```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
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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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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