1. In some categories, the inverse limit does not exist. 2. Any inverse limit of residually finite groups is residually finite. 3. The inverse limit will also belong to that category. 4. Thus shouldn't there be an inverse limit of this system as well? 5. The inverse limit and the natural projections satisfy a universal property described in the next section. 6. The terminology is somewhat confusing : inverse limits are limits, while direct limits are colimits. 7. One can then define the �tale fundamental group as an inverse limit of finite automorphism groups. 8. The inverse limit can be defined abstractly in an arbitrary category by means of a universal property. 9. This is by nature not an algebraic group, but an inverse limit of algebraic groups ( pro-algebraic group ). 10. It is also observed that the result can be described as an inverse limit in the category of " graded " rings.
