Diffeological submanifolds and their friends

TL;DR

This paper compares various notions of submanifolds in differential geometry, highlighting the categorical advantages of diffeological submanifolds and providing new insights into their properties and distinctions.

Contribution

It demonstrates that diffeological submanifolds are categorically superior and clarifies the differences among various submanifold concepts, including a counterexample to a longstanding question.

Findings

Diffeological submanifolds are initial morphisms in the category of manifolds.

A theorem of Joris provides a diffeological submanifold whose inclusion is not an immersion.

Local inductions are characterized as pseudo-immersions that are locally injective.

Abstract

A smooth manifold hosts different types of submanifolds, including embedded, weakly-embedded, and immersed submanifolds. The notion of an immersed submanifold requires additional structure (namely, the choice of a topology); when this additional structure is unique, we call the subset a uniquely immersed submanifold. Diffeology provides yet another intrinsic notion of submanifold: a diffeological submanifold. We show that from a categorical perspective diffeology rises above the others: viewing manifolds as a concrete category over the category of sets, the initial morphisms are exactly the (diffeological) inductions, which are the diffeomorphisms with diffeological submanifolds. Moreover, if we view manifolds as a concrete category over the category of topological spaces, we recover Joris and Preissmann's notion of pseudo-immersions. We show that these notions are all different. In particular, a theorem of Joris from 1982 yields a diffeological submanifold whose inclusion is not an immersion, answering a question that was posed by Iglesias-Zemmour. We also characterize local inductions as those pseudo-immersions that are locally injective. In appendices, we review a proof of Joris' theorem, pointing at a flaw in one of the several other proofs that occur in the literature, and we illustrate how submanifolds inherit paracompactness from their ambient manifold.