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nLab > Latest Changes: cylinder object

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 8th 2016
    • (edited Apr 8th 2016)

    I added to cylinder object a pointer to a reference that goes through the trouble of spelling out the precise proof that for XX a CW-complex, then the standard cyclinder X×IX \times I is again a cell complex (and the inclusion X⊔X→X×IX \sqcup X \to X\times I a relative cell complex).

    What would be a text that features a graphics which illustrates the simple idea of the proof, visualizing the induction step where we have the cylinder over X nX_n, then the cells of X n+1X_{n+1} glued in at top and bottom, then the further (n+1)(n+1)-cells glued into all the resulting hollow cylinders? (I’d like to grab such graphics to put it in the entry, too lazy to do it myself. )

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeMar 9th 2021

    There is some bicategorical generalization and weakening of a cylinder object, under the same name, which does not factorize identity 1-cell but arbitrary 1-cell and have some other weaker properties in more general context.

    • M.E. Descotte E.J., Dubuc M. Szyld, A localization of bicategories via homotopies, Theory and appl. of categories 35:23, abs, arxiv/1805.05248

    The left homotopies defined using these cylinders do not compose, but one can consider formal finite sequences of left homotopies to define certain localizations.

    I could write few details about it, but I am not sure if this belongs to this entry, despite the name being the same. Should it be here or having a separate entry ?

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeMar 9th 2021

    I’d probably be inclined towards a separate entry, like bicategorical cylinder object perhaps.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeMar 9th 2021
    • (edited Mar 9th 2021)

    formal finite sequences of left homotopies to define certain localizations

    I mean considering equivalence classes of sequences.

    OK, I will (gradually) write a separate entry and link from cylinder object.

    My feeling is that this notion is better thought of as relative to 1-cell. The classical case is an example of rather very special kind.

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeMar 17th 2021
    • (edited Mar 17th 2021)

    I did not write the entry yet but I reviewed the paper where the construction appears (without going into the details about the cylinder object itself). The review is in a draft form (very soon to be submitted), so corrections, improvements or suggestions are very welcome.

    MR4112764 (zoranskoda)

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeMar 18th 2021

    The review has been improved,

    MR4112764 (zoranskoda)

    and a variant of it electronically submitted (changes can still be made for few days). Quite an interesting construction.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJul 13th 2021

    added pointer to:

    diff, v16, current

    • CommentRowNumber8.
    • CommentAuthorHurkyl
    • CommentTimeJul 9th 2024

    Added the lawvere cylinder as an example.

    diff, v17, current

1 to 8 of 8

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