From mboxrd@z Thu Jan 1 00:00:00 1970 Return-Path: homotopytypetheory+bncBCQ657FO5MEBBRXG7POAKGQENXR4CKA@googlegroups.com X-Spam-Checker-Version: SpamAssassin 3.4.2 (2018-09-13) on inbox.vuxu.org X-Spam-Level: X-Spam-Status: No, score=-0.7 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-oi0-x23b.google.com (mail-oi0-x23b.google.com [IPv6:2607:f8b0:4003:c06::23b]) by inbox.vuxu.org (OpenSMTPD) with ESMTP id 899b39e4 for ; Mon, 17 Sep 2018 00:20:24 +0000 (UTC) Received: by mail-oi0-x23b.google.com with SMTP id w185-v6sf16694615oig.19 for ; Sun, 16 Sep 2018 17:20:24 -0700 (PDT) DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=googlegroups.com; s=20161025; h=sender:date:from:to:message-id:subject:mime-version :x-original-sender:precedence:mailing-list:list-id:list-post :list-help:list-archive:list-unsubscribe; bh=bngdnMB9Ljv2ME7NFYlkiMQcM0uHmtSjItvOmL/peHo=; b=HUhxTZAimeVs6BihCGTaXW8m4bRbsFTEPgMtYLoPVUwFOFMgdV0jM8kYZFzxALbW+a 9fGmQAG/iCPn0aCKQLEqxQE6Hmuq2yfbtPsOmROsRWIdaeXaoXwPu3b3eOVEQgP7e4CD X1YrhYRB4Eb1bo4cMQeR0JfsStuEnDzWBE7TpmZnKMLQChvFvV3MJcPauJLuqM9LDy+E 7LaIcHXx6ifxmKjkhalMN0BiuZl1sL6mQrvt7OSzK23nMLujKuuYij16B17+QOw9Rquy CE1D5p7AuJFf4k+NKiRQXV2uvTzhvfZ0HdKNvR8NPa3QZqO6e991N69Q8i2pM0+e25f8 PaSw== DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=gmail.com; s=20161025; h=date:from:to:message-id:subject:mime-version:x-original-sender :precedence:mailing-list:list-id:list-post:list-help:list-archive :list-unsubscribe; bh=bngdnMB9Ljv2ME7NFYlkiMQcM0uHmtSjItvOmL/peHo=; b=GIykD1UAIqLsa64qjz1VkBJBhJPG/7pyj2n0qCQhNVL3w+lg0yuKoDZ1ax1fEHHM8X 7OhuTwOyDJ6XB5SiJm3ksabg42jsRkgVUeneKkf4lfpiwiMhDby2BgLnvIRAOpoWDkoc 8GgUq4c+OVnkkp+2y2uZCEDkVxwmfmBBYc4oc9NGYjfCVAcFHgfDqYpI3cCIIJckFSyq ugYSv1xCX+eWBqb2R4ogRsKUINBOrR8eE2Zcgic1LLstWwUJF7W4x3qppPXDcd+bM1/8 bMaPhim0zbQRquCDGJ+EkNXxy9+X7eJ0tGRWIEErhkpjFqzxhNHKdeN0TR58JQM0Ug9P T4ZQ== 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: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=bngdnMB9Ljv2ME7NFYlkiMQcM0uHmtSjItvOmL/peHo=; b=ipikY+ShoeApYjPdZ+9bVmt9JAMBCWgi+p4cXaWdHEygMuDy5eKC/XStQ5ud2JKHhS UMXxnGSliTSCkBMewTe3d9B1Bdu08lVnxuKRAauRhWAeCTkgN6PsO4jNBRIvM35G+kx6 K4n4qTZtRF8pJjjsAjB3hClzr0eJCnX4j6z5YCd3eL8OabaiUHtlA1cwoBELIDHhLA3s qEwbqp27lF3XbDBv/agPayrkPoMpL985GXN6QizZDb3++LQZ1K00J/jLXw4UZMcVGO0P hU0jdA4Q5pCHOBa9L7odHd2fe720qeXeAcca9cNzI7CNZdLi+Ho2F1UswtG1gnC7EN9M Lteg== Sender: homotopytypetheory@googlegroups.com X-Gm-Message-State: APzg51CWOzIM9qWcnnzLECMnuQbmz+rPF6ft0Gs+kGtCvPTnxc5XjF2N KFTIpL6fbD8fXT7LByRNLBI= X-Google-Smtp-Source: ANB0VdZzr4uWGSoz+IXQG0KCOXby25bnK/yBLs7G9J0Fgq6lEDTTQLeL6R6z+ltd/Z1uDKSzbOxX7w== X-Received: by 2002:aca:f5cc:: with SMTP id t195-v6mr122333oih.0.1537143623060; Sun, 16 Sep 2018 17:20:23 -0700 (PDT) X-BeenThere: homotopytypetheory@googlegroups.com Received: by 2002:a9d:5504:: with SMTP id l4-v6ls3427523oth.18.gmail; Sun, 16 Sep 2018 17:20:22 -0700 (PDT) X-Received: by 2002:a9d:6309:: with SMTP id q9-v6mr159961otk.5.1537143622504; Sun, 16 Sep 2018 17:20:22 -0700 (PDT) Date: Sun, 16 Sep 2018 17:20:21 -0700 (PDT) From: Ali Caglayan To: Homotopy Type Theory Message-Id: <5c3f2b44-1213-4cfc-bad6-93481116a175@googlegroups.com> Subject: [HoTT] Orthogonal groups and grassmannians in HoTT MIME-Version: 1.0 Content-Type: multipart/mixed; boundary="----=_Part_1408_1630523090.1537143621922" 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_1408_1630523090.1537143621922 Content-Type: multipart/alternative; boundary="----=_Part_1409_2021303900.1537143621922" ------=_Part_1409_2021303900.1537143621922 Content-Type: text/plain; charset="UTF-8" There is a wonderful paper by Rijke detailing an interesting construction called the "join construction". I would argue that interesting is a vast understatement because it details how to construct images of maps in HoTT. This encapsulates many interesting constructions in HoTT: set quotients, propositional truncation and Rezk completion. Now suppose we have a "oo-group" which can be taken as a pointed conntected type BG with map pt : 1 --> BG. The iterated join construction on pt gives the classical Milnor-construction. In the case where BG = K(Z/2,1) the real projective spaces are obtained (synthetic homotopically mind you) which is expanded on elsewhere with Buchholtz . When BG = K(Z,2) we obtain a definition for the complex projectve spaces. It is hinted at that this construction may be modified to obtiain grassmannians, I would like to know some more details about this. But it would seem to me that BO(n) would need to be known which is kindof circular if one defines it as an infinite grassmannian. Mike has said on Mathoverflow that SU(n) would probably be constructed as BSU(n) in HoTT and obiously loop spaced to get an automatic group structure. If we are to define BSU(n) as a complex grassmannian then I don't see how we can do it the Rijke-Buchholtz way. Because then it seems we would have to know BSU(n) anyway. I would like to hear the communities thoughts on this problem and whether there is any proposed solution. - Ali Caglayan -- 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. For more options, visit https://groups.google.com/d/optout. ------=_Part_1409_2021303900.1537143621922 Content-Type: text/html; charset="UTF-8" Content-Transfer-Encoding: quoted-printable
There is a wonderful paper by Rijke=C2=A0detailing an interesting construction c= alled the "join construction". I would argue that interesting is = a vast understatement because it details how to construct images of maps in= HoTT. This encapsulates many interesting constructions in HoTT: set quotie= nts, propositional truncation and Rezk completion.

Now s= uppose we have a "oo-group" which can be taken as a pointed connt= ected type BG with map pt : 1 --> BG. The iterated join construction on = pt gives the classical Milnor-construction. In the case where BG =3D K(Z/2,= 1) the real projective spaces are obtained (synthetic homotopically mind yo= u) which is expanded on elsewhere=C2=A0with Buchholtz. When BG =3D K(Z,2) we obtain a definition = for the complex projectve spaces.

It is hinted at = that this construction may be modified to obtiain grassmannians, I would li= ke to know some more details about this. But it would seem to me that BO(n)= would need to be known which is kindof circular if one defines it as an in= finite grassmannian.

Mike has said on Mathoverflow=C2=A0that SU(n) wo= uld probably be constructed as BSU(n) in HoTT and obiously loop spaced to g= et an automatic group structure. If we are to define BSU(n) as a complex gr= assmannian then I don't see how we can do it the Rijke-Buchholtz way. B= ecause then it seems we would have to know BSU(n) anyway.
I would like to hear the communities thoughts on this problem a= nd whether there is any proposed solution.

- Ali C= aglayan

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