Contents

# Contents

## Idea

There are various different paradigms for the interpretation of predicate logic in type theory. In “logic-enriched type theory”, there is a separate class of “propositions” from the class of “types”. But we can also identify propositions with particular types. In the propositions as types-paradigm, every proposition is a type, and also every type is identified with a proposition (the proposition that it is an inhabited type).

By contrast, in the paradigm that may be called propositions as some types, every proposition is a type, but not every type is a proposition. The types which are propositions are generally those which “have at most one inhabitant” — in homotopy type theory this is called being of h-level 1 or being a (-1)-type. This paradigm is often used in the categorical semantics of type theory, such as the internal logic of various kinds of categories.

Under “propositions as types”, all type-theoretic operations represent corresponding logical operations (dependent sum is the existential quantifier, dependent product the universal quantifier, and so on). However, under “propositions as some types”, not every such operation preserves the class of propositions; this is particularly the case for dependent sum and disjunction(or). Thus, in order to obtain the correct logical operations, we need to reflect these constructions back into propositions somehow, finding the “underlying proposition”, corresponding to the (-1)-truncation/h-level 1-projection. This operation in type theory is called the bracket type (when denoted $[A]$); in homotopy type theory it can be identified with the higher inductive type $isInhab$.

## Definition

### In homotopy type theory

In homotopy type theory, the propositional truncation of a type $A$ is defined as the higher inductive type generated by the two constructors

$isinhab\colon A \to [A]$
$proptrunc \colon \prod_{x:A} \prod_{y:A} isinhab(x) =_{[A]} isinhab(y)$

### More generally

In any dependent type theory with identity types, given type $A$, the support of $A$ denoted $supp(A)$ or $isInhab(A)$ or $\tau_{-1} A$ or $\| A \|_{-1}$ or $\| A \|$ or, lastly, $[A]$, is the higher inductive type defined by the two constructors

$a \colon A \;\vdash \; isinhab(a) \colon supp(A)$
$x \colon supp(A) \;,\; y \colon supp(A) \;\vdash \; inpath(x,y) \colon (x = y) \,,$

where in the last sequent on the right we have the identity type. (Voevodsky, HoTTLibrary)

This says that $supp(A)$ is the type which is universal with the property that the terms of $A$ map to it and that any two term of $A$ become equivalent in $supp(A)$.

In Agda syntax this is

data isinhab {i : Level} (A : Set i) : Set i where
inhab : A → isinhab A
inhab-path : (x y : isinhab A) → x ≡ y

The recursion principle for $supp(A)$ says that if $B$ is a mere proposition and we have $f: A \to B$, then there is an induced $g : supp(A) \to B$ such that $g(isinhab(a)) \equiv f(a)$ for all $a:A$. In other words, any mere proposition which follows from (the inhabitedness of) $A$ already follows from $supp(A)$. Thus, $supp(A)$, as a mere proposition, contains no more information than the inhabitedness of $A$.

### With a type of all propositions

Suppose the dependent type theory has a univalent type of all propositions $(\mathrm{Prop}, \mathrm{El})$. Then the bracket type of $A$ could be defined as the type

$[A] \coloneqq \prod_{P:\mathrm{Prop}} (A \to \mathrm{El}(P)) \to \mathrm{El}(P)$

### As localization

Let $\mathbb{1}$ be the unit type and let $\mathbb{2}$ be the two-valued type. The bracket type of a type $A$ is the localization of $A$ at the unique function $\mathbb{2} \to \mathbb{1}$.

$\left[A\right] \coloneqq L_\mathbb{2}(A)$

By definition, the type of functions $(\mathbb{1} \to \left[A\right]) \to (\mathbb{2} \to \left[A\right])$ is an equivalence of types.

This is the special case of the n-truncation modality as the n-truncation modality is localization at the unique map from the $(n + 1)$-dimensional sphere type to the unit type, and $\mathbb{2}$ is the zero-dimensional sphere type.

For more see at n-truncation modality.

### As a sequential colimit

Let $A * B$ denote the join type of $A$ and $B$, and let $A^{*(n)}$ be the iterated join type, inductively defined on the natural numbers by $A^{*(0)} \coloneqq \emptyset$ and $A^{*(n + 1)} \coloneqq A^{*(n)} * A$.

Then the propositional truncation of $A$ is the sequential colimit of the sequence of functions

$A^{*(0)} \overset{\mathrm{inr}}\to A^{*(1)} \overset{\mathrm{inr}}\to A^{*(2)} \overset{\mathrm{inr}}\to \ldots$

## Semantics

One presentation of the internal type theory of regular categories consists of dependent type theory with the unit type, strong extensional equality types?, strong dependent sums, and bracket types. (The internal logic of a regular category can alternatively be presented as a logic-enriched type theory?.)

The semantics of bracket types in a regular category $C$ is as follows.

A dependent type (a type in context $X$)

$x\colon X \vdash A(x) \colon Type$

is interpreted in $C$ as an arbitrary morphism

$\array{ A \\ \downarrow \\ X } \,.$

The corresponding bracket type

$x\colon X \vdash [A(x)] \colon Type$

is interpreted then as the image-factorization

$\array{ A &&\to&& [A] & := im(A \to X) \\ & \searrow && \swarrow \\ && X \,. }$

Therefore $[A] \to X$ is a monomorphism, and hence the interpretation of a proposition about the elements of $X$.

homotopy leveln-truncationhomotopy theoryhigher category theoryhigher topos theoryhomotopy type theory
h-level 0(-2)-truncatedcontractible space(-2)-groupoidtrue/​unit type/​contractible type
h-level 1(-1)-truncatedcontractible-if-inhabited(-1)-groupoid/​truth value(0,1)-sheaf/​idealmere proposition/​h-proposition
h-level 20-truncatedhomotopy 0-type0-groupoid/​setsheafh-set
h-level 31-truncatedhomotopy 1-type1-groupoid/​groupoid(2,1)-sheaf/​stackh-groupoid
h-level 42-truncatedhomotopy 2-type2-groupoid(3,1)-sheaf/​2-stackh-2-groupoid
h-level 53-truncatedhomotopy 3-type3-groupoid(4,1)-sheaf/​3-stackh-3-groupoid
h-level $n+2$$n$-truncatedhomotopy n-typen-groupoid(n+1,1)-sheaf/​n-stackh-$n$-groupoid
h-level $\infty$untruncatedhomotopy type∞-groupoid(∞,1)-sheaf/​∞-stackh-$\infty$-groupoid

The original articles are

• M.E. Maietti, The Type Theory of Categorical Universes PhD thesis, Università Degli Studi di Padova, 1998

(which speaks of “mono types”) and

• Frank Pfenning, Intensionality, extensionality, and proof irrelevance in modal type theory, In Proceedings of the 16th Annual Symposium on Logic in Computer Science (LICS’01), June 2001.

• Steve Awodey, Andrej Bauer, Propositions as $[$Types$]$, Journal of Logic and Computation, 14 (2004) 447-471 [doi:10.1093/logcom/14.4.447, pdf]

Discussion in the context of homotopy type theory:

Exposition:

Formalization:

Discussion in the more general context of $n$-truncations:

More on the universal property of propositional truncation:

For propositional truncations as sequential colimits, see section 26.5 of

as well as section 16.2 of the lecture notes:

For n-truncations as localizations at sphere types, see: