Given any category $C$, we can form the corresponding category of pro-objects in $C$, which is denoted by $\mathrm{pro}$-$C$. Since the category $\mathrm{\pi \x9d\x9f\x98}$ with one morphism is a coflitered category, within $\mathrm{pro}$-$C$, we have all pro-objects of the form $X:\mathrm{\pi \x9d\x9f\x98}\beta \x86\x92C$. Clearly such a functor is completely determined by the single object, $X(*)$, of $C$ to which it corresponds. This gives a functor:

which embeds the category $C$ in $\mathrm{pro}$-$C$. (This is really the Yoneda embedding in disguise.)

Definition

Any pro-object isomorphic in $\mathrm{pro}$-$C$ to one of the form, $c(X)$, for $X$ an object of $C$, is called stable or essentially constant.

Stability problem

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 $C$ is an abelian category, it is relatively simple to give necessary and sufficient βinternalβ conditions for a given pro-object to be essentially constant. It must be both essentially epimorphic? and essentially monomorphic?.

Revised on March 12, 2010 07:41:43
by Tim Porter
(95.147.237.205)