This paper addresses the challenge of mapping polygonal buildings from remote sensing images and introduces a novel algorithm, the Global Collinearity-aware Polygonizer (GCP). GCP, built upon an instance segmentation framework, processes binary masks produced by any instance segmentation model. The algorithm begins by collecting polylines sampled along the contours of the binary masks. These polylines undergo a refinement process using a transformer-based regression module to ensure they accurately fit the contours of the targeted building instances. Subsequently, a collinearity-aware polygon simplification module simplifies these refined polylines and generate the final polygon representation. This module employs dynamic programming technique to optimize an objective function that balances the simplicity and fidelity of the polygons, achieving globally optimal solutions. Furthermore, the optimized collinearity-aware objective is seamlessly integrated into network training, enhancing the cohesiveness of the entire pipeline. The effectiveness of GCP has been validated on two public benchmarks for polygonal building mapping. Further experiments reveal that applying the collinearity-aware polygon simplification module to arbitrary polylines, without prior knowledge, enhances accuracy over traditional methods such as the Douglas-Peucker algorithm. This finding underscores the broad applicability of GCP. The code for the proposed method will be made available at https://github.com/zhu-xlab.