From: Ali Caglayan <alizter@gmail.com>
To: Homotopy Type Theory <HomotopyTypeTheory@googlegroups.com>
Subject: [HoTT] Hurewicz theorem in HoTT
Date: Sat, 27 Jul 2019 06:18:45 -0700 (PDT) [thread overview]
Message-ID: <f50d39d3-0017-4820-9c76-877760449e78@googlegroups.com> (raw)
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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
top of my head.
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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