Discussion of Homotopy Type Theory and Univalent Foundations
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* [HoTT] Hurewicz theorem in HoTT
@ 2019-07-27 13:18 Ali Caglayan
  2019-07-30 15:44 ` Luis Scoccola
  0 siblings, 1 reply; 2+ messages in thread
From: Ali Caglayan @ 2019-07-27 13:18 UTC (permalink / raw)
  To: Homotopy Type Theory

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

I stumbled across this mathoverflow question: 

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.


Ali Caglayan

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2019-07-27 13:18 [HoTT] Hurewicz theorem in HoTT Ali Caglayan
2019-07-30 15:44 ` Luis Scoccola

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