Digital topology is part of the ongoing endeavor to understand and analyze digitized images. With a view to supporting this endeavor, many notions from algebraic topology have been introduced into the setting of digital topology. But some of the most basic notions from homotopy theory remain largely absent from the digital topology literature. We embark on a development of homotopy theory in digital topology, and define such fundamental notions as function spaces, path spaces, and cofibrations in this setting. We establish digital analogues of basic homotopy-theoretic properties such as the homotopy extension property for cofibrations, and the homotopy lifting property for certain evaluation maps that correspond to path fibrations in the topological setting. We indicate that some depth may be achieved by using these homotopy-theoretic notions to give a preliminary treatment of Lusternik–Schnirelmann category in the digital topology setting. This topic provides a connection between digital topology and critical points of functions on manifolds, as well as other topics from topological dynamics.

数字拓扑是理解和分析数字化图像的持续努力的一部分。为了支持这一努力，代数拓扑的许多概念已被引入到数字拓扑的设置中。但是同伦理论中的一些最基本的概念在数字拓扑文献中基本上没有。我们着手开发数字拓扑中的同伦理论，并在此设置中定义函数空间、路径空间和共纤维等基本概念。我们建立了基本同伦理论属性的数字模拟，例如共纤维的同伦扩展属性，以及与拓扑设置中的路径纤维相对应的某些评估图的同伦提升属性。我们指出，通过使用这些同伦理论概念在数字拓扑设置中对 Lusternik-Schnirelmann 类别进行初步处理，可以达到一定的深度。本主题提供了数字拓扑与流形上函数临界点之间的联系，以及拓扑动力学中的其他主题。