```Discussion of Homotopy Type Theory and Univalent Foundations
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```From: Ali Caglayan <alizter@gmail.com>
Subject: [HoTT] Hurewicz theorem in HoTT
Date: Sat, 27 Jul 2019 06:18:45 -0700 (PDT)	[thread overview]

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Is there any progress on proving the Hurewicz theorem in HoTT?

I stumbled across this mathoverflow question:
https://mathoverflow.net/questions/283199/an-abstract-nonsense-proof-of-the-hurewicz-theorem

I wonder if we can adapt the following argument:

Define H_n(X; Z) as [S^{n+t}, X /\ K(X, t)] for some large t and pointed
space X. The Hurewicz map is induced by a generator g : S^t->K(Z, t) of
H_n(S^n). Given by postcomposition with (id_X /\ g).

H : [S^{n+t}, X /\ S^t] ---> [S^{n+k}, X /\ K(Z, t)]

Now since X is (n-1)-connected and it can be shown that g is n-connected
(an (n+1)-equivalence in the answer), then it follows that (id_X /\ g)_* is
an isomorphism.

The only trouble I see with this argument working is the definition of
homology. Instead of having a large enough t floating around we would have
to use a colimit and that might get tricky. Showing that g is n-connected
is possible I think using some lemmas about modalities I can't name of the

Do you think this argument will work? Let me know what you all think.

Thanks,

Ali Caglayan

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```next             reply	other threads:[~2019-07-27 13:18 UTC|newest]

Thread overview: 2+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2019-07-27 13:18 Ali Caglayan [this message]
2019-07-30 15:44 ` Luis Scoccola
```

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