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nLab > Latest Changes: finite ∞-group

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    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 15th 2017

    I started finite ∞-group, and added that same reference to Sylow p-subgroup.

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 15th 2017

    We have the related homotopy type with finite homotopy groups which is about

    The concept of a homotopy type (homotopy n-type) all of whose homotopy groups are finite groups

    Should this have the extra condition that only finitely many homotopy groups are non-trivial?

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 15th 2017

    I would think so, I’m sure I’ve seen theorems about such homotopy types, but can’t recall what or where.

    • CommentRowNumber4.
    • CommentAuthorTim_Porter
    • CommentTimeAug 16th 2017

    I was not sure what DavidC was asking. There are two ideas here, one with only finitely many non-trivial homotopy groups and the other without that restriction. These correspond to the bit in brackets (homotopy n-type)' and thehomotopy type’ to the other. Looking at Lurie does not help on this point as the example referred to only looks at the ‘n-type’ bit. Both concepts are studied. Graham Ellis’s paper is a nice one for the n-type result.

    I used the n-type idea in papers on the Yetter model (TQFTs), so perhaps that was one place that DavidR had seen these. They also relate to homotopy cardinality.

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 16th 2017

    Hmm. If this page is really covering two ideas, it had better not begin

    The concept of a homotopy type (homotopy n-type) all of whose homotopy groups are finite groups does not have an established name. Sometimes it is called π\pi-finiteness.

    It would surely be better to say

    The property of a homotopy type (homotopy n-type) that all of its homotopy groups are finite groups does not have an established name. Sometimes it is called π\pi-finiteness.

    Isn’t the second of these much more important?

    By the way, is there a quick way to speak of a homotopy type with trivial homotopy groups after some point, so in the union over nn of nn-types? The two references at that page have to go to the lengths of writing ’Spaces with finitely many non-trivial homotopy groups’.

    • CommentRowNumber6.
    • CommentAuthorTim_Porter
    • CommentTimeAug 16th 2017
    • (edited Aug 16th 2017)

    That is a good question! Perhaps ind-finite homotopy type? The condition without finiteness could also sensibly be called ’coskeletal’ (see Beke ’Higher Cech theory’) so perhaps ’finitely coskeletal’ or something like that for this one might work. Alternatively ’strongly homotopy finite’ would be simpler.

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 16th 2017
    • (edited Aug 16th 2017)

    We have at K(n)-local stable homotopy theory:

    Say that an ∞-groupoid is strictly tame or of finite type (Hopkins-Lurie 14, def. 4.4.1) or maybe better is a truncated homotopy type with finite homotopy groups if it has only finitely many nontrivial homotopy groups each of which is furthermore a finite group.

    Rather a mouthful the ’better’ version.

    And as for ’truncated homotopy type’, the use of the past participle sounds to me that it’s being to compared to what it was in some pristine untruncated form, as though we called abelian groups, abelianised groups.

1 to 7 of 7

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