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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?
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.
@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 $\pi_n(S^n)$ (by Licata and Brunerie), the proof of the freudenthal suspension theorem (by Lumsdaine), and — I believe — also the classical van Kampen theorem!
Dan Licata’s “encode/decode” technique
What’s that technique?
It’s described for $\pi_1(S^1)$ 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. (-:
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”.
That sounds right, I guess they are thinking of the topos-theoretic definition of $\pi_1$ in terms of covering spaces or locally constant sheaves.
Added exactness as a (partial) example, based on a discussion with Mike Shulman in the HoTT Zulip.
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.
Thanks, this is clearer. I’ve fixed $G^{\triangledown} \,\colon\, \mathcal{D}^{\triangledown} \longrightarrow \mathcal{C}^{\triangledown}$ to $G^{\triangledown} \,\colon\, \mathcal{D}^{\triangledown} \longrightarrow \mathcal{C}$: the target category doesn’t need to be extended. I also wonder if we should instead use $\mathcal{D}^\triangleright$, following the notation in Higher Topos Theory (notation 1.2.8.4).
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.]
I guess it’s fine then.
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