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