In this paper, we propose a novel super-resolution and infection edge detection co-guided learning network for COVID-19 CT segmentation (CogSeg). Our CogSeg is a coherent framework consisting of two branches. Specifically, we use image super-resolution (SR) as an auxiliary task, which assist segmentation to recover high-resolution representations. Moreover, we propose an infection edge detection guided region mutual information (RMI) loss, which uses the edge detection results of segmentation to explicitly maintain the high order consistency between segmentation prediction and ground truth around infection edge pixels. Our CogSeg network can effectively maintain high-resolution representation and leverages edge details to improve the segmentation performance. When evaluated on two publicly available COVID-19 CT datasets, our CogSeg improves 10.63 and 13.02 points than the established baseline method (i.e. U-Net) w.t.r mIoU. Moreover, our CogSeg achieves more appealing results both quantitatively and qualitatively than the state-of-the-art methods.