CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
A topological space $X$ is contractible if the canonical map $X \to \ast$ is a homotopy equivalence. It is weakly contractible is this map is a weak homotopy equivalence, hence if all homotopy groups of $X$ are trivial.
Where the Whitehead theorem does not apply, we may find examples of weakly contractible but not contractible spaces, such as the double comb space in Top.
Since the Whitehead theorem applies in ∞Grpd (and generally in any hypercomplete (∞,1)-topos), being weakly equivalent to the point is the same as there being a contraction. So an ∞-groupoid is weakly contractible if and only if it is contractible.
In this context one tends to drop the “weakly” qualifier.
Sometimes one allows also the empty object $\emptyset$ to be contractible. To distinguish this, we say
an $\infty$-groupoid is (-1)-truncated (is a (-1)-groupoid) if it is either empty or equivalent to the point;
an $\infty$-groupoid is (-2)-truncated (is a (-2)-groupoid) if it is equivalent to the point.
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 |