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nLab > Latest Changes: H-space

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 25th 2012

    I have touched H-space, slightly expanded here and there and slightly reorganized it.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 25th 2012

    A question on terminology:

    in the spirit of the terms “H-monoid” and “H-group” it would be natural to say “H-category” for a category object internal to Ho(Top)Ho(Top). Are there any texts that say so, explicitly? Or else, is there any term which is semi-standard at least in some circles for categories internal to Ho(Top)Ho(Top)?

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeNov 26th 2012

    Ho(Top) doesn’t have a lot of pullbacks, so I’m not exactly sure what you would want to mean by a category internal to it. Do you just mean a category-like thing internal to Top where the associativity and unitality only hold up to incoherent homotopy? I don’t think I remember ever seeing such a thing; do you have some example in mind?

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 26th 2012

    Or perhaps a weak category object in the 2-category with underlying 1-category TopTop and with equivalence classes of homotopies for 2-arrows? This probably only makes sense for cofibrant topological spaces, though.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeNov 26th 2012
    • (edited Nov 26th 2012)

    Right, I didn’t say that well.

    What I mean is the structure in ∞Grpd\infty Grpd given by a type X 0:TypeX_0 \colon Type, a dependent type x 0,x 1:X 0⊢X 1(x 0,x 1):Typex_0,x_1 \colon X_0 \vdash X_1(x_0,x_1) \colon Type, a function x 0,x 1,x 2:X 0⊢comp x 0,x 1,x 2:X 1(x 0,x 1)×X 1(x 1,x 2)→X 1(x 0,x 2)x_0,x_1,x_2 \colon X_0 \vdash comp_{x_0,x_1,x_2} \colon X_1(x_0,x_1) \times X_1(x_1,x_2) \to X_1(x_0,x_2) such that there exists an equivalence x 0,x 1,x 2,x 3:X 0⊢(comp x 0,x 2,x 3∘comp x 0,x 1,x 2≃comp x 0,x 1,x 3∘comp x 1,x 2,x 3)x_0,x_1,x_2,x_3 \colon X_0 \vdash (comp_{x_0,x_2,x_3} \circ comp_{x_0,x_1,x_2} \simeq comp_{x_0,x_1, x_3} \circ comp_{x_1,x_2,x_3}) (and similarly for units).

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeNov 26th 2012
    • (edited Nov 26th 2012)

    Or maybe that would rather deserve to be called an “H-Segal space” or the like. The term “H-category” would probably best fit for Ho(Top)Ho(Top)-enriched categories. Any opinions?

    I am asking because I am having discusison with somebody on how to say category internal to HoTT and I would like to make the point that the coherence is important. I kept saying: “notice that for the special 1-object case there is the crucual distinction between an H-monoid and an A ∞A_\infty-space” but I would like to be able to drop the “for the special case” and just say “there is the crucial distinction between an H-category and an A ∞A_\infty-category” – in such a way that I can make “H-category” a pointer to an nnLab entry whose existence and title receives a minimum of consensus.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeNov 26th 2012

    Well, A ∞A_\infty category usually also refers to the enriched case. I’m dubious of distinguishing any of these notions with a name like “H-category”, unless someone actually has a use for one of them. (Also, it’s a little unfortunate that in HoTT we sometimes prefix things with an “h-” without meaning homotopy incoherence, e.g. h-set and h-prop and h-level.)

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeNov 26th 2012
    • (edited Nov 26th 2012)

    Well, A ∞A_\infty category usually also refers to the enriched case.

    You’d never say A ∞A_\infty for Ho(Top)Ho(Top)-enrichment!

    unless someone actually has a use for one of them.

    It appears all over the place, for instance the H-category underlying a Segal space is one way to define what an equivalence in a Segal space is.

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeNov 27th 2012

    My point was that since A ∞A_\infty refers to something enriched in spaces, if you wanted to say ’there is the crucial distinction…’ comparing them to H-categories, then H-categories would have to be enriched in, rather than ’internal’ to, Ho(Top).

    What about: an H-category is a monad in the homotopy bicategory of spans?

    • CommentRowNumber10.
    • CommentAuthorjim_stasheff
    • CommentTimeNov 27th 2012
    Seems to me we have an h-virus and an H-virus.
    Last time I checked, there were 25 other lettes, cf. why they are called
    the J-homomorphism and K-theory
    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeNov 27th 2012

    25 other lettes

    One day I’ll invent an Umlaut-theory: äu-cohomology for äußerst interessante Kohomologie.

    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeDec 4th 2012

    What about something like “A 3A_3-category”? (Or maybe A 4A_4? I’m not sure how the indexing is aligned for A nA_n.)

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeDec 4th 2012

    Ah, that’s a good pont! Yes. I’ll try to get back to that later…

    • CommentRowNumber14.
    • CommentAuthorjim_stasheff
    • CommentTimeDec 5th 2012
    What about something like “A_3-category”? (Or maybe A_4? I’m not sure how the indexing is aligned for A_n.)

    If the reference is to A_n-spaces, the n refers to the number of variables:
    A_2 is an H-space, A_3 has just an associating homotopy...
    • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTimeDec 5th 2012

    Thanks! Then “A 3A_3-category” would be what I meant.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeOct 10th 2019

    added brief pointer (here) to the Pontrjagin ring.

    diff, v35, current

  17. In general, internalization is different from enrichment, and the definitions above for i.e. H-monoid, H-group, H-ring made it fairly clear that the definition of H-category considered here are category objects internal to Ho(Top), rather than Ho(Top)-enriched categories.

    Anonymous

    diff, v46, current

  18. Removed the link to the Arkowitz paper, since it was local filesystem link rather than a web url.

    Anonymous

    diff, v48, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeNov 2nd 2022

    Thanks for the alert. But then let’s just add a working link:

    diff, v49, current

    • CommentRowNumber20.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 9th 2023

    Added the recent

    Will add to these authors’ pages.

    diff, v50, current

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeJul 4th 2023

    added pointer to:

    diff, v52, current

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeJul 4th 2023

    and pointer to:

    diff, v52, current

    • CommentRowNumber23.
    • CommentAuthorperezl.alonso
    • CommentTimeDec 21st 2023

    rational homotopy type of H-spaces

    diff, v53, current

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeDec 21st 2023

    Thanks. This used to be mentioned in passing at Sullivan model of loop space but without a reference.

    I have expanded a little (here) and cross-linked the two paragraphs.

    diff, v54, current

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