Discussion of Homotopy Type Theory and Univalent Foundations
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* [HoTT] Weaker Freudenthal
@ 2019-08-04  9:59 Ali Caglayan
  2019-08-04 10:43 ` [HoTT] " Ali Caglayan
  0 siblings, 1 reply; 2+ messages in thread
From: Ali Caglayan @ 2019-08-04  9:59 UTC (permalink / raw)
  To: Homotopy Type Theory

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Can we prove that the map A -> loop susp A is (k+1)-connected for a 
k-connected A, without having to invoke Freudenthal?

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* [HoTT] Re: Weaker Freudenthal
  2019-08-04  9:59 [HoTT] Weaker Freudenthal Ali Caglayan
@ 2019-08-04 10:43 ` " Ali Caglayan
  0 siblings, 0 replies; 2+ messages in thread
From: Ali Caglayan @ 2019-08-04 10:43 UTC (permalink / raw)
  To: Homotopy Type Theory

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Here is an observation I had made:

We have that A is k-connected, we can show that susp A is (k+1)-connected 
and then show that loop susp A is k-connected. This gives us a k-connected 
map A -> 1 and a k-connected map loop susp A -> 1. This gives us a diagram 
which commutes with A and loop susp A in the top corners and 1 in the 
bottom. The LHS composition is homotopic to the RHS composition hence 
naming eta : A -> loop susp A, we have eta o unitmap being k-connected 
hence eta must also be k-connected. This isn't quite there.

Now I was hoping to use the fact that loop spaces are pullbacks hence there 
are maps coming out of 1s hence (k+1)-connectedness appears, but I couldn't 
get it to work.

On Sunday, 4 August 2019 12:59:09 UTC+3, Ali Caglayan wrote:
>
> Can we prove that the map A -> loop susp A is (k+1)-connected for a 
> k-connected A, without having to invoke Freudenthal?
>

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Discussion of Homotopy Type Theory and Univalent Foundations

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