From mboxrd@z Thu Jan 1 00:00:00 1970 Return-Path: <> X-Spam-Checker-Version: SpamAssassin 3.4.4 (2020-01-24) on inbox.vuxu.org X-Spam-Level: * X-Spam-Status: No, score=1.2 required=5.0 tests=DKIM_ADSP_CUSTOM_MED, DKIM_SIGNED,DKIM_VALID,FREEMAIL_FROM,HTML_MESSAGE,NML_ADSP_CUSTOM_MED, RCVD_IN_DNSWL_NONE,RCVD_IN_MSPIKE_H2,T_SCC_BODY_TEXT_LINE autolearn=no autolearn_force=no version=3.4.4 Received: from SY2PR01CU004.outbound.protection.outlook.com (mail-australiaeastazhn15011004.outbound.protection.outlook.com [52.102.200.4]) by inbox.vuxu.org (Postfix) with ESMTP id A2732201A3 for ; Fri, 22 Mar 2024 10:55:58 +0100 (CET) ARC-Seal: i=4; a=rsa-sha256; s=arcselector9901; d=microsoft.com; cv=pass; b=PCtTsi5nJ3FUIgEI2LObMrkWCZlOVnyigBFQv9tlv75E//SPww0Eb+yeC+88BogAQuc1rrVVG13zMJR/uV+PzdW6Bd9i+q/L5v9BeJLo0jGF+8t/RrVPrjmsUhtT0txTo44IYi+PlT/shPhWGVDNaOdnnk/U1lbiHIh5sbcXj6evMmoRJuqSkG1BsB5Rw6pXo7u96YsPAWNLe6XZXiwfLu/8GZdZMPgRdD26w2DRJk5UOsi0ClDRISDx6drRojGCOuiyuVuPeZbKJG2OLhoMn6PJuhfcCvccPKxpQ2XXkC6rAwzeCB0hucHpAwBLlDE7w8i9nZVTDn+NVimMHDjbCA== 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=2EV69Nq/ucAc8AR651Y6cgQY2go4gfIswvDMxOFitls=; b=gaut0brNeOGZZduI3bRLJyUa0ARQ4mxnzxE4rBxR2yPFqtct5BaMBr/fpoig1aNx56hzgm4xsqZ9VXcFtHSk7tGG0LaROqx1y+DvRb58yyykOaBs93jgZfZj634zNg1IsGM4gVN2TQ9GQvplU8AmGRRlWL3jwU2jZ7LB653ZDwRvLROB1H3cSeM1jkkeAlbjfxCcQ7iNaUnlic1AwK0HfIP82K28Nzbw3R/nHP8csNFkj4039Ptof/my/jktXVvtnwp1+dNF/CNPKBfywzeQ1U380WmZKmANNQRwZd9i+G1sgIyknJthlxDAZpyccZ0hXbvPr3I+nEw55IHA5B5Nmw== ARC-Authentication-Results: i=4; mx.microsoft.com 1; spf=pass (sender ip is 209.85.219.175) smtp.rcpttodomain=mq.edu.au smtp.mailfrom=gmail.com; dmarc=pass (p=none sp=quarantine pct=100) action=none header.from=gmail.com; dkim=fail (body hash did not verify) header.d=gmail.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=2EV69Nq/ucAc8AR651Y6cgQY2go4gfIswvDMxOFitls=; b=eXEEJVyLI4yAQSj5UyWhj5lwdRruPO9E37oWSiEygWsSyA9MDj7DUwEK9lKIQMbtaAqGg84efVsdT8SzyJc6RDy+P+ehpyzLfKLpfDoE5c13gPmyIfqlp2tJrGYbyAYpjyJ1Mgj8RphgjEe0BjI040Kst2SYaGERNjS+px4NXE5u53hKWZmRUj9/WpnbIlCVhabcG+8vARMKZRz+Fo5qaiOqmESZp8VoV8BQnyfsFxarmnroiQOKGo/FzfncAJ+7co6lf7F2FLyVOY+zwg6TB9HpKgpAIy1wtj5yRzYiWCjNfuwtCOlH84btAGkJ379VbCi59SmacpNfsY2bi5msEg== Received: from SmtpServer.Submit by SYZPR01MB7487 with Microsoft SMTP Server id 15.20.7409.24; Fri, 22 Mar 2024 09:55:56 +0000 ARC-Seal: i=3; a=rsa-sha256; s=arcselector9901; d=microsoft.com; cv=pass; b=DgcsHQCFY47tVAM2bcZxTH3qzjA03vdeP5OZEXBpqqgWZcEsgC6UvNFq9jASWCZ7MCpqjn/DxH7hK7KS8VwN5k1wOxNiYK1FOVQI/G9U/ckgzno17+i0HAiGsILLUL9pAdZGugcMy7lz+NetG64F1Cz3N0sfx39bwOSfYYliLY73PTPuPKeguYzUnsr+ggY8TPtMWih3VD8E2CJZod01fsxjWOLeOHhGRQIgXpcamBikpyDj56NLhHBf79IF8cTZMr4hslCHjW5227T8eXu4541yT2ooApQlrUPEmdTXCbn9XMB2wPiCqD3/W3sGoN/tNTAROXPT2QPiu+rlLoMnSA== 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=0qaNW56TZb5zaYGtRnfHNOfDTwU8DS0GpeKYtPRxmso=; b=iyzczWux/6T9N9gv+I+H8672ycJ28WLwNwhKD3W72yBW90jQF18FUDkSa22nO2qim/VNVbvne+r5f0wl1oum5bb5dLC11hEuHArntSvvh1RApUsrDEiqlCqVhMbb+qaUhgFt3gBMPlGRnBoHE8cSxBa/JO5q1bTf5TkENhWq03C6/2jmjrS1BbuoTQx/ZiLzPz1L70GjOCXhw9PLIUyvlsBRJKYjyx5UuJQW/S/MrvzKBryhh2IPpzS5zSwiE8luHJUZwFe5H3/NGAf9mllZNjcmbnm36R9bPBjXz2zmIbx7U7gbaPwRpaXlbSrrix7oC+CJrw8N5Q2R3fMk2kSFrw== ARC-Authentication-Results: i=3; mx.microsoft.com 1; spf=pass (sender ip is 209.85.219.175) smtp.rcpttodomain=mq.edu.au smtp.mailfrom=gmail.com; dmarc=pass (p=none sp=quarantine pct=100) action=none header.from=gmail.com; dkim=fail (body hash did not verify) header.d=gmail.com; arc=pass (0 oda=0 ltdi=0 93) Received: from SY4PR01MB6868.ausprd01.prod.outlook.com (2603:10c6:10:141::5) by SYZPR01MB7487.ausprd01.prod.outlook.com (2603:10c6:10:16c::19) with Microsoft SMTP Server (version=TLS1_2, cipher=TLS_ECDHE_RSA_WITH_AES_256_GCM_SHA384) id 15.20.7409.24; Fri, 22 Mar 2024 09:55:51 +0000 Received: from MEAPR01MB4358.ausprd01.prod.outlook.com (2603:10c6:220:16::20) by SY4PR01MB6868.ausprd01.prod.outlook.com (2603:10c6:10:141::5) with Microsoft SMTP Server (version=TLS1_2, cipher=TLS_ECDHE_RSA_WITH_AES_256_GCM_SHA384) id 15.20.7409.24; Fri, 22 Mar 2024 09:55:48 +0000 Received: from MEAPR01MB4358.ausprd01.prod.outlook.com ([fe80::9ab9:787d:5b4c:34e3]) by MEAPR01MB4358.ausprd01.prod.outlook.com ([fe80::9ab9:787d:5b4c:34e3%5]) with mapi id 15.20.7409.023; Fri, 22 Mar 2024 09:55:47 +0000 ARC-Seal: i=2; a=rsa-sha256; s=arcselector9901; d=microsoft.com; cv=pass; b=H8wOIrCNJw+ajb4xqWgsKJFWG2ptr5ge0g/cOSTVg5Kr4Sm4LC+CBYutVRsnl4kIKaIkuICE37aaJtzzwfEjRSgYMl9qGLPL9bf+WlZfkhGXbynDP30OnZOiimguHf/YmvzD1l7rOpTtePYBxht8fDBQiim+62AtWS3k6MCCKm6Zj3Lvf13GKDqUGi5PYFHv2LoBIr5qyi1aRm6CZYD4fQsCabIsdLv5XWg1CwMy5HjpoXyW1hmtSTO2ZvCI8eUb31EHa3t+Y/WEOGXfO5csxp244wlS23M+Wvl/VjZ0AR8H7qq/qQ/SArDXbdcE+/HjIvDJadnKgMiHM1/DdpDE4Q== ARC-Message-Signature: i=2; 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=0qaNW56TZb5zaYGtRnfHNOfDTwU8DS0GpeKYtPRxmso=; b=O1aeEBs5p+axC/xk9HNajNlEoMkkoKePtFKC0ZixSCKdhURbv1uu9/7UjJJBRdv47MNBGOfe9jsdxO7blJMXBs0VCiaWczH/dZFhYvzDwI2+RFv1S4AhRKQ+TwsbjhssI7VlYfkAn+bFsqTcYv6MdW9NVVml5im9mHlVjkUs3jKOpjjCETzPj+JmwU0mvc0W/1T6U2yPij6pojbBW4fu7gqF7e3Nz/0MMTvpxfsz6tj+CQxUw3YaoUR7iOoKUzC4jh23qFr+nsVA+t2BspklM2lRztbn/RKxlVw7nL976SqfeB7KmGdcCHdyYvGOkRiQ01Q8zOnkDlflJhKX7aQY7Q== ARC-Authentication-Results: i=2; mx.microsoft.com 1; spf=pass (sender ip is 209.85.219.175) smtp.rcpttodomain=mq.edu.au smtp.mailfrom=gmail.com; dmarc=pass (p=none sp=quarantine pct=100) action=none header.from=gmail.com; dkim=fail (body hash did not verify) header.d=gmail.com; arc=pass (0 oda=0 ltdi=0 93) Received: from MEWPR01CA0002.ausprd01.prod.outlook.com (2603:10c6:220:1e3::8) by SYBPR01MB5679.ausprd01.prod.outlook.com (2603:10c6:10:e1::9) with Microsoft SMTP Server (version=TLS1_2, cipher=TLS_ECDHE_RSA_WITH_AES_256_GCM_SHA384) id 15.20.7409.24; Thu, 21 Mar 2024 23:19:01 +0000 Received: from ML1PEPF0000F178.ausprd01.prod.outlook.com (2603:10c6:220:1e3:cafe::2d) by MEWPR01CA0002.outlook.office365.com (2603:10c6:220:1e3::8) with Microsoft SMTP Server (version=TLS1_2, cipher=TLS_ECDHE_RSA_WITH_AES_256_GCM_SHA384) id 15.20.7409.13 via Frontend Transport; Thu, 21 Mar 2024 23:19:01 +0000 Authentication-Results: spf=pass (sender IP is 209.85.219.175) smtp.mailfrom=gmail.com; dkim=fail (body hash did not verify) header.d=gmail.com;dmarc=pass action=none header.from=gmail.com; Received-SPF: Pass (protection.outlook.com: domain of gmail.com designates 209.85.219.175 as permitted sender) receiver=protection.outlook.com; client-ip=209.85.219.175; helo=mail-yb1-f175.google.com; pr=C Received: from au-smtp-inbound-delivery-1.mimecast.com (103.96.20.101) by ML1PEPF0000F178.mail.protection.outlook.com (10.167.241.69) with Microsoft SMTP Server (version=TLS1_2, cipher=TLS_ECDHE_RSA_WITH_AES_256_GCM_SHA384) id 15.20.7409.10 via Frontend Transport; Thu, 21 Mar 2024 23:19:01 +0000 ARC-Message-Signature: i=1; a=rsa-sha256; c=relaxed/relaxed; d=dkim.mimecast.com; s=201903; t=1711063141; h=from:from:reply-to:subject:subject:date:date:message-id:message-id: to:to:cc:cc:mime-version:mime-version:content-type:content-type: in-reply-to:in-reply-to:references:references:dkim-signature; bh=0qaNW56TZb5zaYGtRnfHNOfDTwU8DS0GpeKYtPRxmso=; b=rA/i0oTYjuazkvsPZWn8wQFw8VitCO1cBgGdw2cYdC9wHO1AQ9ikLasc9nnw3h1J5ORroz o/92/s7ODI23WtKiuDOO6V4e9UlFCPngxyTtGE0J+dTGju1zuj3QwzA8118wvUUSfTPzlP QyxDavozVIwyEBfuPcqcWiDkivEBnjAak8U2kuvLCVxlXdougukou7ZuWxJ+NcENg30LG0 gqOVHIuHRZcJJEn2CWJfWR5db1usolB8ijOslcszv81oXYPd4Qd3yfrj0MM31sUlUEVZde 7ydbuipn4moAglxAjrgN9q8hmPJptb0uRmYnCJu9R2z1G4Azy5S6wcBhlwFBYw== ARC-Seal: i=1; s=201903; d=dkim.mimecast.com; t=1711063141; a=rsa-sha256; cv=none; b=OgXBh9nMjo4np1mTBza//Xo09ViGfhjJTkVxertdTjNJsiMNdxEc4pE7iL2V2jvoVlJ3AK 9HbqBTlBFSGpiWvzNbxl5VWkr121SUq/GY8Vwy+E9KxnirCPhn+iJOHJva4/ik2nAQWYnt YYGIr+m44JGOu/GomyJeTvlpeYj3J0PvvpDhjTwPWOubKnjlkZR9lM1hQ10b0jyFbWCJtY tlSILE4sEuwAP06z+D+cWhTYtjYikiS1KM2XHckmT41KHvjXWHa2dDQgC9wGEW7jaLSnya hhsIw6b2WCph9WpjkZ9aQj/KLjtXhaXVfi5KjmW+vB6a6Qb4oULf+KEFDzIJ3g== ARC-Authentication-Results: i=1; relay.mimecast.com; dkim=pass header.d=gmail.com header.s=20230601 header.b=Bavdfdaj; dmarc=pass (policy=none) header.from=gmail.com; spf=pass (relay.mimecast.com: domain of duskgoo@gmail.com designates 209.85.219.175 as permitted sender) smtp.mailfrom=duskgoo@gmail.com Received: from mail-yb1-f175.google.com (mail-yb1-f175.google.com [209.85.219.175]) by relay.mimecast.com with ESMTP with STARTTLS (version=TLSv1.3, cipher=TLS_AES_256_GCM_SHA384) id au-mta-71-0omNWkbRM_GYjd37uUB62w-1; Fri, 22 Mar 2024 10:18:51 +1100 X-MC-Unique: 0omNWkbRM_GYjd37uUB62w-1 Received: by mail-yb1-f175.google.com with SMTP id 3f1490d57ef6-dc23bf7e5aaso1552067276.0 for ; Thu, 21 Mar 2024 16:18:51 -0700 (PDT) DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=gmail.com; s=20230601; t=1711063129; x=1711667929; darn=mq.edu.au; h=cc:to:subject:message-id:date:from:in-reply-to:references :mime-version:from:to:cc:subject:date:message-id:reply-to; bh=ds/sKsSuV+1zO+ztBo7cQmctpAZ/ufYqd/3TnKuOKK8=; b=Bavdfdaj+6U7ZhEUS2NsqfYRfqIAPakcWIRW0K9TF0qqylEmDQf/0caZ8UrlJHc1pO swIk0husBKT+Cuh8VjslN0ju2VFxzJCmeNO+GXv0+Nu39B/0iB/ecHQayvC8sZKnws/Y x2uNDVrPrTkyrSwz3PpUogcXi1I2W0WqSBUzUrMucDk+9BkBJ9SFMvp09AtxVVlbrjQ/ gidLTy53jObDbQoTmBfW4R9JqZrsbz9wfDwaxH4CoqPYIIjpOgE52IP+X+wAbmi+W0dK O0sJNE+au0Rg7eCxDGWsGjsVfhFYXXCSoynLvJ3XL6OzxXlR06t0F+6dXT87k99Tqo2G AOGg== X-Google-DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=1e100.net; s=20230601; t=1711063129; x=1711667929; h=cc:to:subject:message-id:date:from:in-reply-to:references :mime-version:x-gm-message-state:from:to:cc:subject:date:message-id :reply-to; bh=ds/sKsSuV+1zO+ztBo7cQmctpAZ/ufYqd/3TnKuOKK8=; b=IWHJy+nNlDRQhQpr8QsfQYDbVle8eRD59zLOdwywvNorUZudDIxnvZMZa9CrgA66Ym DoxomZwQq92WASAwcYbvAf85BJNzjgSp/EtpYW7X/fB24MWRqpBrIt7sJGp+8sfYfjfA VqeHnxjfzSiQkz5uyHSWsv44LMdMpPogbxXjVgFA4hTxjXyMpkD0U7x++itx25YvpINx ENmphkHDYiKoi7VJ0uo0m7rPORQzEtVK4e9D/INOMX3mTVcA2vuZ8XBx6HIWZ+Ik8tey mRg7lj8IekzmYpu22wUH8cuCrpd7viOnfo4UOL+wAOmZdzf40aszSi3ZtbeCgVdIWYAx kzPw== X-Gm-Message-State: AOJu0YwIloNSTA8erOzrye4Jk5OA0Qi/pAd8PhJEhSwEjOBdCYs1hfkz 2LjOXzBJlGK7iuWKBX6nRxu+EZOzWzcUPaswfBiFxxeIvl1Jyd6Zq39DlC2PWW+CS5skL9ES8/p RKxmjZdLxuW4iDZNj+CJhfW4MQySpPHEk X-Google-Smtp-Source: AGHT+IEXAZVYD5PnnO/LBI9Bs1dTMD6WDConN3tIIdqqOUebAqxFoF4cZ87WUZBYhUXNIwzFeKk23frc3gL/dgvl4tw= X-Received: by 2002:a25:ba90:0:b0:dcc:aa1f:b418 with SMTP id s16-20020a25ba90000000b00dccaa1fb418mr592331ybg.1.1711063129343; Thu, 21 Mar 2024 16:18:49 -0700 (PDT) MIME-Version: 1.0 References: In-Reply-To: From: Dusko Pavlovic Date: Thu, 21 Mar 2024 13:18:38 -1000 Message-ID: Subject: Re: T-algebras in CAT v. categories in T-alg To: David Yetter CC: Categories mailing list Authentication-Results-Original: relay.mimecast.com; dkim=pass header.d=gmail.com header.s=20230601 header.b=Bavdfdaj; dmarc=pass (policy=none) header.from=gmail.com; spf=pass (relay.mimecast.com: domain of duskgoo@gmail.com designates 209.85.219.175 as permitted sender) smtp.mailfrom=duskgoo@gmail.com X-Mimecast-Spam-Score: 0 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-Type: multipart/alternative; boundary="000000000000035440061433ee4c" X-EOPAttributedMessage: 0 X-EOPTenantAttributedMessage: 82c514c1-a717-4087-be06-d40d2070ad52:0 X-MS-Exchange-SkipListedInternetSender: ip=[209.85.219.175];domain=mail-yb1-f175.google.com X-MS-Exchange-ExternalOriginalInternetSender: ip=[209.85.219.175];domain=mail-yb1-f175.google.com X-MS-PublicTrafficType: Email X-MS-TrafficTypeDiagnostic: ML1PEPF0000F178:EE_|SYBPR01MB5679:EE_|SY4PR01MB6868:EE_|SYZPR01MB7487:EE_ X-MS-Office365-Filtering-Correlation-Id: 68a05ae3-051f-4079-9419-08dc49fd4b3e X-Moderation-Data: 3/22/2024 9:55:42 AM X-LD-Processed: 82c514c1-a717-4087-be06-d40d2070ad52,ExtAddr,ExtAddr X-MS-Exchange-AtpMessageProperties: SA X-MS-Exchange-SenderADCheck: 0 X-MS-Exchange-AntiSpam-Relay: 0 X-Microsoft-Antispam: BCL:0; X-Microsoft-Antispam-Message-Info: 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 X-Forefront-Antispam-Report: CIP:103.96.20.101;CTRY:US;LANG:en;SCL:1;SRV:;IPV:NLI;SFV:NSPM;H:mail-yb1-f175.google.com;PTR:mail-yb1-f175.google.com;CAT:NONE;SFS:(13230031)(35950700004)(82310400014)(32650700005)(48200799009)(7416005)(376005)(1800799015)(41110700001)(41080700001)(41090700016);DIR:OUT;SFP:1501; X-Auto-Response-Suppress: DR, RN, NRN, OOF, AutoReply X-MS-Exchange-CrossTenant-Network-Message-Id: 68a05ae3-051f-4079-9419-08dc49fd4b3e X-MS-Exchange-CrossTenant-Id: 82c514c1-a717-4087-be06-d40d2070ad52 X-MS-Exchange-CrossTenant-AuthSource: ML1PEPF0000F178.ausprd01.prod.outlook.com X-MS-Exchange-CrossTenant-AuthAs: Anonymous X-MS-Exchange-CrossTenant-FromEntityHeader: Internet X-MS-Exchange-CrossTenant-OriginalArrivalTime: 22 Mar 2024 09:55:47.4125 (UTC) X-MS-Exchange-CrossTenant-MailboxType: HOSTED X-MS-Exchange-CrossTenant-UserPrincipalName: 4jURz38Q+5e2R8brheB9UKo5QiCm0z8lBCJQ0Ahong7SYEVlhyh5dUaOLTCnXM1jSm4Gi9tfExg2kTMiW2QqhZ1bMclBQGKLrvD/WZgf76pLGFVebQ2HODfMkTxqxIXAcQENek58RTMVI+nCGEIMboA151oJfvUhhEk35Bvw4D2D2hbB4wE941KlMFE5us+B X-MS-Exchange-Transport-CrossTenantHeadersStamped: SYZPR01MB7487 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: =?utf-8?B?amFaSUd6ZHJ4U3MwUWNVN2tBMXlwYUtkTHMyb1hodnNBMFhzeG1xVHVRcHM3?= =?utf-8?B?UzUrSVQwbXdmaFRzeVZzUC9NcTY3QzRkSjZZcHdaVVZXRFk3STNtYm5UdHZ5?= =?utf-8?B?eEFCaUxkZHRNS0ZoZXFPekFDVUZ0ZFU4RmtZNFhESTJpMVNjdFFwMVlCajcy?= =?utf-8?B?ZENGTElTRXJKcDExSldDY2s3dyt5UEFjUE9LVUcxSGUvUUEvS0NEM2RuUDBv?= =?utf-8?B?aGVnZ2ZxMEFwRTl1cTMrU3VQdzdPMmxEc1czQUFONG9KS3FxWFVJa1RpaXdC?= =?utf-8?B?SHBXQnNVdzhxU01qR0VZbWR3cXhEeXFvMURjSDhvZS9lUGVmakZDSjAwMG9U?= =?utf-8?B?ZUtXemN1a0h0aHNEMFl3OUkrcFJUZjIrMkNUNjE5cmtjaW1HdTkyVkhyRGU2?= =?utf-8?B?Vm51ZnBUUS90U0NtZjBNdGlEMHlwUDU4TkpHbHFyRjJjSS95aXR0Y2FNTVcr?= =?utf-8?B?bVc3OW1GdU9EbTZtcDkwV2RCMk1HTElOcFlRQU1ERmM0TFlDNU1hc0tKa1My?= =?utf-8?B?T3A1Y1ZMWjk5d1M1aDYyVmR5NEJEejZHelh1dkFrQ1lBbkM5WGNNQ2pIYXRV?= =?utf-8?B?QWZzaHZQc3VxQUtGNW9OZU56OXRaT1J6NFZVc3lhdm5Qc05kRDVEZVF5U2po?= =?utf-8?B?RDc4ZVJTMGphNHdzcmpJeWwvdTYwQkt0VEd5M3FsSzJuTFd0QXlETTkvSHpu?= =?utf-8?B?dCtoUnQvRGpWcnNRelpqWnU2YmZoK3lJdUpOZ0drQ3hMVk9OdFpoaUNoNElF?= =?utf-8?B?ZVFYbC9MYXRQVlNJOThDM0RJc0kwV3psTzlkVjhvKzZtRmx1dHExOURXMVI5?= =?utf-8?B?ZlhDbnJ4S0Y2UFhrVm1vMDlwWUI2THlSV09kT2ZlTEhvcnBCOWV4M095d1Iy?= =?utf-8?B?WUwvQ3V5L0I2Zy95RUNVdjVzUi9zSmpVMjdUWlpvZGgwak1LRTVLM3lRalVB?= =?utf-8?B?Wk82TU94SmpxOE1PMm1YMVBqWDZMT29QM3g3djRjWWRrcy9vOE1OK1I0Y0Ji?= =?utf-8?B?VlNzVEVtczRrQTNYZkMreEc3aUY0eXdLRHN0cU9LREFzOXlUam5RRU40cE5N?= =?utf-8?B?Q3hMNWFpTUU4c2toa25Fa2ZEME1qck5MT3daT1FMWlFEMkZBcXZyWlUySDZu?= =?utf-8?B?WmVCdU1YUmNCTXpyUFBrd3BmQkdmNGpteUM2ZDdCNDVJemxnZ0g5ejd4bGJT?= =?utf-8?B?OEJ1eUE1ZVhZYkZ4R1ZCb1gyUjZELzlzeGNydmJ2dCtLdkh4R3hYcXdNc1Jp?= =?utf-8?B?L2RjVlFHQVVOeFg3WmM1TzNrbk9qSDN0aUFtb2hwTUt3TDNMRXlTREJIWCtl?= =?utf-8?B?K1RUM2Z2eHhFNlhaZjU0VDV2encrRW5WYitoODczTmFXVEZoRjBpNEx6dFRE?= =?utf-8?B?YjVQL2Zqa09BcTZGUmVONkFCVFNqQzV0NXVmd3M4N1dTUDR3M2RTK0ZwdElz?= =?utf-8?B?aWN3Nm5halZzRE1KbmQwWndGWGN2cjdWWVlzTG9qS3Bwd3pxaTVIcVdzbWhx?= =?utf-8?B?eVh6RWtBZzlSdWczNTJocHVOQkxRT2o4UEtUREFPN09CUXdCSzR3RXFIV3Nz?= =?utf-8?B?dys1MTBLbndteWlKL0swTk1FZHBBQzg5K0Fvbi81dkFYeHZtREFEN1IwNmM0?= =?utf-8?B?c0ZId1F1SXdUNjJTc0JyTVUyWFJ6eEczUTZYS2duZ0syZjZLNXA5QjZZQmxT?= =?utf-8?B?YUZRSW4rTEJhZWNrOHVZS2cwNmRmd3d2QmY2MnVhZlNoS1JKb3BiRGtaYUIw?= =?utf-8?B?d3diSDBYeHd1VFdzbEdpdTVvMkRuLytTMUtuZlVkZm5YMWptcm1SVjVlWDZJ?= =?utf-8?B?REpLdy9TYUMveThEcElyRWoxUFdNMWtrY1dyRE9odWNzZW1zUkhReG91WEwz?= =?utf-8?B?UXJFUGFkdDFNYUQ3cFMxZVMzVkh2dnRVZ0ovZ1JoVndEVW9HQW4wN0F6eU00?= =?utf-8?Q?RTNlj6pGRIKby5scm1fPf2pHpvoEXFUV?= X-OriginatorOrg: mq.edu.au --000000000000035440061433ee4c Content-Type: text/plain; charset="utf-8" Content-Transfer-Encoding: quoted-printable maybe i am missing something, but it sounds like a variation on the theme o= f "a sheaf of rings is a ring of sheaves", which got grothendieck from sche= mas to toposes. but categories are simpler than sheaves. if a category is v= iewed as a left-exact functor from, say, a finite limit sketch C into Set, = then the question becomes: for which D are the D-preserving functors into t= he category of C-preserving functors equivalent to the category of C-preser= ving functors into D-preserving functors? ie which functors preserve the C-= limits? is there a subtlety that i am missing? -- dusko On Thu, Mar 21, 2024 at 11:28=E2=80=AFAM David Yetter > wrote: Dear Colleagues: This is surely something well-known, but it is also opaque to search-engine= queries. It is well-known (and I've both used the result and proved it by= hand) than group objects in Cat and category object in Groups are the same= thing: strict monoidal categories in which every object and every arrow h= ave an inverse with respect to \otimes. What class of theories (e.g. finite product, left-exact, finitely axiomatiz= able equational,...) have the property that category objects in their categ= ory of models are the same as models of the theory in Cat? Citations would= be welcomed. The question came up in work with an old student of mine, and rather than s= pending time proving the result we'd like for the particular theory at hand= , I thought it best to see if it followed from something in the literature.= Alas, all sensible keyword combinations give pages of irrelevant search r= esults, so asking the community seemed the best way to proceed. Thanks in advance. Best thoughts, D.Y. 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 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 --000000000000035440061433ee4c Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable
maybe i am missing something, but it sounds like a variation on the th= eme of "a sheaf of rings is a ring of sheaves", which got grothen= dieck from schemas to toposes. but categories are simpler than sheaves. if = a category is viewed as a left-exact functor from, say, a finite limit sketch C into Set, then the question becomes: fo= r which D are the D-preserving functors into the category of C-preserving f= unctors equivalent to the category of C-preserving functors into D-preservi= ng functors? ie which functors preserve the C-limits? is there a subtlety that i am missing? -- dusko

On Thu, Mar 21, 2024 at 11:28=E2=80= =AFAM David Yetter <dyetter@ksu.edu> wrote:
Dear Colleagues:

This is surely something well-known, but it is also opaque to search-engine= queries.  It is well-known (and I've both used the result and proved = it by hand) than group objects in Cat and category object in Groups are the= same thing:  strict monoidal categories in which every object and every arrow have an inverse with respect to \oti= mes.

What class of theories (e.g. finite product, left-exact, finitely axiomatiz= able equational,...) have the property that category objects in their categ= ory of models are the same as models of the theory in Cat?  Citations = would be welcomed.

The question came up in work with an old student of mine, and rather than s= pending time proving the result we'd like for the particular theory at hand= , I thought it best to see if it followed from something in the literature.=   Alas, all sensible keyword combinations give pages of irrelevant search results, so asking the community seemed th= e best way to proceed.

Thanks in advance.

Best thoughts,
D.Y.
 
 
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&n= bsp;group files = ;  |   Leave group   |   Learn more about Microsoft 365 Groups
 
 
 
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
 
--000000000000035440061433ee4c--