gamma \vdash K_{x:A.B(x)}(f, a, b, p, q):\mathrm{Id}_{\mathrm{hId}_{x:A.B(x)}(a, b, p, f(a), f(b))}(q, \mathrm{apd}_{x:A.B(x)}(f, a, b, p))}$$

Axiom K for identification types follows from axiom K for dependent heterogeneous identification types by defining the type $A$ to be the dependent type $C(\mathrm{pt})$ for type family $x:\mathbb{1} \vdash C(x)$ indexed by the unit type $\mathbb{1}$, and taking the canonical element $\mathrm{pt}:\mathbb{1}$, identification $\mathrm{refl}_{\mathbb{1}}(\mathrm{pt}):\mathrm{Id}_{\mathbb{1}}(\mathrm{pt}, \mathrm{pt})$, and elements $a:C(\mathrm{pt})$ and $b:C(\mathrm{pt})$. Since the two types are equivalent:

$$\mathrm{Id}_{A}(a, b) \equiv \mathrm{Id}_{C(\mathrm{pt})}(a, b) \simeq \mathrm{hId}_{x:\mathbb{1}.C(x)}(\mathrm{pt}, \mathrm{pt}, \mathrm{refl}_{\mathbb{1}}(\mathrm{pt}), a, b)$$

and because $\mathrm{hId}_{x:\mathbb{1}.C(x)}(\mathrm{pt}, \mathrm{pt}, \mathrm{refl}_{\mathbb{1}}(\mathrm{pt}), a, b)$ is always a contractible type with center of contraction $\mathrm{apd}_{x:\mathbb{1}.C(x)}(f, \mathrm{pt}, \mathrm{pt}, \mathrm{refl}_{\mathbb{1}}(\mathrm{pt}))$ for element $a:A$, and dependent function $f:\prod_{x:\mathbb{1}} C(x)$, by axiom K for heterogeneous identification types, it follows that $\mathrm{Id}_{C(\mathrm{pt}}(a, a)$ and thus by definition $\mathrm{Id}_{A}(a, a)$ is also always a contractible type, with center of contraction $\mathrm{refl}_A(a)$.

Leibniz’s identity of indiscernibles

In dependent type theory with power sets, Leibniz's identity of indiscernibles is given by the following inference rule:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, x:A, y:A \vdash \mathrm{idOfInd}_A(x, y):\mathrm{Id}_A(x, y) \simeq \left(\prod_{P:\mathcal{P}(A)} P(x) \simeq P(y)\right)}$$

This implies that all types $A$ are sets, because the type

$$\prod_{P:\mathcal{P}(A)} P(x) \simeq P(y)$$

is always a proposition, in the context of weak function extensionality, and thus by the equivalence of types, $\mathrm{Id}_A(x, y)$ is also a proposition for all $x:A$ and $y:A$, which means that $A$ is a set.

Set-theoretic principle of unique choice

In dependent type theory with power sets, the set-theoretic principle of unique choice is given by the following rule:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash \mathrm{puc}_{A, B}:\left(\sum_{R:A \times B \to \mathcal{P}(\mathbb{1})} \mathrm{isTotal}(R) \times \mathrm{isFunctional}(R)\right) \simeq (A \to B)}$$

Because the type $A \times B \to \mathcal{P}(\mathbb{1})$ is a set, the type

$$\sum_{R:A \times B \to \mathcal{P}(\mathbb{1})} \mathrm{isTotal}(R) \times \mathrm{isFunctional}(R)$$

is also a set, and the equivalence only exists if the function type $A \to B$ is a set, which only occurs if $B$ is a set. Thus, this rule implies that all types are sets.

Boundary separation

In cubical type theory, boundary separation is given by the following rule:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash r:I \quad \Gamma, (r =_I 0) \vee (r =_I 1) \; \mathrm{true} \vdash a \equiv b:A}{\Gamma \vdash a \equiv b:A}$$

Axiom of existence

In dependent type theory, the type-theoretic axiom of existence states that every type has a choice operator, a function from its bracket type to the type itself, and is given by the following rule:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \epsilon_A:[A] \to A}$$

Indeed, it suffices for the identity types to have choice operators:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, x:A, y:A \vdash \epsilon_{\mathrm{Id}_A}(x, y):[\mathrm{Id}_A(x, y)] \to \mathrm{Id}_A(x, y)}$$

Equality reflection

In dependent type theory, equality reflection is given by the following rule:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash p:\mathrm{Id}_A(a, b)}{\Gamma \vdash a \equiv b:A}$$

Strong pattern matching

…

$S^1$-localization

In dependent type theory, the axiom of $S^1$-localization is given by the following rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash f:S^1 \to A}{\Gamma \vdash \mathrm{circlocal}_{A}:\mathrm{isEquiv}(\mathrm{const}_{A, S^1})}$$

 Axiom of finiteness

In dependent type theory with dependent product types, dependent sum types, identity types, function extensionality, and a type of all propositions, the axiom of finiteness is given by the following rule:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{finWitn}_A:\mathrm{isFinite}(A)}$$

where

$$\mathrm{isFinite}(A) \equiv \begin{array}{c} \prod_{S:(A \to \mathrm{Prop}) \to \mathrm{Prop}} (((\lambda x:A.\bot) \in S) \times \prod_{P:A \to \mathrm{Prop}} \prod_{Q:A \to \mathrm{Prop}} (P \in S) \\ \times (\exists!x:A.x \in Q) \times (P \cap Q =_{A \to \mathrm{Prop}} \lambda x:A.\bot) \to (P \cup Q \in S)) \to ((\lambda x:A.\top) \in S) \end{array}$$

The membership relation and the subtype operations used above are defined in the nLab article on subtypes.

Circle type axioms

Any one of the following axioms on the circle type implies axiom K or uniqueness of identity proofs for all types in the dependent type theory, and thus is an axiom of set-truncation:

• That $S^1$ is a set,
  $$\mathrm{trunc}_0^{S^1}:\mathrm{isSet}(S^1)$$
• That $S^1$ is a proposition or a contractible type, which are the same condition as $S^1$ being a set because $S^1$ is a pointed connected type
  $$\mathrm{trunc}^{S^1}:\mathrm{isProp}(S^1) \qquad \mathrm{contr}_{S^1}:\mathrm{isContr}(S^1)$$
• That there is an identification $K:\mathrm{refl}_{S^1}(\mathrm{base}) =_{S^1} \mathrm{loop}$

Kuratowski-finite implies finite

In dependent type theory, in general there are Kuratowski-finite types which are not h-sets, such as the circle type. However, every finite type is an h-set, and hence why they are usually called finite sets.

A naive translation into dependent type theory of the traditional set-theoretic theorem in classical mathematics that “Kuratowski-finite implies finite” results in the statement that every Kuratowski-finite type is finite. This implies that the circle type is a finite type, which implies that the circle type is an h-set, which implies that every type is an h-set. Thus, that every Kuratowski-finite type is finite is an axiom of set truncaiton.

The statement “Kuratowski-finite implies finite” in classical mathematics is really a statement about h-sets, similar to how the axiom of choice is really a statement about h-sets. This is because it is still true in constructive mathematics that every Kuratowski-finite type with decidable equality is a finite type, and in dependent type theory, every type with decidable equality is an h-set.

See also

• h-set
• set-level type theory