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[2607:f8b0:4864:20::b34]) by gmr-mx.google.com with ESMTPS id u18si157748qtk.2.2020.02.21.05.23.48 for (version=TLS1_3 cipher=TLS_AES_128_GCM_SHA256 bits=128/128); Fri, 21 Feb 2020 05:23:48 -0800 (PST) Received-SPF: pass (google.com: domain of b.a.w.spitters@gmail.com designates 2607:f8b0:4864:20::b34 as permitted sender) client-ip=2607:f8b0:4864:20::b34; Received: by mail-yb1-xb34.google.com with SMTP id p123so1086364ybp.2 for ; Fri, 21 Feb 2020 05:23:48 -0800 (PST) X-Received: by 2002:a25:3813:: with SMTP id f19mr32576258yba.316.1582291428196; Fri, 21 Feb 2020 05:23:48 -0800 (PST) MIME-Version: 1.0 From: Bas Spitters Date: Fri, 21 Feb 2020 14:23:37 +0100 Message-ID: Subject: [HoTT] Joyal's definition of elementary higher topos To: homotopytypetheory Content-Type: text/plain; charset="UTF-8" Content-Transfer-Encoding: quoted-printable X-Original-Sender: b.a.w.spitters@gmail.com X-Original-Authentication-Results: gmr-mx.google.com; dkim=pass header.i=@gmail.com header.s=20161025 header.b=WGtY84Bp; spf=pass (google.com: domain of b.a.w.spitters@gmail.com designates 2607:f8b0:4864:20::b34 as permitted sender) smtp.mailfrom=b.a.w.spitters@gmail.com; dmarc=pass (p=NONE sp=QUARANTINE dis=NONE) header.from=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: , In 2014, Andra Joyal proposed a definition of an elementary higher topos. "This lecture contains a proposed definition that is not an (=E2=88=9E,1)-category but a presentation of one by a model category-like structure; this is closer to the type theory, but further from the intended examples. In particular, there are unresolved coherence questions even as to whether every Grothendieck (=E2=88=9E,1)-topos can be presented by a model in Joyal=E2=80=99s sense (in particular, how strict ca= n a universe be made, and can the natural numbers object be made fibrant)." https://ncatlab.org/nlab/show/elementary+%28infinity%2C1%29-topos Has there been any progress on these coherence questions? --=20 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 e= mail to HomotopyTypeTheory+unsubscribe@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/= HomotopyTypeTheory/CAOoPQuQc317E8qUEOWJ2JQD_iXAFyE%3DWcttruqd5A8tRpHqttg%40= mail.gmail.com.