Object re-identification (ReID) aims to find instances with the same identity as the given probe from a large gallery. Pairwise losses play an important role in training a strong ReID network. Existing pairwise losses densely exploit each instance as an anchor and sample its triplets in a mini-batch. This dense sampling mechanism inevitably introduces positive pairs that share few visual similarities, which can be harmful to the training. To address this problem, we propose a novel loss paradigm termed Sparse Pairwise (SP) loss that only leverages few appropriate pairs for each class in a mini-batch, and empirically demonstrate that it is sufficient for the ReID tasks. Based on the proposed loss framework, we propose an adaptive positive mining strategy that can dynamically adapt to diverse intra-class variations. Extensive experiments show that SP loss and its adaptive variant AdaSP loss outperform other pairwise losses, and achieve state-of-the-art performance across several ReID benchmarks. Code is available at https://github.com/Astaxanthin/AdaSP.