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Atrium > Mathematics, Physics & Philosophy: Clarify the definition of a pointwise kan extension?

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    • CommentRowNumber1.
    • CommentAuthorHarry Gindi
    • CommentTimeDec 16th 2009
    • (edited Dec 16th 2009)
    Currently, the definition at kan extension says that the only difference is that they are computed by limits and colimits. Why would the method of computation matter? Shouldn't the universal property guarantee the uniqueness up to unique isomorphism?

    Also, the definition of a pointwise kan extension isn't even given explicitly, it's only mentioned in a few remarks.
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 16th 2009
    This comment is invalid XHTML+MathML+SVG; displaying source. <div> <blockquote> Why would the method of computation matter? </blockquote> <p>It's not meant to say that the computations yield different results, if both exist.</p> <blockquote> Also, the definition of a pointwise kan extension isn't even given explicitly, it's only mentioned in a few remarks. </blockquote> <p>Please, feel encouraged to help improve the entry. I know (and probably we all know) that it is far from being perfect or even satisfactory. If you feel you care about it, try to improve it., And be it only by adding a query box somewhere saying for instance "At this point such and such should be stated".</p> <p>Apart from that general remark, isn't the pointwise definition explicitly given in the sections</p> <p><a href="http://ncatlab.org/nlab/show/Kan+extension#byColimits">pointwise in terms of colimits</a></p> <p>and</p> <p><a href="http://ncatlab.org/nlab/show/Kan+extension#byCoends">pointwise in terms of coends</a></p> <p>?</p> </div>
    • CommentRowNumber3.
    • CommentAuthorHarry Gindi
    • CommentTimeDec 16th 2009
    Those look like formulas, not definitions. I also don't understand them, so I can't really help fix the article. All I'm saying is that there are a few remarks saying that pointwise is more important, but not even explaining the difference or giving an example of how the pointwise kan extension can fail to exist.
    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeDec 16th 2009

    Oh. These formulas are the definition. The pointwise Kan extension is defined to be the (co)limit or (co)end as given there.

    Okay, so we neeed to explain this better on the page.

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