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Has anyone formally calculated the étale homotopy type of $\text{Spec}(\mathbb{Z}) \cup \{ \text{place}_{\infty} \}$?

According to arithmetic topology, $\text{Spec}(\mathbb{Z}) \cup \{ \text{place}_{\infty} \}$ is formally analogous to $S^3$, so I predict that this is the étale homotopy type. We should have $\pi_3(\text{Spec}(\mathbb{Z})\cup \{ \text{place}_{\infty} \}) \cong \mathbb{Z}$, $\pi_2(\text{Spec}(\mathbb{Z}) \cup \{ \text{place}_{\infty} \}) \cong 0$, and $\pi_1(\text{Spec}(\mathbb{Z}) \cup \{ \text{place}_{\infty} \}) = 0$.

Ronald J. Zallman
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    And $\pi_1=0$ ? – David Roberts Sep 22 '20 at 20:53
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    Ok,. Try this: https://mathoverflow.net/a/186140/4177 – David Roberts Sep 22 '20 at 20:56
  • Shoot, I intended to compactify. – Ronald J. Zallman Sep 22 '20 at 21:01
  • shall we close this as duplicate? – David Roberts Sep 22 '20 at 23:44
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    Does this answer your question? What are the higher homotopy groups of Spec Z ? – David Roberts Sep 22 '20 at 23:48
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    @DavidRoberts I intended to compactify it first; I hope you don't mind if I change it to $\text{Spec}(\mathbb{Z}) \cup { \text{place}_{\infty} }$. – Ronald J. Zallman Sep 23 '20 at 00:50
  • no probs, that would make it a different question :-) – David Roberts Sep 23 '20 at 03:18
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    What is your definition of etale homotopy type after adding the infinite place? – naf Sep 23 '20 at 05:34
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    What topos corresponds to $\operatorname{Spec}(\mathbb{Z})\cup{\infty}$? I know of a couple of proposals, but they're both in a rather embrional stage... – Denis Nardin Sep 23 '20 at 06:14
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    This question has already 5 upvotes but still doesn't make sense. Please clarify what you mean by $\operatorname{Spec} \mathbf{Z} \cup {{\rm place}_\infty}$ or its etale homotopy type. – Piotr Achinger Sep 24 '20 at 13:15
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    I like to think in terms of the pro-smooth topos (filtered limits of smooth maps). This might not have a candidate for a weakly contractible cover, though (I haven’t found one). Certainly the ordinary smooth topos does not. I respect your concern for a grounded question. But it seems from what @DenisNardin is saying that in choosing the topos, I might only get one or a few of the proposals and a limited perspective. I would much rather hear a broad answer overviewing the current state of the literature than limit myself to a partial view by committing myself to a particular topos. – Ronald J. Zallman Sep 24 '20 at 22:53
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    @DeanYoung Can you give a pointer to where the pro-smooth topos of $\mathrm{Spec},\mathbb{Z}\cup{\infty}$ is defined? I have no idea of what you're referring to. Note that $\mathrm{Spec},\mathbb{Z}\cup{\infty}$ is not a scheme, so the general definitions do not apply. – Denis Nardin Sep 25 '20 at 05:08
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    I'm okay with a broad scope, I was more concerned with the set of possible precise formulations of your question being empty. – Piotr Achinger Sep 25 '20 at 07:29
  • how many connected components does your thing have? –  Sep 27 '20 at 12:19

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