We give a formulation of the Nielsen-Schreier theorem (subgroups of free groups are free) in homotopy type theory using the presentation of groups as pointed connected 1-truncated types. We show the special case of finite index subgroups holds constructively and the full theorem follows from the axiom of choice. We give an example of a boolean infinity topos where our formulation of the theorem does not hold and show a stronger "untruncated" version of the theorem is provably false in homotopy type theory.
翻译:我们用单一式理论的表述方式,用直截了当的直截了当的组合来表述尼尔森-施赖尔理论(自由团体的分组是自由的)。我们用直截了当的组合来表述单一式理论。我们展示了有限指数分组的特殊情况,这种特殊情形具有建设性,完全的定理源自选择的轴心。我们举了一个布林无穷无穷的例子,我们的定理的构思并不坚固,并展示了更强的“无节制的”定理版本在同质类型理论中是完全虚假的。