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nLab > Latest Changes: attaching space

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 2nd 2017

    had occasion to give attaching space its own little entry. Cross-linked with Top and CW-complex.

    • CommentRowNumber2.
    • CommentAuthorTim_Porter
    • CommentTimeMay 3rd 2017
    • (edited May 3rd 2017)

    Just a comment / grumble / gripe: I do not really like that term, ’attaching map’, as the English construction used seems to me very awkward. I know that Hatcher uses it, but, looking at another text that I have at hand, Ronnie uses ’adjunction space’ as does Wikipedia (though that may not be a coincidence!). The term is on a par for ugliness with ‘matching space’ as used in simplicial homotopy theory.

    My argument against it is that an ’attaching map’ is the map used for attaching one space to another, i.e. the map is involved in the process of attaching. In no way is the use of the present participle ‘attaching’, similarly justified in ’attaching space’.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 3rd 2017
    • (edited May 3rd 2017)

    “adjunction space” is redirecting to “attaching space”.

    Google says that the second term is used slightly more than the first, and not just by Hatcher, but also in other textbooks.

    • CommentRowNumber4.
    • CommentAuthorTim_Porter
    • CommentTimeMay 3rd 2017

    … that may be so, but it does not mean that it is a good terminology!;-) It may be an English English v. American English problem. (There is a grammatical slip in the present entry, so I will adjust it and also mention the alternative (better, ;-) terminology.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeMay 3rd 2017

    I’ve never heard this terminology before, and I’m not sure why we should perpetuate it. Why do we need a special word for pushouts along monos in Top?

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 3rd 2017

    Certainly “attaching map” is in wide circulation, esp. when referring to construction of CW complexes.

    I’m not crazy about “attaching space” either, but the nLab can perform a useful descriptive function by pointing out that what some people call an attaching space is nothing but a special sort of pushout. Doesn’t imply that we approve of the terminology (and I wouldn’t repeat the term myself anywhere else, I don’t think).

    • CommentRowNumber7.
    • CommentAuthorTim_Porter
    • CommentTimeMay 3rd 2017

    I did a non-representative search via Google and many of the sites caught by that had ’attaching’ and ’space’ separated by one or more word, e.g. ’attaching a space along a map’. I agree with Todd that as it is merely a special type of pushout I think it is probably best to say that and also to note any special properties that having one of the ’arms’ an inclusion gives it. Note however that Hatcher uses it so some mention is probably needed.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMay 3rd 2017

    I did a non-representative search via Google and many of the sites caught by that had ’attaching’ and ’space’ separated by one or more word,

    Put quotation marks

      "attaching space"

    then Google only returns hits that have this combination. It returns at least three textbooks that use the term, some lecture notes and various further articles or notes.

    But, by all means, please rename the entry if that helps. Maybe people here prefer “pushout along monomorphisms in topological spaces”? :-)

    • CommentRowNumber9.
    • CommentAuthorTim_Porter
    • CommentTimeMay 3rd 2017
    • (edited May 3rd 2017)

    I was merely complaining about very ugly (and grammatically doubtful) established terminology. The current entry with both the used terms and the link with pushouts seems fine as it the redirects will handle any searches or links. That however does not stop me feeling the terminology is awkward and ugly. I can live with it!

    My reason for saying it is awkward is, more or less, that in ’attaching map’ the map is attaching something, but in ’attaching space’ the space is not attaching anything. The adjective ’attaching’ does not really qualify the noun.

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeMay 3rd 2017

    I don’t think “adjunction space” is any better; it may be grammatically less twisted, but it conveys much less intuition, and there’s no relationship to an adjunction. Why do people feel the need for a special word for pushouts along monos in Top? Can’t we just talk about “pushouts”?

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMay 3rd 2017
    • (edited May 3rd 2017)

    How about naming the entry “space attachment” as a generalization of the standard “cell attachment”.

    Why do people feel the need for a special word for pushouts along monos in Top?

    Unsure what you really mean with this question. In topology it is standard to consider subspaces A⊂XA \subset X and maps f:A→Yf \colon A \to Y, and to use the notation X∪ fYX \cup_f Y for the corresponding gluing construction. This appears all over the place, and typically in contexts where the reader is not assumed to be versed in category theory. So they call it by some other name.

    Can’t we just talk about “pushouts”?

    You can, I can, “we” can. But there are people who are familiar with the construction known as X∪ fYX \cup_f Y in topology, and I want to tell them: look, this is a special case of a general construction known as a pushout.

    There is unbounded room on the nnLab. There is room for being erudite, and there is room for explaining something to those who are not yet erudite. Let’s use both opportunities.

    • CommentRowNumber12.
    • CommentAuthorTim_Porter
    • CommentTimeMay 3rd 2017

    I think the idea of a title “space attachment” is very good. I also had my doubts about ’adjunction space’ here. The wording “space attachment” emphasises the action of attaching one space to another and that is what is the essence here.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeMay 3rd 2017

    Okay, I have renamed the entry to “space attachment”, and I have tweaked the first sentence accordingly.

    • CommentRowNumber14.
    • CommentAuthorTim_Porter
    • CommentTimeMay 3rd 2017

    That reads much better now.

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