From mboxrd@z Thu Jan 1 00:00:00 1970 Return-Path: X-Spam-Checker-Version: SpamAssassin 3.4.2 (2018-09-13) on inbox.vuxu.org X-Spam-Level: X-Spam-Status: No, score=-0.9 required=5.0 tests=DKIM_SIGNED,DKIM_VALID, DKIM_VALID_AU,DKIM_VALID_EF,FREEMAIL_FORGED_FROMDOMAIN,FREEMAIL_FROM, HEADER_FROM_DIFFERENT_DOMAINS,HTML_MESSAGE,MAILING_LIST_MULTI, RCVD_IN_DNSWL_NONE autolearn=ham autolearn_force=no version=3.4.2 Received: from mail-ot1-x337.google.com (mail-ot1-x337.google.com [IPv6:2607:f8b0:4864:20::337]) by inbox.vuxu.org (OpenSMTPD) with ESMTP id 29abbe51 for ; Sun, 4 Aug 2019 10:43:44 +0000 (UTC) Received: by mail-ot1-x337.google.com with SMTP id l7sf44335436otj.16 for ; Sun, 04 Aug 2019 03:43:44 -0700 (PDT) DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=googlegroups.com; s=20161025; h=sender:date:from:to:message-id:in-reply-to:references:subject :mime-version:x-original-sender:precedence:mailing-list:list-id :list-post:list-help:list-archive:list-unsubscribe; bh=Jgc+t7A47OncnLExTKNFvIwOiQ3BJpd/NCzXtBJyLLI=; b=ApCObLZI89npkfy5PXAwyrb4gY/lBY3EjfODn5/S0Rl72S/HiuEcj5x5hd5wakRLB6 S1qCqrpAQuh/AuBIIAUB8RYQwLwP8mFEC4GkSIv4MA4Vx9QQasFVjqqTE10b2gaszF8z uaUFntdWY0RO9YksnfhbgCV6XeqVfbJ3kaBO81VDHdsSNYUJ6z1L57yHmZgXPReO2z0l NwoooG3rl9sGLHT59FSGKRz90uL7Pf8RYd4fZ1w3MDf4X/LGIDHEZTTyOnlG0lNV0h5V I5wDNPK7H55bD2Q+56LjIS5MVWkwijSQH0iEKkB5fGqTph6Q7usuBkR2kHbMFdvOCAUH u1eA== DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=gmail.com; s=20161025; h=date:from:to:message-id:in-reply-to:references:subject:mime-version :x-original-sender:precedence:mailing-list:list-id:list-post :list-help:list-archive:list-unsubscribe; bh=Jgc+t7A47OncnLExTKNFvIwOiQ3BJpd/NCzXtBJyLLI=; b=s0+HbavqCJZvsb4gNt9bVmmbhU/QHI4MwT83eFFKSd2/+A+40sQd0qmrGxVnBN1wqF WU1g6mBUuQWQ3Cu8FYoWH/7UO4I97+6a4STMt3ckg/Yd1zrNbO2dEaPQKbXMsnfaGGJQ LfnVpzt5a+J8NMPuKDCTvcWKV6NobrACmuEoiLXNp/L9GO+l8DX5WgWplSdCyIchNEPE qPieFfjVT9JUmJ/fEtAB1MbFNvRiWQ9JbfM7aBgjZcqK0PGRuCro6LS3NFaJ62UyE7G9 OroKfXRFh/OwBCdchfOJ6kzgNX4zs6SEpI37ynV5APSyq68627L+OZJxIwqZ5WENOnus Dw8w== X-Google-DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=1e100.net; s=20161025; h=sender:x-gm-message-state:date:from:to:message-id:in-reply-to :references:subject:mime-version:x-original-sender:precedence :mailing-list:list-id:x-spam-checked-in-group:list-post:list-help :list-archive:list-unsubscribe; bh=Jgc+t7A47OncnLExTKNFvIwOiQ3BJpd/NCzXtBJyLLI=; b=POyWq7+dir9WXdyC28zcWxqbQjXig3fMHUXsuQWGek4Yp/bnzZ20G3sANikcd9aADP 3fES2AKIREqRSdpHVlkWL2UsxqDp6VxbFXQQl6xzYJQKXR5YZQIJqMwrcDvXODpQjWuQ r1IuL1oBGhi0wygfhqkBv4P0icY8cUNHUAIph7/etIgazN+szl7Dnxb3Y9P9WJ2mh8+A V/WX2TXWfLcUCJAwLimV9HyqO8smd+FTrm90QugJPjEBPPw2mNlxJjQADHbascHwtIE7 jQGDaaHt/2fEUfbZ+jm2Dxke/BPgTuGeXZrHxC/pCGpRzyKAFlDJCmyloy9CG5IWiHE6 pI+A== Sender: homotopytypetheory@googlegroups.com X-Gm-Message-State: APjAAAVRN3/X/uobirZDDJC91qjbaDOfDdSA4AVSuDJyOgYzlyhK9Ntx RxRJnCVRyF+CBvb+JdrXBAE= X-Google-Smtp-Source: APXvYqyMnFmqVsffMwVw5/z73+HvEJl4BVjQl+jM7QPZBfIx53g4I+nxX7wOV3XH1yysbRdmmILgIw== X-Received: by 2002:aca:1713:: with SMTP id j19mr7961820oii.63.1564915423437; Sun, 04 Aug 2019 03:43:43 -0700 (PDT) X-BeenThere: homotopytypetheory@googlegroups.com Received: by 2002:aca:3a44:: with SMTP id h65ls12479181oia.1.gmail; Sun, 04 Aug 2019 03:43:43 -0700 (PDT) X-Received: by 2002:aca:4306:: with SMTP id q6mr8055673oia.39.1564915422946; Sun, 04 Aug 2019 03:43:42 -0700 (PDT) Date: Sun, 4 Aug 2019 03:43:41 -0700 (PDT) From: Ali Caglayan To: Homotopy Type Theory Message-Id: In-Reply-To: <81b245ba-e70f-4a13-8d0c-4eaad69f3da8@googlegroups.com> References: <81b245ba-e70f-4a13-8d0c-4eaad69f3da8@googlegroups.com> Subject: [HoTT] Re: Weaker Freudenthal MIME-Version: 1.0 Content-Type: multipart/mixed; boundary="----=_Part_1383_839624104.1564915422018" X-Original-Sender: alizter@gmail.com Precedence: list Mailing-list: list HomotopyTypeTheory@googlegroups.com; contact HomotopyTypeTheory+owners@googlegroups.com List-ID: X-Google-Group-Id: 1041266174716 List-Post: , List-Help: , List-Archive: , ------=_Part_1383_839624104.1564915422018 Content-Type: multipart/alternative; boundary="----=_Part_1384_1121748383.1564915422018" ------=_Part_1384_1121748383.1564915422018 Content-Type: text/plain; charset="UTF-8" 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? > -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/a4549eab-f563-42ad-a95a-10166e8e0664%40googlegroups.com. ------=_Part_1384_1121748383.1564915422018 Content-Type: text/html; charset="UTF-8" Content-Transfer-Encoding: quoted-printable
Here is an observation I had made:

We h= ave that A is k-connected, we can show that susp A is (k+1)-connected and t= hen 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 bott= om. The LHS composition is homotopic to the RHS composition hence naming et= a : A -> loop susp A, we have eta o unitmap being k-connected hence eta = must also be k-connected. This isn't quite there.

<= div>Now I was hoping to use the fact that loop spaces are pullbacks hence t= here are maps coming out of 1s hence (k+1)-connectedness appears, but I cou= ldn't get it to work.

On Sunday, 4 August 2019 12:59:09 UTC+3, A= li 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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