Diffeological submanifolds and their friends (Q6614921)

Three intrinsic notions of submanifolds are known in the literature, namely, (embedded) submanifolds, weakly-embedded submanifolds and diffeological submanifolds. This paper proposes a fourth such notion of \textit{uniquely immersed submanifolds}. It is shown that these four notions of submanifolds are all different.\N\NViewing manifolds as a concrete category over the category of sets, the \textit{initial morphisms} are exactly the (diffeological) \textit{inductions}, which are the diffeomorphisms with diffeological submanifolds. From this standpoint, \textit{H. Joris} and \textit{E. Preissmann}'s [Ann. Inst. Fourier 37, No. 2, 195--221 (1987; Zbl 0596.58004)] notion of \textit{psuedo-immersions} is recovered.\N\NA theorem of \textit{H. Joris} [Arch. Math. 39, 269--277 (1982; Zbl 0504.58007)] yields a diffeological submanifold whose inclusion is not an immersion. This answers a question posed by \textit{P. Iglesias-Zemmour} [Diffeology. Providence, RI: American Mathematical Society (AMS) (2013; Zbl 1269.53003)]. Local inductions are characterized as those pseudo-immersions that are locally injective.\N\NAppendix A provides a proof of Joris' theorem along the lines of [\textit{I. Amemiya} and \textit{K. Masuda}, Kodai Math. J. 12, No. 1, 92--97 (1989; Zbl 0683.58009)].