CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
n-category = (n,n)-category
n-groupoid = (n,0)-category
An ∞-groupoid or a topological space or another realization of the concept (∞,0)-category is contractible if it is weakly equivalent to the point.
Sometimes one allows also the empty object to be contractible. To distinguish this, we say
an -groupoid is (-1)-truncated (is a (-1)-groupoid) if it is either empty or equivalent to the point;
an -groupoid is (-2)-truncated (is a (-2)-groupoid) if it is equivalent to the point.
Notice that since the Whitehead theorem applies in ∞Grpd, being weakly equivalent to the point is the same as there being a contraction.
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see also