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nLab > Latest Changes: van Kampen colimit

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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 23rd 2013
    • CommentRowNumber2.
    • CommentAuthorKarol Szumiło
    • CommentTimeFeb 23rd 2013

    I am puzzled by the choice of “van Kampen” as a name for this concept. Do you know where it comes from? Is it supposed to refer to the van Kampen theorem? In its most basic form it is a theorem about homotopy pushouts of spaces which happen to be “van Kampen” in this sense. But as far as I can tell there is no direct relation between the “van Kampen theorem” and “van Kampen colimits”. So is there some other justification for this name? Or maybe there is some relation to the van Kampen theorem I overlooked?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeFeb 24th 2013

    Thanks, nice entry.

    Where you point to HTT I added also a pointer to around example 1.2.3 in 2Cats+Goodwillie, where this kind of characterization of ∞\infty-toposes reappears.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 25th 2013

    @Karol, I think the name in this context originates from the theory of adhesive categories, where it was indeed used for the pushouts there which have this property. In Lack and Sobocinski’s original paper “Adhesive categories”, they say

    The name van Kampen derives from the relationship between these squares and the van Kampen theorem in topology, in its “coverings version”, as presented for example in [2]. This relationship is described in detail in [18].

    The reference [18] is their paper “Toposes are adhesive”, which makes a connection to an abstract van Kampen theorem of Brown and Janelidze which I have not read.

    However, I find the name highly appropriate in retrospect, because it is precisely this property of colimits (and more generally higher inductive types) which underlies Dan Licata’s “encode/decode” technique for computing with homotopy groups in homotopy type theory. This technique has recently been responsible for many breakthroughs in formalizing homotopy theory in HoTT, including the calculation of π n(S n)\pi_n(S^n) (by Licata and Brunerie), the proof of the freudenthal suspension theorem (by Lumsdaine), and — I believe — also the classical van Kampen theorem!

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeFeb 25th 2013

    Dan Licata’s “encode/decode” technique

    What’s that technique?

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 25th 2013

    It’s described for π 1(S 1)\pi_1(S^1) here.

    Sorry for the terse messages — I’m fighting a couple of deadlines right now and probably shouldn’t be showing up here at all. (-:

    • CommentRowNumber7.
    • CommentAuthorKarol Szumiło
    • CommentTimeMar 4th 2013

    I’ve taken a look at the paper by Brown and Janelidze and I think I understand now (though I’m not completely clear on the details.) The definition of a van Kampen colimit morally says that the category of “fibrations” over the colimit is equivalent to the category of compatible families of “fibrations” over the original diagram whatever “fibration” should mean in the context of interest. The classical van Kampen theorem can be phrased in this language if we take “fibration” to mean “covering space”.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeMar 4th 2013

    That sounds right, I guess they are thinking of the topos-theoretic definition of π 1\pi_1 in terms of covering spaces or locally constant sheaves.

  9. I think in the proof of the theorem the two directions of implication were the wrong way round. If I’m not mistaken, the counit of the adjunction should be an isomorphism iff colimits are pullback-stable.

    Jonas Frey

    diff, v4, current

    • CommentRowNumber10.
    • CommentAuthorncfavier
    • CommentTimeFeb 3rd 2023
    • (edited Nov 11th 2023)
    • CommentRowNumber11.
    • CommentAuthorncfavier
    • CommentTimeNov 11th 2023
    • (edited Nov 11th 2023)

    I’ve added to the examples the observation that the loop space of the circle is ℤ\mathbb{Z}, by applying the descent property to the circle seen as a coequalizer. This would have helped motivate the concept of van Kampen colimits for me when I was starting to learn about it.

    diff, v9, current

    • CommentRowNumber12.
    • CommentAuthorncfavier
    • CommentTimeDec 13th 2023

    Added exactness as a (partial) example, based on a discussion with Mike Shulman in the HoTT Zulip.

    diff, v11, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeDec 13th 2023

    In the new example, I found it hard to discern what was being derived from what, so I have now reworded a little to make it clearer (here). Please check if you agree.

    In the course of this, I tried to polish-up the notation and formatting in the section “Universality and descent” (here), for instance by using different fonts for categories and functors, and by using a more suggestive notation for the result of adjoining a terminal object.

    • CommentRowNumber14.
    • CommentAuthorncfavier
    • CommentTimeDec 13th 2023

    Thanks, this is clearer. I’ve fixed G ▿:𝒟 ▿⟶𝒞 ▿G^{\triangledown} \,\colon\, \mathcal{D}^{\triangledown} \longrightarrow \mathcal{C}^{\triangledown} to G ▿:𝒟 ▿⟶𝒞G^{\triangledown} \,\colon\, \mathcal{D}^{\triangledown} \longrightarrow \mathcal{C}: the target category doesn’t need to be extended. I also wonder if we should instead use 𝒟 ▹\mathcal{D}^\triangleright, following the notation in Higher Topos Theory (notation 1.2.8.4).

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeDec 13th 2023
    • (edited Dec 13th 2023)

    I’ve fixed

    Right, thanks.

    I also wonder

    Feel free to change it. (But cocones are typically displayed in the shape ▿\triangledown not in so much in the shape ▹\triangleright :-)

    [edit: Generic cocones, at least – of course the coequalizers in the example are the exception from the rule. Anyway, feel free to change it or not.]

    • CommentRowNumber16.
    • CommentAuthorncfavier
    • CommentTimeDec 13th 2023

    I guess it’s fine then.

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