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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeDec 21st 2009
    • (edited Dec 21st 2009)

    created an entry mapping cocone, following a suggestion by Zoran, that this is the right technical term for what is discussed in more detail at generalized universal bundle.

    (the examples section needs more attention, though...)

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeDec 22nd 2009

    I think algebraic topologists would call that the "mapping path space".

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeDec 22nd 2009
    • (edited Dec 22nd 2009)

    Whitehead had complained to cocone terminology in old times, because of the seeming (though false) double dualization, so he used mapping path space, what his teaching school (most of US for example) followed later. But he himself was not confident in that terminology. For example there is a table in his book where he lists the dual notions and at the place where mapping cocone/mapping path space should fit he puts just the symbol for the construction while on the dual side he puts the whole name. Similarily for the mapping cocylinder.

    Somebody, could be Eilenberg (I should check) answered to Whitehead to call "ne" rather than cocone, effectively cancelling the double dualization.

    Postnikov uses the mapping cocylinder, while for the Whitehead's complaint he comments: "we do not see a particular criminal in the cocone terminology, but will anyway not use it".

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeDec 22nd 2009

    I made mapping path space a redirect and added a section Note on terminology with Zoran's comment above, slightly edited.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeDec 22nd 2009

    What a good thing we don't talk about colimits as being representing objects for path spaces (rather than cocones). (-:

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeDec 22nd 2009
    • (edited Dec 15th 2011)

    Yes, a successful case of co-communication. ;-)

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeDec 15th 2011

    I’d like to register a disagreement with myself of 2 years ago. I don’t know what I was thinking at the time, but it seems clear to me now that the term “mapping path space” (or “mapping path fibration”) refers to the object also called the mapping cocylinder, not to the mapping cocone. I just took a survey of a handful of algebraic topology books and they all agree with myself of today. Zoran, were you saying that Whitehead actually used “mapping path space” to mean the mapping cocone?

    • CommentRowNumber8.
    • CommentAuthorzskoda
    • CommentTimeDec 16th 2011

    I should check, but my memory is also that Whitehead meant the mapping cocylinder.

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeDec 16th 2011

    Okay, I’ve updated mapping cocone and cocylinder to reflect this. I left the “terminology” section at mapping cocone as-is, since I wasn’t sure what to do with the sentence “Whitehead complained about the term cocone back in the old days, because of the seeming (though false) double dualization, so he used mapping path space” which suggests to me that Whitehead used “mapping path space” to mean “mapping cocone”.

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