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Promising results have been achieved recently in category-level manipulation that generalizes across object instances. Nevertheless, it often requires expensive real-world data collection and manual specification of semantic keypoints for each object category and task. Additionally, coarse keypoint predictions and ignoring intermediate action sequences hinder adoption in complex manipulation tasks beyond pick-and-place. This work proposes a novel, category-level manipulation framework that leverages an object-centric, category-level representation and model-free 6 DoF motion tracking. The canonical object representation is learned solely in simulation and then used to parse a category-level, task trajectory from a single demonstration video. The demonstration is reprojected to a target trajectory tailored to a novel object via the canonical representation. During execution, the manipulation horizon is decomposed into longrange, collision-free motion and last-inch manipulation. For the latter part, a category-level behavior cloning (CatBC) method leverages motion tracking to perform closed-loop control. CatBC follows the target trajectory, projected from the demonstration and anchored to a dynamically selected category-level coordinate frame. The frame is automatically selected along the manipulation horizon by a local attention mechanism. This framework allows to teach different manipulation strategies by solely providing a single demonstration, without complicated manual programming. Extensive experiments demonstrate its efficacy in a range of challenging industrial tasks in highprecision assembly, which involve learning complex, long-horizon policies. The process exhibits robustness against uncertainty due to dynamics as well as generalization across object instances and scene configurations. The supplementary video is available at https://www.youtube.com/watch?v=WAr8ZY3mYyw