# nLab n-truncation modality

Contents

### Context

#### Higher category theory

higher category theory

# Contents

## Idea

For all $n \in \{-2, -1, 0,1,2,3, \cdots\}$, n-truncation is a modality (in homotopy type theory).

## Properties

### In low degree

$(-2)$-truncation is the unit type modality (constant on the unit type).

$(-1)$-truncation is given by forming bracket types, turning types into genuine propositions.

Classically, this is the same as the double negation modality; in general, the bracket type ${\|A\|_{-1}}$ only entails the double negation $\neg(\neg{A})$:

there is a canonical function

${\|A\|_{-1}} \longrightarrow \neg(\neg{A})$

and this is a 1-epimorphism precisely if the law of excluded middle holds.

### In homotopy type theory

(…)

The $(-1)$-truncation in the context is forming the bracket type hProp. $n$-truncation is given by a higher inductive type.

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