Abstract
We fix a proper class of [Formula: see text]-triangles [Formula: see text] in an extriangulated category [Formula: see text]. Let [Formula: see text] be a class of objects in [Formula: see text] such that [Formula: see text], [Formula: see text] and [Formula: see text] for all [Formula: see text] and all [Formula: see text]. In this paper, we study [Formula: see text]-Gorenstein objects, and use the relative homological methods to show that the stability of the Gorenstein category [Formula: see text] in [Formula: see text]. Moreover, we give some equivalent characterizations of [Formula: see text]-Gorenstein objects and finite [Formula: see text]-(co)resolution dimensions for any object in [Formula: see text]. As an application, we consider the equivalent characterizations of [Formula: see text]-[Formula: see text]projective objects, [Formula: see text]-[Formula: see text]projective dimension and their duals.
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