着重研究了M-纤维式范畴中的同伦论相关概念, 即M-纤维式同伦扩张性质. 通过M-纤维式收缩以及M-纤维式形变收缩的概念，给出了M-纤维式同伦扩张性质的等价描述，从而推广了一般拓扑范畴中同伦扩张性质的相关等价描述. 另外，证明了在M-纤维式范畴中的一些空间构造，如M-纤维式贴附空间的同伦不变形性. 最后, 对于2个M-纤维式映射是否同伦等价的问题，通过M-纤维式映射柱的概念给出了相关的判定定理，这一判定是一般拓扑范畴中两映射是否同伦等价的判定的自然推广.
Some homotopical notions in M-fibrewise category such as M-fibrewise homotopy extension property are studied. With the help of notions such as M-fibreswise retract and M-fibrewise deformation retract, an equivalent description of M-fibrewise homotopy extension property which generalizes the related one for homotopy extension property in general topological category is given. Furthermore the homotopy invariance of some constructions in M-fibrewise category such as M-fibrewise attaching space is shown. Finally, a judgment theorem is given via M-fibrewise mapping cylinder for the problem on whether two M-fibrewise maps are homotopic. This judgment is a natural generalization for the one on whether two maps are homotopic in general topological category.