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Immersions & sumersions are important in differential manifolds. They rely on their definition of the construction of the tangent bundle.

I realise that generalised smooth spaces do not have a canonical tangent bundle.

But they have better categorical properties, but nastier objects. Is it possible to define what a submerssion/immersion is here by a universal property?

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Diffeological spaces are examples of generalized smooth spaces, they give a complete and cocomplete category of spaces including smooth spaces as a full subcategory. A diffeological space is a set together with a diffeology: a collection of plots (maps from a numerical domain) which determines the smooth structure.

For these diffeological spaces you have the concept of inductions and subductions. Let me focus on inductions they are morphisms $f:(X,D)\rightarrow (X',D')$ such that $f^*(D')=D$ ($D$ is the pull-back diffeology of $D'$ along $f$) and $f$ is injective. For smooth manifolds with their induced diffeology, every induction is an immersion but an immersion is not necessarily an induction even it is injective. Rather an immersion is a local induction. Thus at least for diffeological spaces if you want to generalize immersions you have to work locally (with a superset of every point). You also have local inductions, subductions. You can have a look at P. Iglesias-Zemmour's book "Diffeology" (published by the AMS).

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Nice answer. Immersions are defined locally. What does it mean that an immersion is a local induction? That one must reduce its domain of definition further? For subductions, are they the dual of what you've stated about inductions? – Mozibur Ullah Dec 2 '12 at 14:44

A surjective submersion can be characterised in the category of smooth manifolds as being an element in the largest pullback-stable class of regular epis. Also, submersions have local sections through every point in their codomain. Or they look locally like a projection out of a cartesian product with fibre a vector space. This latter is in fact how submersions are defined for infinite-dimensional (well, Fréchet) manifolds. The first definition works fine for smooth spaces, but proofs may not generalise if they need some of the other properties. The second one works if you are considering concrete smooth spaces, but is meaningless otherwise. The third definition implies the second, but restricts the sort of fibres.

Edit: Another property of surjective submersions is that they are the largest class of morphisms which admit sections over open covers (which we can assume are made up of charts) of which all pullbacks exist. This means that surjective submersions are in some sense a 'saturation' of the singleton pretopology consisting of covers of the form $\coprod U_i \to X$.

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Thanks, I wasn't aware of the first definition. Off-topic, it seems to me that the all three definitions are available for topological manifolds, whether any of them are provably equivalent is a different question. Are you aware as to whether they're useful distinctions to make in that category? – Mozibur Ullah Dec 2 '12 at 14:51

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