A dévissage theorem for modular exact categories with weak equivalences

In this note, we will introduce a notion of modularity of exact categories due to Masana Harada [Har05]. The naming is coming from the classical modular lattices theory [Bir48]. We will also state and prove so-called “homotopy Grayson-Staffeldt-Jordan-Hölder theorem” which is implicitly appeared in [Gra87] and [Sta89]. The theorem says contractibility of a simplicial set associated to a certain ordered set. Combining these two idea and utilizing Waldhausen’s technique in [Wal85], we will get a dévissage theorem for modular exact categories with weak equivalences which is a generalization of original Quillen’s one in [Qui73].