nLab
stable pro-object

Given any category CC, we can form the corresponding category of pro-objects in CC, which is denoted by propro-CC. Since the category 𝟘\mathbb{0} with one morphism is a coflitered category, within propro-CC, we have all pro-objects of the form X:πŸ˜β†’CX: \mathbb{0}\to C. Clearly such a functor is completely determined by the single object, X(*)X(*), of CC to which it corresponds. This gives a functor:

c:Cβ†’proβˆ’Cc: C\to pro-C

which embeds the category CC in propro-CC. (This is really the Yoneda embedding in disguise.)

Definition

Any pro-object isomorphic in propro-CC to one of the form, c(X)c(X), for XX an object of CC, 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 CC is an abelian category, it is relatively simple to give necessary and sufficient β€˜internal’ 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)