Talk:Limit (category theory)
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I'm wondering if it would be better to separate out the stuff on colimits. In fact I've already started a colimit page. That way there would be more space to discuss the special cases such as products, equalisers and pushouts (and their duals on the colimit) page. Magnus 17:50 Apr 16, 2003 (UTC)
I think this article needs examples and motivations for the definitions instead of simply giving them out of the blue. Phys 18:08, 1 Sep 2003 (UTC)
Indeed! It also should include pointers to the most prominent limits/colimits and be more detailed and structured. Maybe I add something soon... -- Markus 25/11/2003
Done. But one may still think about a "motivations" section... --Markus 26/11/2003
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[edit] limits in topology versus limits in category theory
Is there a way that you can interpret the limit of a sequence in a topological space as a limit of some appropriate functor between some categories? -Lethe | Talk 23:16, 8 October 2005 (UTC)
One would have to understand a topological space as a category. I don't have any good idea for this... -David 12/02/2006
[edit] Cones as natural transformations
Did you know cone may be views as natural transformations. It could be quite cool to explain limits with that point of view ? -David 12/02/2006
- I've been meaning to incorporate that. I had stated some (minimal) notes at User:Fropuff/Draft 4. There is some more material at Universal morphism#Limits and colimits. -- Fropuff 20:18, 12 February 2006 (UTC)
[edit] Creation of limits
If have encountered in a paper the following definition of creation of limits:
- G is said to create limits for F if whenever (L, φ) is a limit of GF there exists a unique limit (L′, φ′) of F such that G(L′, φ′) = (L, φ).
I consider this to be a rather weird definition of creation of limits, but the author of the paper insists on his definition. Is this definition used in a relevant portion of the literature? --Tillmo (talk) 14:49, 17 January 2008 (UTC)
- Hi Till. Hmm. It's not Mac Lane's definition, but it's close enough to be confusing! Yes, it would be more tasteful if the author had used a word other than "creates". The most famous theorem about the creation of limits is probably Beck's monadicity theorem, and I guess that would fail for this weaker definition. Sam Staton (talk) 15:15, 17 January 2008 (UTC)
- In the terminology used in this article G would be said to lift limits uniquely. This is indeed a useful concept, but not the one that is usually meant by create limits. Mac Lane doesn't talk about lifting of limits, only creation. The book by Adámek et. al. discusses both. (Although they use a slightly different definition of creation then that used here—which agrees with Mac Lane. They insist on a unique preimage source not just a unique preimage cone.) Note that creation of limits is equivalent to the unique lifting of limits plus reflection of limits. -- Fropuff (talk) 17:17, 17 January 2008 (UTC)
[edit] Definition of a cone seems strange. Please comment.
Er .. I miss something in the definition of a cone, which is allegedly supposed to generalize, among other things, cartesian products (think of J as a two-point discrete category): some property to assure that whenever we have a family (pair) of morphisms from an object in C to the diagram J \subset C (for simplicity), these factor through the cone N? Or (how) can one derive that from the functoriality of the diagram J -> C? For example, in a cartesian set product N = X \times Y, we want pairs of arrows A -> X, A -> Y to factor through N, so we need a morphism A -> N, but from this definition I dont see why it should exist. Will learn wiki markup later. Promise. H4nne (talk) 07:51, 30 April 2008 (UTC)
