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
Homotopy level (or h-level) is another name for the notion of truncation (particularly in (∞,1)-categories and their internal language of homotopy type theory) in which the numbering is offset by 2:
a homotopy n-type is a type of homotopy level .
This offset in counting enables it to “start” at 0 rather than (-2), which is convenient when defining it by induction over the natural numbers in type theory. Thus, the correspondence between the various terminologies is indicated in the following table.
| 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 | (-1)-groupoid/truth value | h-proposition | ||
| h-level 2 | 0-truncated | discrete space | 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 | h-2-groupoid | |
| h-level 5 | 3-truncated | homotopy 3-type | 3-groupoid | h-3-groupoid | |
| h-level | -truncated | homotopy n-type | n-groupoid | h--groupoid | |
| h-level | untruncated | homotopy type | ∞-groupoid | (∞,1)-sheaf/∞-stack | h--groupoid |