The compact objects in ∞Grpd are the retracts of finite homotopy types (finite CW-complexes). Not every such retract is itself a finite homotopy type; the vanishing of Wall’s finiteness obstruction is a necessary and sufficient condition for this to happen.
C. T. C. Wall, Finiteness conditions for CW-complexes I pdf
C. T. C. Wall, Finiteness conditions for CW-complexes II pdf
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