objects such that commutes with certain colimits
preserves -filtered (∞,1)-colimits.
For -compact we just say compact.
This appears as (HTT, def. 188.8.131.52).
Let be a regular cardinal.
This is (HTT, cor. 184.108.40.206).
If the (∞,1)-category is a locally presentable (∞,1)-category, then it is the simplicial localization of a combinatorial model category , and one may ask how the 1-categorical notion of compact object in relates to the -categorical notion of compact in .
Since compactness is defined in terms of colimits, the question is closely related to the question which 1-categorical -filtered colimits in are already homotopy colimits (without having to derive them first).
In ∞Grpd, for uncountable , the -compact objects are precisely the -essentially small ∞-groupoids. When , the compact objects in ∞Grpd are the retracts of the -small ∞Grpds, i.e., the retracts of the finite homotopy types (finite CW-complexes). Not every such retract is equivalent to a -small ∞-groupoid; the vanishing of Wall's finiteness obstruction is a necessary and sufficient condition for such an equivalence to exist.
compact object in an -category
The general definition appears as definition 220.127.116.11 in
Compactness in presenting model categories of simplicial sheaves is discussed for instance in
section 4 of