The present paper establishes a homotopy based on Marcus-Wyse (for brevity M-) topology, which can contribute to the classification of 2D digital images in an M-topological approach. Since M-topology is considered in the Euclidean 2D space with integer coordinates, it can be used in studying 2D digital images from the viewpoints of both digital topology and digital geometry such as image processing, image analysis, computer graphics, mathematical morphology and so forth. To develop the homotopy, the present paper uses two maps such as an M-continuous map and Marcus-Wyse adjacency (for brevity MA-) map because they have their own features and merits. Besides, using this homotopy, the present paper proposes an MA-homotopy equivalence and further, MA-contractibility which can be also used in classifying digital images. Finally, the paper establishes an MA-homotopic thinning derived from the above homotopy, which can contribute to the compression of 2D digital images. (C) 2015 Elsevier B.V. All rights reserved.