natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory
For all $n \in \{-2, -1, 0,1,2,3, \cdots\}$, n-truncation is a modality (in homotopy type theory).
$(-2)$-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
and this is a 1-epimorphism precisely if the law of excluded middle holds.
(…)
The $(-1)$-truncation in the context is forming the bracket type hProp. $n$-truncation is given by a higher inductive type.
homotopy level | n-truncation | homotopy theory | higher category theory | higher topos theory | homotopy type theory |
---|---|---|---|---|---|
h-level 0 | (-2)-truncated | contractible space | (-2)-groupoid | true/unit type/contractible type | |
h-level 1 | (-1)-truncated | contractible-if-inhabited | (-1)-groupoid/truth value | (0,1)-sheaf/ideal | mere proposition/h-proposition |
h-level 2 | 0-truncated | homotopy 0-type | 0-groupoid/set | sheaf | h-set |
h-level 3 | 1-truncated | homotopy 1-type | 1-groupoid/groupoid | (2,1)-sheaf/stack | h-groupoid |
h-level 4 | 2-truncated | homotopy 2-type | 2-groupoid | (3,1)-sheaf/2-stack | h-2-groupoid |
h-level 5 | 3-truncated | homotopy 3-type | 3-groupoid | (4,1)-sheaf/3-stack | h-3-groupoid |
h-level $n+2$ | $n$-truncated | homotopy n-type | n-groupoid | (n+1,1)-sheaf/n-stack | h-$n$-groupoid |
h-level $\infty$ | untruncated | homotopy type | ∞-groupoid | (∞,1)-sheaf/∞-stack | h-$\infty$-groupoid |
Univalent Foundations Project, section 6.9 of Homotopy Type Theory -- Univalent Foundations of Mathematics