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?



are immersions/submersions captured in generalised smooth spaces by some universal property?
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). 


