Talk:Inverse limit

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Mathematics rating:
C Class
Mid Importance
 Field: Foundations, logic, and set theory

Derived functor[edit]

There should be some mention of the derived functor, {\varprojlim}^{1}. Likewise for the direct limit, if somebody happens to know how that works.

Done long ago.


I'm a little lost on the Formal Definition. Theres 7 different 'i's. Which 'i's are bounded to which expressions? Could someone perhaps use a different letter for different expressions? —Preceding unsigned comment added by (talk) 18:56, 7 October 2010 (UTC)

I've had a go at clarifying this. Is it better now? maybe some editors think it is now too fussy. ComputScientist (talk) 19:07, 7 October 2010 (UTC)

"Unlike for algebraic objects, the inverse limit might not exist in an arbitrary category."[edit]

The article says "Unlike for algebraic objects, the inverse limit might not exist in an arbitrary category." This can give the impression that inverse limits always exist in a category of algebraic objects. I don't know to which extent this is true, but here is a counterexample if the poset is not required to be directed : let I = {a, b} be a set with two distinct elements, ordered by equality (thus not directed); let Ka a field with 2 elements, let Kb a field with 3 elements. The (only) corresponding projective system has no inverse limit in the category of fields, because there would then exist field homomorphisms from the limit field to Ka and to Kb, which is impossible since field homomorphims are injective. Additionnally, if the poset I is empty, the inverse limit doesn't exist in the category of fields, since there are no final objects in this category. Marvoir (talk) 07:58, 7 October 2012 (UTC)

"Short exact sequence of inverse systems"[edit]

Morphisms in C^I are not defined in the article. Is it a full subcategory? Or are the morphisms homomorphisms that commute with the transition maps? Surely the short exact sequence must have something to do with the structure of the transition maps?! (talk) 13:54, 26 April 2013 (UTC)

Gluing or ungluing?[edit]

The article at the present states that the inverse limit amounts to a glue together operation. Perhaps some people can feel this way. I worked (did research) with inverse limits quite a bit, and my intuition is just the opposite: the inverse limit, when projections are surjections or epimorphisms (a typical case), is a result of gradual ungluing consecutive spaces--the approximating spaces are glued in themselves, but less and less so in the limit. When projections are not onto then a different intuition applies; in particular the intersection is a special case of an inverse limit. Once again, there is no gluing here, not at all. Wlod (talk) —Preceding undated comment added 08:40, 16 May 2013 (UTC)