Submersions, immersions, and \'etale maps in diffeology

Although structural maps such as subductions and inductions appear naturally in diffeology, one of the challenges is providing suitable analogous for submersions, immersions, and \'{e}tale maps (i.e., local diffeomorphisms) consistent with the classical versions of these maps between manifolds. In this paper, we consider diffeological submersions, immersions, and \'{e}tale maps as an adaptation of these maps to diffeology by a nonlinear approach. In the case of manifolds, there is no difference between the classical and diffeological versions of these maps. Moreover, we study their diffeological properties from different aspects in a systematic fashion with respect to the germs of plots. We also discuss notions of embeddings of diffeological spaces and regard diffeological embeddings similar to those of manifolds. In particular, we show that diffeological embeddings are inductions. In order to characterize the considered maps from their linear behaviors, we introduce a class of diffeological spaces, so-called diffeological \'etale manifolds, which not only contains the usual manifolds but also includes irrational tori. We state and prove versions of the rank and implicit function theorems, as well as the fundamental theorem on flows in this class. As an application, we use the results of this work to facilitate the computations of the internal tangent spaces and diffeological dimensions in a few interesting cases.