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nLab > Latest Changes: Completion

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    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 23rd 2009
    Rather than ask whether it's worth it and have Urs say "do it, don't talk about it!", I started a page to compare different notions of completion. Fortunately, we are well supplied with experts on the subject. What would be great would be a comparison of different completion processes. How widely they are applicable, e.g., to the enriched case? In which situations two or more coincide, etc.
    • CommentRowNumber2.
    • CommentAuthorJonAwbrey
    • CommentTimeOct 23rd 2009

    I think you should call it n-telechy …

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 24th 2009

    Thanks, David.

    Ignoring my own advice I'll remark here, instead of implementing it immediately, that there are also notions of "completions" of things that are not categories, of course. Maybe the entry should be renamed to "completion of a category"?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeOct 24th 2009

    Jon,

    just for your information, I didn't get that one either. Probably my German lack of humour, or possibly my German lack of English language skill. I have a guess, but even with that I don't see how this is at all related to David's message.

    But in any case, can't we agree that we try to reduce the irrelevant exchanges here a bit, at least as long as it isn't clear that we all mostly share the same enjoyment for them?

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 24th 2009
    I agree with Urs. Just because entelechy means the movement towards fulfilment, and so vaguely relates to completion, and begins with the same sound as 'n-category' doesn't make this amusing. If this sort of comment formed 1% of a parcel of useful contributions, it might be tolerable.
    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 24th 2009
    This comment is invalid XHTML+MathML+SVG; displaying source. <div> Urs wrote <br/><br/><blockquote><br/><br/>Maybe the entry should be renamed to "completion of a category"?<br/><br/></blockquote><br/><br/>Some of the entries apply to enriched categories, so it could be "completion of an enriched category". On the other hand, the one current example applying to something more special than a category is the profinite completion of a group. But one can profinitely complete a wider range of things - isn't this pro-completion? And we need ind-completion too.<br/><br/>Also, aren't there completion constructions which don't have 'completion' in their names? </div>
    • CommentRowNumber7.
    • CommentAuthorTobyBartels
    • CommentTimeOct 24th 2009

    Also, aren't there completion constructions which don't have 'completion' in their names?

    Sure; in general, I would consider any regular mono-reflector to be a completion. And then there's algebraic completion, which I would not consider really a completion (because of its inferior form of uniqueness, which stops it from being reflective).

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeOct 26th 2009

    I added some remarks to completion about completions in general and how the distinction between "completion" and "free cocompletion" relates to "property" vs "property-like structure". The organization is suboptimal but I don't know how to fix it right now; please refactor if anyone has a better idea.

    • CommentRowNumber9.
    • CommentAuthorTobyBartels
    • CommentTimeOct 26th 2009

    I added a couple of examples and links, including a counterexample to argue that a ‘completion’ should be monic.

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 26th 2009
    Looking good! This could be a great page to show how category theory systematises a concept. Should there be pro-completion?

    I wonder what some good concrete examples are to show differences.
    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeOct 26th 2009

    I added pro-completion of a category and Stone-Cech compactification. Is the unit of the reflection for the profinite completion of a group monic?

    David, are you wondering about more examples of borderline cases? Like reflectors that aren't monic or property-like structure that isn't a property?

    • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 26th 2009
    I think I meant is there anything approaching the ideal case of a category where different completions give different categories, all of which are recognisably important.

    Does the list continue to anything interesting:
    free cocompletion: free completion :: Ind-completion: Pro-completion:: ideal completion: ???, the free completion under cofiltered limits of epimorphisms.
    • CommentRowNumber13.
    • CommentAuthorTobyBartels
    • CommentTimeOct 26th 2009

    I want to call the last item ‘filter completion’, although ‘filter’ can be rather overloaded.

    • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeOct 26th 2009

    "filter completion" is of course the obvious dual of "ideal completion." Not sure whether I think it's a good name though.

    I don't think I've ever seen anyone write about the dual of ideal completion, but I seem to recall that maybe pro-objects whose transition maps are epi are sometimes better behaved than arbitrary ones? Perhaps sometimes when people work with "good pro-objects" they are really working in this "filter completion"?

    • CommentRowNumber15.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 27th 2009
    We could make this page more cohesive. It's not clear when in the 'List of completions' we have examples of enriched category completions: Cauchy completion of a metric space (yes) of a uniform space (no?), Dedekind completion of a linear order (yes for 2-enriched categories?)

    We have a long list of enriched categories. Does anything worth noting happen when we complete these?
    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeOct 27th 2009

    this is becoming a great page, guys, neat

    • CommentRowNumber17.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 28th 2009
    Asked yet another question, and added MacNeille completion.

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