Given any category , we can form the corresponding category of pro-objects in , which is denoted by -. Since the category with one morphism is a coflitered category, within -, we have all pro-objects of the form . Clearly such a functor is completely determined by the single object, , of to which it corresponds. This gives a functor:
c: C\to pro-C
which embeds the category in -. (This is really the Yoneda embedding in disguise.)
Any pro-object isomorphic in - to one of the form, , for an object of , is called stable or essentially constant.
In any given categorical context, the so-called stability problem is the problem of deciding what internal criteria can be applied to check if a given pro-object in that context, is or is not stable,
If is an abelian category, it is relatively simple to give necessary and sufficient ‘internal’ conditions for a given pro-object to be essentially constant. It must be both essentially epimorphic? and essentially monomorphic?.