From mboxrd@z Thu Jan 1 00:00:00 1970 X-Spam-Checker-Version: SpamAssassin 3.4.4 (2020-01-24) on inbox.vuxu.org X-Spam-Level: X-Spam-Status: No, score=0.0 required=5.0 tests=DKIMWL_WL_HIGH,DKIM_SIGNED, DKIM_VALID,HTML_MESSAGE,HTTPS_HTTP_MISMATCH,RCVD_IN_DNSWL_NONE, RCVD_IN_MSPIKE_H2,T_SCC_BODY_TEXT_LINE autolearn=ham autolearn_force=no version=3.4.4 Received: (qmail 21356 invoked from network); 26 Dec 2023 14:15:27 -0000 Received: from mail-me3aus01hn2215.outbound.protection.outlook.com (HELO AUS01-ME3-obe.outbound.protection.outlook.com) (52.100.209.215) by inbox.vuxu.org with ESMTPUTF8; 26 Dec 2023 14:15:27 -0000 ARC-Seal: i=5; a=rsa-sha256; s=arcselector9901; d=microsoft.com; cv=pass; b=Ro1rEa1yE40NRippp5cB/Q9r1pydBb5QAtsGSQapcxP2raDk77TngQOgzksG14NLdSLSvGi2rmHW5uy/eFIA7Xo5DTEh53RKktObK+oZ4vHMe2wIsyVnx+lh0TIvESuAikeV6f20II70nKLvJx3hjTAEN1KNgJbsj7vKGyI40eQ7yL6xq9wAfpLM9DDiFRKsIf+08Xqv19xwnwHFbXRucuYmKXpr8P6lSbfDV++pZMH1Fd3VAX15oNAnlewUkuOqr9pGHSDIE7QWrEGuFMpGvGLpBvGN1qjoqRK8Zau2T3y3NKlSSPER7O94P4TX5OgaQ3obkkGGdWUvJoPhgFthtA== ARC-Message-Signature: i=5; a=rsa-sha256; c=relaxed/relaxed; d=microsoft.com; s=arcselector9901; h=From:Date:Subject:Message-ID:Content-Type:MIME-Version:X-MS-Exchange-AntiSpam-MessageData-ChunkCount:X-MS-Exchange-AntiSpam-MessageData-0:X-MS-Exchange-AntiSpam-MessageData-1; bh=+6OBOzltz6zxp7AFTVrd/Y1E3D7W13EdjyCbSGmC3+s=; b=TA0nI87Jo2oh8tNg+r3uLff8l/pq7LqDsCFyrxqSl7+oZh0dxFBBLU1okU5icB6H+Q4dd4JsGlg/Ha5+QkxFuXyNfi/6jgehHshO+7te1eKbmwEyECIku10dl++Ao7JjdpurPtjo+QVaxogStfKfYeS1G5Hf6ul7ko7YaxaViTp6Xfq9C2AT6YD2OWJkyyFWpIOL4QRDbu44YYWcfiZlSXOk6pJF3CaLpUUTWNaVB7mq4kJQZEw6F7P5aFFoZOEry8W6Wv2/EblGUc18I1zae9IScVP6BRNJRxi5SktNsJSmswhSN8WUzmYJeI7MvMr/HmKw3owpsDOroe+5etbgbg== ARC-Authentication-Results: i=5; mx.microsoft.com 1; spf=pass (sender ip is 40.107.116.83) smtp.rcpttodomain=mq.edu.au smtp.mailfrom=uqam.ca; dmarc=pass (p=none sp=none pct=100) action=none header.from=uqam.ca; dkim=fail (body hash did not verify) header.d=uqam.onmicrosoft.com; arc=pass (0 oda=0 ltdi=0 93) DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=groups.office.net; s=selector1; h=From:Date:Subject:Message-ID:Content-Type:MIME-Version:X-MS-Exchange-SenderADCheck; bh=+6OBOzltz6zxp7AFTVrd/Y1E3D7W13EdjyCbSGmC3+s=; b=Fgs0oCNh8Z20RzkgSU19Aixrs+UL3r5trz46gqt43cx35rlO1J8JQ5mtJe6ip9ktTQQTcIEXLtVgfsN1h4ASwoS2MZyDPDIXZ+RWKEy/lbFVd+3vIoFlp8BgAQ3D5iHPNXe/0Wj57L+EuWwDkHAoYbvgLCRlgzYxL+AOhNiK0Ho8sTAVzOgrbGRHXv1UUgkMVqwVsxbxg0mB0+ct1WErjxCjeDB6aap9Hv0MD+8mh5OWf3KiuW24LIxyYxk2uekbY5CmzoChk1cgnjmFpoami+XoJVulTMOkD+GdIMZra6F0nO6Z8jF4vpU6pxr8NF+tQJzogJph8W7HpOs7WcxTyw== Received: from SmtpServer.Submit by SY4PR01MB6380 with Microsoft SMTP Server id 15.20.7113.27; Tue, 26 Dec 2023 14:15:23 +0000 ARC-Seal: i=4; a=rsa-sha256; s=arcselector9901; d=microsoft.com; cv=pass; b=WOkjyGFULxISU+ONIWb8SPSmziCFQGtzv5p9P4vj6wKoJWEYU+rFDTVWcfJAqtkw4miMMOfs7G/xbNaekhAW3epsLRl0k1JxEHhRFTwtyjMc6zg2KMPKOTjtmUI0AwEOn255LSqmxIDULL1Fxn1H0AKUTu0f6rxeLL9J6dVn94TX5sGEMvzOOTxqFb4vKrmNwtFi409G2GIm+n0pSrP93g6OWWBaMxNzYl8jpI86rZh4UOuqudSP1BxZ5O9E/DtIzi9IWy/EXduoLFKDV6SgU16/6sXKOFwK7/CfAjoMHqhMwM4F/lCZZzZH/rX4Hc8zzKwd8mGfb3wCD7+sn7zeAw== ARC-Message-Signature: i=4; a=rsa-sha256; c=relaxed/relaxed; d=microsoft.com; s=arcselector9901; h=From:Date:Subject:Message-ID:Content-Type:MIME-Version:X-MS-Exchange-AntiSpam-MessageData-ChunkCount:X-MS-Exchange-AntiSpam-MessageData-0:X-MS-Exchange-AntiSpam-MessageData-1; bh=ANsE/Tis42Pu9BCVHFOCmJf/ab+XFz1SOEkmn5RhLow=; b=AWOSnAoRlczYF3C7QGJXivl+i83sxtLJqY7LinGO4MFkiuBcUOO2IcHnNJ2EWmGYqDe6hbRWTxuULkeqRjxQXqsdWsDkIHuMfNBUGiV3fpi78zhilC5demTUjMe/IKCY1Qa9PgRq4r+oJN5baBk4Zlo2mVkunXN/GbMW9LI1AIuU8TCpvyKXMdu81YJrN1WXUlvIJ0ZTJhr2FujKCMeDfHwChjgTz4n+mwDxZ4eBupcLqi082Y89e2aAVVrBx2hF2Oy/bvJCLEr9NdGFSGC6Mnl23e6JCym1d6F3ZHDGaf1GcLHYNRng5W6Xjj7z4Mpa4FtWKC6fVFNwsjApLCaEjA== ARC-Authentication-Results: i=4; mx.microsoft.com 1; spf=pass (sender ip is 40.107.116.83) smtp.rcpttodomain=mq.edu.au smtp.mailfrom=uqam.ca; dmarc=pass (p=none sp=none pct=100) action=none header.from=uqam.ca; dkim=fail (body hash did not verify) header.d=uqam.onmicrosoft.com; arc=pass (0 oda=0 ltdi=0 93) Received: from ME3PR01MB5447.ausprd01.prod.outlook.com (2603:10c6:220:ca::12) by SY4PR01MB6380.ausprd01.prod.outlook.com (2603:10c6:10:10e::14) with Microsoft SMTP Server (version=TLS1_2, cipher=TLS_ECDHE_RSA_WITH_AES_256_GCM_SHA384) id 15.20.7113.27; Tue, 26 Dec 2023 14:15:18 +0000 Received: from SY2PR01MB2986.ausprd01.prod.outlook.com (2603:10c6:1:22::21) by ME3PR01MB5447.ausprd01.prod.outlook.com (2603:10c6:220:ca::12) with Microsoft SMTP Server (version=TLS1_2, cipher=TLS_ECDHE_RSA_WITH_AES_256_GCM_SHA384) id 15.20.7113.27; Tue, 26 Dec 2023 14:15:14 +0000 Received: from SY2PR01MB2986.ausprd01.prod.outlook.com ([fe80::d301:121b:e5e8:e979]) by SY2PR01MB2986.ausprd01.prod.outlook.com ([fe80::d301:121b:e5e8:e979%4]) with mapi id 15.20.7113.027; Tue, 26 Dec 2023 14:15:13 +0000 ARC-Seal: i=3; a=rsa-sha256; s=arcselector9901; d=microsoft.com; cv=pass; b=kGBr0N4wiYPjSkkvif9ZoU7Qny92N4l4RPbdJrQ5zuC6FQRNu/5LRRqEnYUFOvZYIX56qElTXdztSrONr6qxF6wuDQVYlXGkGlFHspuc/wdaGNpcxzD9WOF9DhKAv7MVtP5jcrCCDyukk/WuMi7QKKq1t3SaQBSs56DS8aDp7ipxHVQ3oXS7O0hZ95YXwupo7DDaA17T7u52S+tQD1DgWSWWZCOmKL9q8D9qch16I9uC8oVbz4uDIBexxVrKIz/hTeCT5qjj8DI07H6jJTLbI0b/KCb5B0e+6xsE8kfE7GNkI3kXess2rZwgoK3aPXwZ0gmBanKP3LFj1phcipcz/A== ARC-Message-Signature: i=3; a=rsa-sha256; c=relaxed/relaxed; d=microsoft.com; s=arcselector9901; h=From:Date:Subject:Message-ID:Content-Type:MIME-Version:X-MS-Exchange-AntiSpam-MessageData-ChunkCount:X-MS-Exchange-AntiSpam-MessageData-0:X-MS-Exchange-AntiSpam-MessageData-1; bh=ANsE/Tis42Pu9BCVHFOCmJf/ab+XFz1SOEkmn5RhLow=; b=aLxq/rfPikOP/tLIGHAi+Ylx+pbq+BS89FPA8sn2/exG0XpM37+Xph59q9OvSZPkGud2G59feOkRMlqscorIvRs4KuH2h57VFynjCrgvICC+kgZMY/SrQpU/SB8xlUIwN9Jvz0OKnMc//aFmtaJGpQoQ65AGAiI/xlHlsuKrMfF1xAX0jPf4hNxUGPhxxcr6hn/c+jOycg2N0oYvq5QPvZhnHWBZE9OhyHvCGKDH+myL03igAJt7PyWVrGfLCpfNsj+yOA1pm9+q3YJL0oQVqBnN3yttkcOH8s8GOs7o2PfthP4J+TkY6rvrB2giy4AOrxW19+xomoEDEU7dbDlm9Q== ARC-Authentication-Results: i=3; mx.microsoft.com 1; spf=pass (sender ip is 40.107.116.83) smtp.rcpttodomain=mq.edu.au smtp.mailfrom=uqam.ca; dmarc=pass (p=none sp=none pct=100) action=none header.from=uqam.ca; dkim=fail (body hash did not verify) header.d=uqam.onmicrosoft.com; arc=pass (0 oda=0 ltdi=0 93) Received: from SY5P282CA0015.AUSP282.PROD.OUTLOOK.COM (2603:10c6:10:208::14) by ME3PR01MB5559.ausprd01.prod.outlook.com (2603:10c6:220:ce::6) with Microsoft SMTP Server (version=TLS1_2, cipher=TLS_ECDHE_RSA_WITH_AES_256_GCM_SHA384) id 15.20.7113.27; Mon, 25 Dec 2023 23:56:38 +0000 Received: from SY4AUS01FT013.eop-AUS01.prod.protection.outlook.com (2603:10c6:10:208:cafe::56) by SY5P282CA0015.outlook.office365.com (2603:10c6:10:208::14) with Microsoft SMTP Server (version=TLS1_2, cipher=TLS_ECDHE_RSA_WITH_AES_256_GCM_SHA384) id 15.20.7113.27 via Frontend Transport; Mon, 25 Dec 2023 23:56:38 +0000 Authentication-Results: spf=pass (sender IP is 40.107.116.83) smtp.mailfrom=uqam.ca; dkim=fail (body hash did not verify) header.d=UQAM.onmicrosoft.com;dmarc=pass action=none header.from=uqam.ca; Received-SPF: Pass (protection.outlook.com: domain of uqam.ca designates 40.107.116.83 as permitted sender) receiver=protection.outlook.com; client-ip=40.107.116.83; helo=CAN01-YQB-obe.outbound.protection.outlook.com; pr=C Received: from au-smtp-inbound-delivery-1.mimecast.com (103.96.20.101) by SY4AUS01FT013.mail.protection.outlook.com (10.114.156.177) with Microsoft SMTP Server (version=TLS1_2, cipher=TLS_ECDHE_RSA_WITH_AES_256_GCM_SHA384) id 15.20.7135.17 via Frontend Transport; Mon, 25 Dec 2023 23:56:37 +0000 ARC-Message-Signature: i=2; a=rsa-sha256; c=relaxed/relaxed; d=dkim.mimecast.com; s=201903; t=1703548597; h=from:from:reply-to:subject:subject:date:date:message-id:message-id: to:to:cc:mime-version:mime-version:content-type:content-type: in-reply-to:in-reply-to:references:references:dkim-signature; bh=ANsE/Tis42Pu9BCVHFOCmJf/ab+XFz1SOEkmn5RhLow=; b=LDHnQAlqW60sHPzGfrKoNQV+0ShXLabMBdZl9V4AsQyJVL+dIZ0tVichFDUmGASNQrWaFw Bh0kbnpKt8CVfjH2LHRPqjC3OaF6f51APPSH+D6dYKytO24HCaI8fTBecjfXN99pINQrAW q+s8RF7DJn7K5MAndP/48evrm3VPSP3i32mmcmqvbsJYhME7bus3vY3JNp3d9wrI/15rMl FSDJi4uqOAC3Uix8ehYDHKaUwdDJI3Zqn1JJlCFY8+HwRLPZ6yPYaXg09p6mupl0C/ETVw /U8cIkz9YcDct3ejv7oHzf8uahslJtzzkVd/gFvPas46nt1k1/WBk4KQ+4Zfxg== ARC-Seal: i=2; s=201903; d=dkim.mimecast.com; t=1703548597; a=rsa-sha256; cv=pass; b=qFGpBGkIXa3x8yTNT01JMgDL2mw3T9SAeI93PtclurwD52uVQlfBw6jYpeWXMw3/3DUkAf aHUYC1aZcHUWQ+diNevrCBZSda/52UloK6eQVfKIwl7QSh2pNaMaiSPrSOnEPREY7gyUII G69HqUSs1RCLAsexsO6c/zPFxq1ww+u4SLcnwh7PTwZhpxB7j0zvgjh7OdAgHJAtgZ0pvc WTBQX8BKd7SFYjJSGCyXweKWF8PjQYxSjHUXwtFOzskoN4UHMUX1xms2mBGF9OxOEM5fmw LC3P2k54+C9JQtX9QBIAxp2ndGcrC3i8wQOCL4oTdOL5/VxOA8pgGjA9uUEknQ== ARC-Authentication-Results: i=2; relay.mimecast.com; dkim=pass header.d=UQAM.onmicrosoft.com header.s=selector2-UQAM-onmicrosoft-com header.b=KMCLW0IG; arc=pass ("microsoft.com:s=arcselector9901:i=1"); dmarc=pass (policy=none) header.from=uqam.ca; spf=pass (relay.mimecast.com: domain of joyal.andre@uqam.ca designates 40.107.116.83 as permitted sender) smtp.mailfrom=joyal.andre@uqam.ca Received: from CAN01-YQB-obe.outbound.protection.outlook.com (mail-yqbcan01on2083.outbound.protection.outlook.com [40.107.116.83]) by relay.mimecast.com with ESMTP with STARTTLS (version=TLSv1.2, cipher=TLS_ECDHE_RSA_WITH_AES_256_GCM_SHA384) id au-mta-17-OxM_4YjkMXSZ9Bcsewv95g-1; Tue, 26 Dec 2023 10:56:33 +1100 X-MC-Unique: OxM_4YjkMXSZ9Bcsewv95g-1 ARC-Seal: i=1; a=rsa-sha256; s=arcselector9901; d=microsoft.com; cv=none; b=m06W5JBYFrCJGFG2m4DcbH1dXNjWIkPQVHOPCdx4D2xtKAPIeQv4mfeAPBkK80zktL26yPVdRUNj42CNpVRaBKUemlJq+NOB4TdXbIKpb0gbWFvn5Os9g6abXo0aXGufeTsog6Bk4yEkfo95zIqGIWPjibTS5ByieOnTLMkVlQX3VtuxbCmWtVrHWM/YheGe26pPaOi+WSSE/mjmClGarhqiqAS5Puob+XTbIzXx/n5R/l37YhzrSs1oJtVF5nP/LsQmpogU0S78dSFCIOKgq/sct9MJXPgBJ3SmltfOubx+1TCHPObEZPvTxYqp6DEowNhaQueUzeYTtoN+cE5Apg== ARC-Message-Signature: i=1; a=rsa-sha256; c=relaxed/relaxed; d=microsoft.com; s=arcselector9901; h=From:Date:Subject:Message-ID:Content-Type:MIME-Version:X-MS-Exchange-AntiSpam-MessageData-ChunkCount:X-MS-Exchange-AntiSpam-MessageData-0:X-MS-Exchange-AntiSpam-MessageData-1; bh=BwMBDYCj8H83Pt1OcHMpfzNAYsIpyX1k5fAmqiClua0=; b=lTUt6kkLSMGu9Sl0fewzxhDTPtHiNLLtvbA6hF7FvDeElXCqRvcwl3EGYml0EVNYXal8KnGMIlCE66PXImnVqsjY4FQI7TH6qxJLnM5Rfr+ahCEADcKkZ5JH830xMZwZ0ZplIIl4zF1gmzCQe/3vnW9sSnP4G1mGQPRrmSZU71P8I+23wUMe0sEYR1ryWJG1wRnn1GqoLimaN1DdDRXroKjAiO9lqipHMjd9NUt72D5R9gpUx4bMgfYnNtoextw7henJbXchVL8WSB7MLzb7QagtFlcoP4/4++L3cXo2jDEN8fjSm/5clHfm1SfEMbCt8Jb1PQm1o5Ha7mdwjiIZJA== ARC-Authentication-Results: i=1; mx.microsoft.com 1; spf=pass smtp.mailfrom=uqam.ca; dmarc=pass action=none header.from=uqam.ca; dkim=pass header.d=uqam.ca; arc=none DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=UQAM.onmicrosoft.com; s=selector2-UQAM-onmicrosoft-com; h=From:Date:Subject:Message-ID:Content-Type:MIME-Version:X-MS-Exchange-SenderADCheck; bh=BwMBDYCj8H83Pt1OcHMpfzNAYsIpyX1k5fAmqiClua0=; b=KMCLW0IGuYIPv8TwVJ6aDrNuBCTQ7BqdwBZXLPyLVHmz0pO+L0rXRNA/lGIQBVq9DAEcXpSb0SIg06NGsGdcvyAe3K5d/oxdq00SPUs5mTPS8vEiy+C1AJx61PjTRpWIwSQG/JkF/NqUoYoxFaOkvB3+TbiRBGz182mvMfhUOvU= Received: from QB1PR01MB2948.CANPRD01.PROD.OUTLOOK.COM (2603:10b6:c00:38::33) by YT3PR01MB8385.CANPRD01.PROD.OUTLOOK.COM (2603:10b6:b01:9b::11) with Microsoft SMTP Server (version=TLS1_2, cipher=TLS_ECDHE_RSA_WITH_AES_256_GCM_SHA384) id 15.20.7113.27; Mon, 25 Dec 2023 23:56:29 +0000 Received: from QB1PR01MB2948.CANPRD01.PROD.OUTLOOK.COM ([fe80::b0dc:3de3:38:de08]) by QB1PR01MB2948.CANPRD01.PROD.OUTLOOK.COM ([fe80::b0dc:3de3:38:de08%4]) with mapi id 15.20.7113.027; Mon, 25 Dec 2023 23:56:28 +0000 From: =?Windows-1252?Q?Joyal=2C_Andr=E9?= To: "Michael Barr, Prof." , "categories@mq.edu.au" Subject: Re: History: The Eilenberg-Mac Lane collaboration Thread-Topic: History: The Eilenberg-Mac Lane collaboration Thread-Index: AQHaMVJNgkw8N+F07E2RRH3sCiRqFbCvMYLe Date: Mon, 25 Dec 2023 23:56:28 +0000 Message-ID: References: In-Reply-To: Accept-Language: fr-CA, en-US X-MS-Has-Attach: X-MS-TNEF-Correlator: msip_labels: x-ms-traffictypediagnostic: QB1PR01MB2948:EE_|YT3PR01MB8385:EE_|SY4AUS01FT013:EE_|ME3PR01MB5559:EE_|ME3PR01MB5447:EE_|SY4PR01MB6380:EE_ X-MS-Office365-Filtering-Correlation-Id: 0270160c-fc99-49b2-7efc-08dc05a52216 X-MS-Exchange-SenderADCheck: 0 X-MS-Exchange-AntiSpam-Relay: 0 X-Microsoft-Antispam-Untrusted: BCL:0 X-Microsoft-Antispam-Message-Info-Original: 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 X-Forefront-Antispam-Report-Untrusted: CIP:255.255.255.255;CTRY:;LANG:en;SCL:1;SRV:;IPV:NLI;SFV:NSPM;H:QB1PR01MB2948.CANPRD01.PROD.OUTLOOK.COM;PTR:;CAT:NONE;SFS:(13230031)(39860400002)(346002)(136003)(366004)(376002)(396003)(230922051799003)(186009)(451199024)(64100799003)(1800799012)(166002)(38070700009)(83380400001)(110136005)(76116006)(316002)(296002)(66946007)(66556008)(66476007)(66446008)(64756008)(786003)(6506007)(52536014)(8936002)(8676002)(71200400001)(9686003)(7696005)(45080400002)(66574015)(478600001)(26005)(1015004)(966005)(38100700002)(41300700001)(5660300002)(2906002)(122000001)(19627405001)(33656002)(15974865002)(86362001)(55016003);DIR:OUT;SFP:1101 MIME-Version: 1.0 X-MS-Exchange-Transport-CrossTenantHeadersStamped: YT3PR01MB8385 Authentication-Results-Original: relay.mimecast.com; dkim=pass header.d=UQAM.onmicrosoft.com header.s=selector2-UQAM-onmicrosoft-com header.b=KMCLW0IG; arc=pass ("microsoft.com:s=arcselector9901:i=1"); dmarc=pass (policy=none) header.from=uqam.ca; spf=pass (relay.mimecast.com: domain of joyal.andre@uqam.ca designates 40.107.116.83 as permitted sender) smtp.mailfrom=joyal.andre@uqam.ca X-Mimecast-Spam-Score: -4 X-Mimecast-Impersonation-Protect: Policy=MQ - Tag Header Only on Default Settings;Similar Internal Domain=false;Similar Monitored External Domain=false;Custom External Domain=false;Mimecast External Domain=false;Newly Observed Domain=false;Internal User Name=false;Custom Display Name List=false;Reply-to Address Mismatch=false;Targeted Threat Dictionary=false;Mimecast Threat Dictionary=false;Custom Threat Dictionary=false Content-Language: fr-CA Content-Type: multipart/alternative; boundary="_000_QB1PR01MB2948EE8068748C7333AFE2D4FD90AQB1PR01MB2948CANP_" Return-Path: <> X-EOPAttributedMessage: 0 X-EOPTenantAttributedMessage: 82c514c1-a717-4087-be06-d40d2070ad52:0 X-MS-Exchange-SkipListedInternetSender: ip=[40.107.116.83];domain=CAN01-YQB-obe.outbound.protection.outlook.com X-MS-Exchange-ExternalOriginalInternetSender: ip=[40.107.116.83];domain=CAN01-YQB-obe.outbound.protection.outlook.com X-MS-Exchange-Transport-CrossTenantHeadersStripped: SY4AUS01FT013.eop-AUS01.prod.protection.outlook.com X-MS-PublicTrafficType: Email X-MS-Office365-Filtering-Correlation-Id-Prvs: 0582dde2-9025-47a5-0235-08dc05a51cab X-Moderation-Data: 12/26/2023 2:15:10 PM X-LD-Processed: 82c514c1-a717-4087-be06-d40d2070ad52,ExtAddr,ExtAddr X-MS-Exchange-AtpMessageProperties: SA X-Microsoft-Antispam: BCL:0; X-Microsoft-Antispam-Message-Info: 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 X-Forefront-Antispam-Report: CIP:103.96.20.101;CTRY:CA;LANG:en;SCL:1;SRV:;IPV:NLI;SFV:NSPM;H:CAN01-YQB-obe.outbound.protection.outlook.com;PTR:mail-yqbcan01on2083.outbound.protection.outlook.com;CAT:NONE;SFS:(13230031)(4636009)(396003)(376002)(39860400002)(346002)(136003)(64100799003)(1800799012)(451199024)(8000799017)(48200799006)(82310400011)(40470700004)(316002)(8936002)(8676002)(45080400002)(7696005)(498600001)(33656002)(88732003)(86362001)(6506007)(89122003)(76576003)(966005)(26005)(9686003)(1015004)(156005)(7636003)(35950700001)(166002)(82740400003)(32850700003)(32820700001)(66574015)(83380400001)(336012)(32650700002)(41300700001)(110136005)(786003)(70586007)(68406010)(5660300002)(40480700001)(19627405001)(2906002)(7416002)(7406005)(7336002)(7366002)(7276002)(55016003)(52536014)(30864003)(40460700003)(41080700001);DIR:OUT;SFP:1501; X-Auto-Response-Suppress: DR, RN, NRN, OOF, AutoReply X-MS-Exchange-CrossTenant-Network-Message-Id: 0270160c-fc99-49b2-7efc-08dc05a52216 X-MS-Exchange-CrossTenant-Id: 82c514c1-a717-4087-be06-d40d2070ad52 X-MS-Exchange-CrossTenant-AuthSource: SY4AUS01FT013.eop-AUS01.prod.protection.outlook.com X-MS-Exchange-CrossTenant-AuthAs: Anonymous X-MS-Exchange-CrossTenant-FromEntityHeader: Internet X-MS-Exchange-CrossTenant-OriginalArrivalTime: 26 Dec 2023 14:15:13.9711 (UTC) X-MS-Exchange-CrossTenant-MailboxType: HOSTED X-MS-Exchange-CrossTenant-UserPrincipalName: 6FJkuuMBLoYzAjlwWa0WlCSbGQwSmFeM9z8Fduj+Uc46ed3t7VPQQU6/5/bf3ax5ZjP5NG25NeJMFqe0f/Oe7Sh3FOOSrPxFOL+cTfyi4PqBwV7FO7Hqpg6PpdF7zCq1h+ebi+BE7kiKIGpkcniKif2Jmk/2d/aH+Ix4Cmu2pAHutA1hUn+szx+58sfmAiYEmXUISeI5Vy6BwK4D+as+8QSmAYa+JZRZtX/q7sShq/M= X-MS-Exchange-Transport-CrossTenantHeadersStamped: SY4PR01MB6380 X-MS-Exchange-UnifiedGroup-DisplayName: Categories mailing list X-MS-Exchange-UnifiedGroup-Address: categories@mq.edu.au X-MS-Exchange-UnifiedGroup-MailboxGuid: 9c2f954e-92a7-451b-b723-a07075d7adb5 X-MS-Exchange-UnifiedGroup-CustomizedMessage: RedeemedBusinessGuests X-MS-Exchange-Parent-Message-Id: Auto-Submitted: auto-generated X-MS-Exchange-Generated-Message-Source: Throttled Fork Delivery Agent X-MS-Exchange-AntiSpam-MessageData-ChunkCount: 1 X-MS-Exchange-AntiSpam-MessageData-0: =?Windows-1252?Q?gABzBo0i9TCohCkEvpTvVe2mrLiVINhb3BkYAQ2FTqvq1Yj6XQoozD4u?= =?Windows-1252?Q?ul5nNPyyd4XnKPXUBSiuhZ1pUhOwtJyZ+0PhJ1xjymQrrzcIBhxFTxwB?= =?Windows-1252?Q?zFHHiUoGqxoePOj+Ex13DC0L5b6n2hF2p0A3Ll6dUChdQ2eFWVuGtRsV?= =?Windows-1252?Q?TXltXM6nIu8mdBaSE2473gPnQAyR2CMQLM1ZrUC8cRRc8UxgJtj3ZAkZ?= =?Windows-1252?Q?4VbWt/sM3lPTPzZdia+7/NYmiNTcmMru7rVGjXYnsdXh1Vg1t3SKQSwS?= =?Windows-1252?Q?Hd9Q3A3wwpiBqr03v3+x133Qa8iubW9HOQQX7NblgtzJhmvYB1lH6fJW?= =?Windows-1252?Q?aQhLKZxtTfKCoXfUr5dvjw0E+zEs2DavGF7lz4tIpnyi4HuMCvyrf4ua?= =?Windows-1252?Q?nIb25SJ6zXr+g5Tt/N9d+4XGTJ1RcycKtqIw65t9t3J4webGdHzbz+7h?= =?Windows-1252?Q?HwsA2zXmpxPFk9UqLKXhcH2qEjffeRhBIy/Kk5oO41MJ6SOj05lGb7is?= =?Windows-1252?Q?Xv86eDZ9Nai4HULQ0ktmks5eLRnF1Ts3gEu3dbdwDbdGVFDyDi7roht4?= =?Windows-1252?Q?lIxWlDG/+8g2ovWL/sFO1Pn0GU2x1eaBEnVz/Nbuqh5/CU+143e9xccQ?= =?Windows-1252?Q?uWVSKJByyXwp4v9Vg9wEbpoQY9UItzobx3RdG4Xb1tNQQWIZWlV+CvQb?= =?Windows-1252?Q?hie2iO2YnesyiAFy6+M8WjOZEqhkNDRqrIe7FIruUl5+Hn7xonW+0gct?= =?Windows-1252?Q?kyJveJCyaLKaCtLq7lesp9taUrE3m7BionFWKppjhpH5FKhA76MMQpCC?= =?Windows-1252?Q?460FPReeHBRBUe+NEW1FUKMW9c1ayn8vNvp4vN5fCQQ9QHnT5opC3LC/?= =?Windows-1252?Q?pcZCBSWy+5CM5bHZz6FLN6ugRPYI9tN9XkLSZzY1ldqBmTBIM6v6aBW6?= =?Windows-1252?Q?Oaj+1QLDab8PRln5Nr/jolu6FNKc1/cscTu3If2pyHRpM0doquK91W/H?= =?Windows-1252?Q?GsfWXzcORiymniibi2yEmO/u/1m/G3qyMBLJ8Yo7QGGfGt63YehHlVAi?= =?Windows-1252?Q?KMlsf5n3y6mkg+vUUioADdnVBarrxpaCGBYLUF/uax8xkJLo1uLgK+Zl?= =?Windows-1252?Q?uswcJPGhI1r+1P9ct/1SAtEM8SyKkNoFL9YLjHEkBhrwn8O50pqTIzG8?= =?Windows-1252?Q?Kp8o2ei5i1Vh1Onj1V90ldvDSebJsDYNQtG2Wgd0/o9tdEvjv4q8m/EZ?= =?Windows-1252?Q?BGR0MbR3AQz6F9KuiHzuypOv+5F3tO56+y3s0dQKIYxrfsL6QC7kBdJV?= =?Windows-1252?Q?NLk4mMRQdZ8ez5fLMwQQQ24QdYU8axTHsMoMnbTDLAOc141hlPQvujOM?= =?Windows-1252?Q?J5ckH2wn+ZS9m2prbhYXftz8eBC3GE4ZSxd25SQadZ4vw7CoC7Lt7Nij?= =?Windows-1252?Q?7RWa2jrf6uLR3o18fjhBddczR53gbx0gJ14npthoE32bTwL3yXCFIdo8?= =?Windows-1252?Q?nx32tZH8fRnu7gSXEt1qYTbThtKa8dqhH4EDIkGbqLICZ4+I+FZhV91r?= =?Windows-1252?Q?xaH1jrA/KpzfR0zprzSCyPb947uu4l6B/vcbnq8pH0lyqLhC1QCvUqCO?= =?Windows-1252?Q?4J0=3D?= X-OriginatorOrg: mq.edu.au --_000_QB1PR01MB2948EE8068748C7333AFE2D4FD90AQB1PR01MB2948CANP_ Content-Type: text/plain; charset="Windows-1252" Content-Transfer-Encoding: quoted-printable Dear Michael, Thank you for your recollection of collaboration between Sammy and Saunders= . You wrote: They spent five years publishing their work from the war years and then t= heir collaboration ceased. Almost surely because they were no longer in th= e same location. Mail was slow and unsatisfactory and there was no interne= t. When Charles and I were trying to collaborate on TTT, mail between Mont= real and Cleveland took a minimum of two weeks. Then we both got computers= and we discovered data transfers and we were off. Of course, collaboratio= n by mail was possible, but highly unsatisfactory. I do not have a direct knowledge of this, but Myles Tierney told me another= story a few decades ago. As you know, Steenrod discovered his (mod 2) cohomology operations in 1947. Eilenberg and MacLane saw that the operations could be explained in terms o= f the mod 2 cohomology of the Eilenberg-MacLane spaces K(Z/2, n). They decided to compute the cohomology of K(A,n) for any abelian group A and any n >0. They published three notes on the cohomology theory of abelian groups in the Proceedings of the National Academy of Sciences (1950). Followed by three papers on the group H(pi,n) in 1953. Cartan was reading their work closely, and he began to work on the problem with Serre in the early 50. Eilenberg and MacLane had been trying to solve the general problem by iterating some general contruction (the famous bar resolution). In contrast, Cartan and Serre concentrated their effort on the special case of K(Z/p,n) for a prime p. Also, they exploited systematically the fact that K(A, n) is the base space= of a fibration with a contractible total space E(A,n) and fiber K(A,n-1). Eilenberg evetually understood that Cartan's approach was more powerful and after visiting him in Paris 1952 he started a collaboration. I guess that their Homological Algebra is one upshot. According to Myles, MacLane was upset and he never worked with Eilenberg ag= ain. He authored a book on homological algebra using abelian categories years la= ter. Fortunately, Eilenberg and MacLane stayed in contact: they were the founding fathers of the growing school of category theory! The integral homology of K(pi,n) was computed in general by G.J. Decker in his Phd thesis (1974) with MacLane, but his description is very complica= ted. Some finite abelian groups are only described with an infinite presentation= ! For a better description in special cases, see the paper of Larry Breen, R. Mikhailov and A. Touze (2013). > Larry was planning to describe them all explicitly. Unfortunately, he died May 8, 2023 (after having Alzheimer for many years)= . He was a very good mathematician. http://www.ams.org/news?news_id=3D7249 News from the AMS Advancing research. Creating connections. www.ams.org Best wishes, Andr=E9 ________________________________ De : Michael Barr, Prof. Envoy=E9 : 17 d=E9cembre 2023 21:19 =C0 : categories@mq.edu.au Objet : History: The Eilenberg-Mac Lane collaboration I would like to tell some of the story of Eilenberg-Mac Lane. I will start= with a tale told me my a graduate student at Columbia when I began as an i= nstructor in 1962. He said that Sammy had never met Saunders in person until the International= Congress in Cambridge, Mass in 1950 and that after they met, they never co= llaborated again. It is hard to imagine a more wrong story, but the student appeared to belie= ve it. In fact, Sammy arrived in the US in (I could be off by a year) 1939 and got= a one year appointment at U. Michigan. At that time he was working on the= (co)homology of K(pi,1). A space with only one non-zero homotopy group pi= in dimension n is called a K(pi,n) space. When n =3D1, pi can be any grou= p. If n > 1, pi has to be commutative. In any case, the (co)homology grou= ps depend only on pi, not on the space. Saunders was working on group extension theory=97what we now call exact seq= uences 0 ---> A --> Pi --> pi --> 1 where A is abelian. He came to Ann Arb= or to give a talk on his work and Sammy immediately recognized that they we= re making the same kinds of computation. In fact they had both discovered = H^2(pi,A), which eventually got to be called the second Eilenberg-Mac Lane = cohomology group of pi with coefficients in A. (I'm omitting lots of detai= ls here.) Anyway, they had a long discussion about this. How could the sa= me computation arise in algebraic topology and (what eventually got to be c= alled) homological algebra. In order to start explaining this they discovered the idea of a natural tra= nsformation, for which they needed functors, for which they needed categori= es. The following year, Sammy went to Indiana U. where he met, among others, Cl= ifford Truesdell, the finest 19th century physicist of the 20th century, wh= ich had an interesting consequence, see below. Meantime, the war had started. Saunders moved from being a junior fellow a= t Harvard to a war office in New York. I'm not sure what they were doing t= here, but I would speculate they were creating ballistic firing tables. Wh= ere to aim a cannon given muzzle speed and wind velocity. But he somehow a= rranged to have Sammy join the office. Then every night after work, Sammy = went to Saunders's apartment and they worked. On categories, on the Eilenb= erg-Mac Lane cohomology theory and on the the (co)homology of K(pi,n) space= s. The last was doubtless their deepest work. Or any rate, the one I don'= t really understand. Then the war ended. Sammy stayed in NY, spending the rest of his career at= U. Chicago. They spent five years publishing their work from the war year= s and then their collaboration ceased. Almost surely because they were no = longer in the same location. Mail was slow and unsatisfactory and there wa= s no internet. When Charles and I were trying to collaborate on TTT, mail = between Montreal and Cleveland took a minimum of two weeks. Then we both g= ot computers and we discovered data transfers and we were off. Of course, = collaboration by mail was possible, but highly unsatisfactory. I'm not sure when Bill Lawvere college. Best guess would 1955. He went to= Indiana U. and totally impressed Truesdell. In fact, I have heard that he= ended up living at the Truesdell's. At that time, he was just as interest= ed in physics as in math. But Truesdell felt that Bill's true calling was = math. Although he maintained and interest in physics all his life. At any= rate (and I heard this story from Truesdell himself) when he started think= ing where Bill should go to study math, he recalled the mathematician who h= ad impressed him the most=97Sammy. And that explains how Bill ended up doi= ng his graduate work at Columbia. Michael You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group | = Learn more about Microsoft 365 Groups --_000_QB1PR01MB2948EE8068748C7333AFE2D4FD90AQB1PR01MB2948CANP_ Content-Type: text/html; charset=WINDOWS-1252 Content-Transfer-Encoding: quoted-printable

Thank you for your recollection of collaboration= between Sammy and Saunders.

You wrote:

  = They spent five years publishing their work from the war years and then their collaboration ceased.  Almost surely bec= ause they were no longer in the same location.  Mail was slow and unsa= tisfactory and there was no internet.  When Charles and I were trying = to collaborate on TTT, mail between Montreal and Cleveland took a minimum of two weeks.  Then we both got computers an= d we discovered data transfers and we were off.  Of course, collaborat= ion by mail was possible, but highly unsatisfactory.

I do not have a dire= ct knowledge of this, but Myles Tierney told me another story a few decades ago.
As you know, Steenro= d discovered his (mod 2) cohomology operations in 1947. 
Eilenberg and MacLan= e saw that the operations could be explained in terms of the mod 2 cohomology 
of the Eilenberg-Mac= Lane spaces K(Z/2, n).
They decided to comp= ute the cohomology of 
K(A,n) for any abeli= an group A  and any n >0.  
They published three= notes on the cohomology theory of abelian groups
 in the Proceed= ings of the National Academy of Sciences (1950).
Followed by thr= ee papers on the group H(pi,n)  in 1953.
Cartan was reading t= heir work closely, and he began to work on
the problem with Ser= re  in the early 50.
Eilenberg and MacLan= e had been trying to solve the general problem
by iterating some general contruction (the famou= s bar resolution).
In contrast, Cartan = and Serre concentrated their effort on the 
special case of K(Z/= p,n) for a prime p.
Also, they exploited= systematically the fact that K(A, n) is the base space of a fibration 
with a contractible total space E(A,n) and fiber= K(A,n-1).
Eilenberg evetually understood that Cartan's app= roach was more powerful
and after visiting him in Paris 1952 he started = a collaboration.
I guess that their Homological Algebra is one up= shot.
According to Myles, MacLane was upset and he nev= er worked with Eilenberg again.
He authored a book on homological algebra using = abelian categories years later.
Fortunately, Eilenberg and MacLane stayed in con= tact: 
 they were the founding fathers of the grow= ing school of category theory!

The integral homology of K(pi,n) was computed in= general by G.J. Decker
in his Phd thesis (1974) with MacLane, but his d= escription is very complicated.
Some finite abelian groups are only described wi= th an infinite presentation!
For a better description in special cases, see t= he paper of 
Larry Breen, R. Mikhailov and A. Touze (2013).
<https://arxi= v.org/pdf/1312.5676>
Larry was planning to describe them all explicit= ly.
Unfortunately, he died  May 8, 2023 (after = having Alzheimer for many years).
He was a very good mathematician.
http://www.ams.org/news?= news_id=3D7249
Advancing research. Creating connections.

Best wishes,
Andr=E9






De : Michael= Barr, Prof. <barr.michael@mcgill.ca>
Envoy=E9 : 17 d=E9cembre 2023 21:19
=C0 : categories@mq.edu.au <categories@mq.edu.au>
Objet : History: The Eilenberg-Mac Lane collaboration
 
I would like to tell some of the story of Eilenberg-Mac Lane.  I will = start with a tale told me my a graduate student at Columbia when I began as= an instructor in 1962.

He said that Sammy had never met Saunders in person until the International= Congress in Cambridge, Mass in 1950 and that after they met, they never co= llaborated again. 

It is hard to imagine a more wrong story, but the student appeared to belie= ve it.

In fact, Sammy arrived in the US in (I could be off by a year) 1939 and got= a one year appointment at U. Michigan.  At that time he was working o= n the (co)homology of K(pi,1).  A space with only one non-zero homotop= y group pi in dimension n is called a K(pi,n) space.  When n =3D1, pi can be any group.  If n > 1, pi has t= o be commutative.  In any case, the (co)homology groups depend only on= pi, not on the space.

Saunders was working on group extension theory=97what we now call exact seq= uences 0 ---> A --> Pi --> pi --> 1 where A is abelian.  H= e came to Ann Arbor to give a talk on his work and Sammy immediately recogn= ized that they were making the same kinds of computation.  In fact they had both discovered H^2(pi,A), which eventually got to be cal= led the second Eilenberg-Mac Lane cohomology group of pi with coefficients = in A.  (I'm omitting lots of details here.)  Anyway, they had a l= ong discussion about this.  How could the same computation arise in algebraic topology and (what eventually got to b= e called) homological algebra. 

In order to start explaining this they discover= ed the idea of a natural transformation, for which they needed functors, for which they needed categories. 

The following year, Sammy went to Indiana U. wh= ere he met, among others, Clifford Truesdell, the finest 19th century physicist of the 20th&n= bsp;century, which had an interesting consequence, see below.

Meantime, the war had started.  Saunders m= oved from being a junior fellow at Harvard to a war office in New York.  I'm not sure what they were doing there= , but I would speculate they were creating ballistic firing tables.  W= here to aim a cannon given muzzle speed and wind velocity.  But he som= ehow arranged to have Sammy join the office.  Then every night after work, Sammy went to Saunders's apartment and they worked= .  On categories, on the Eilenberg-Mac Lane cohomology theory and on t= he the (co)homology of K(pi,n) spaces.  The last was doubtless their d= eepest work.  Or any rate, the one I don't really understand.

Then the war ended.  Sammy stayed in NY, s= pending the rest of his career at U. Chicago.  They spent five years publishing their work from the war years and then th= eir collaboration ceased.  Almost surely because they were no longer i= n the same location.  Mail was slow and unsatisfactory and there was n= o internet.  When Charles and I were trying to collaborate on TTT, mail between Montreal and Cleveland took a minimum = of two weeks.  Then we both got computers and we discovered data trans= fers and we were off.  Of course, collaboration by mail was possible, = but highly unsatisfactory.

I'm not sure when Bill Lawvere college.  B= est guess would 1955.  He went to Indiana U. and totally impressed Truesdell.  In fact, I have heard that he en= ded up living at the Truesdell's.  At that time, he was just as intere= sted in physics as in math.  But Truesdell felt that Bill's true calli= ng was math.  Although he maintained and interest in physics all his life.  At any rate (and I heard this story from Tr= uesdell himself) when he started thinking where Bill should go to study mat= h, he recalled the mathematician who had impressed him the most=97Sammy.&nb= sp; And that explains how Bill ended up doing his graduate work at Columbia.

Michael
 
 
You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message.
 
View group files   |   Leav= e group   |&n= bsp;  Learn more about Microsoft 365 Groups
 
--_000_QB1PR01MB2948EE8068748C7333AFE2D4FD90AQB1PR01MB2948CANP_--