homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
n-category = (n,n)-category
n-groupoid = (n,0)-category
A $(-2)$-groupoid or (-2)-type is a (−2)-truncated object in ∞Grpd.
There is, up to equivalence, just one $(-2)$-groupoid, namely the point.
Compare the concepts of $(-1)$-groupoid (a truth value) and $0$-groupoid (a set). Compare also with $(-2)$-category and $(-1)$-poset, which mean the same thing for their own reasons.
The point of $(-2)$-groupoids is that they complete some patterns in the periodic tables and complete the general concept of $n$-groupoid. For example, there should be a $(-1)$-groupoid $(-2)\Grpd$ of $(-2)$-groupoids; a $(-1)$-groupoid is simply a truth value, and $(-2)\Grpd$ is the true truth value.
As a category, $(-2)\Grpd$ is a monoidal category in a unique way, and a groupoid enriched over this should be (at least up to equivalence) a $(-1)$-groupoid, which is a truth value; and indeed, a groupoid enriched over $(-2)\Grpd$ is a groupoid in which any two objects are isomorphic in a unique way, which is equivalent to a truth value.
See (−1)-category for references on this sort of negative thinking.
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 | mere proposition, 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 $n+2$ | $n$-truncated | homotopy n-type | n-groupoid | h-$n$-groupoid | |
h-level $\infty$ | untruncated | homotopy type | ∞-groupoid | (∞,1)-sheaf/∞-stack | h-$\infty$-groupoid |