objects such that commutes with certain colimits
the collection of equivalence classes in is -small;
is essentially small if the above conditions hold “absolutely,” i.e. with ”-small” replaced by “small.”
This appears as HTT, def. 22.214.171.124, prop. 126.96.36.199.
This is HTT, prop. 188.8.131.52
The analogous statement holds for ∞-groupoids.
This is (HTT, corollary 184.108.40.206).
Notice that this proposition really requires that be uncountable. When it is not true: the -compact objects of ∞Grpd are the homotopy retracts of finite CW-complexes, while the -small ∞-groupoids are just the finite CW-complexes. Not every retract of a finite CW-complex has the homotopy type of a finite CW-complex: there is an obstruction, defined for a retract of a finite CW-complex, which is an element of , is called Wall’s finiteness obstruction, and vanishes if and only if has the homotopy type of a finite CW-complex. See Wall’s paper in the references.
This is the topic of section 5.4.1 of
Wall’s finiteness obstruction was defined in