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[84.202.247.109]) by smtp.gmail.com with ESMTPSA id l30sm2240322lfk.83.2019.02.17.05.49.42 (version=TLS1_2 cipher=ECDHE-RSA-CHACHA20-POLY1305 bits=256/256); Sun, 17 Feb 2019 05:49:42 -0800 (PST) Date: Sun, 17 Feb 2019 14:49:41 +0100 From: Richard Williamson To: Thomas Streicher Cc: Steve Awodey , Michael Shulman , =?iso-8859-1?B?IkFuZGVycyBNw7ZydGJlcmci?= , Homotopy Type Theory Subject: Re: [HoTT] A unifying cartesian cubical type theory Message-ID: <20190217134941.GA2100@richard.richard> References: <6F861453-7F0E-4FD3-91B7-378B8ED25D7F@cmu.edu> <4d63c003926b2c19e530c107c5b141cd.squirrel@webmail.mathematik.tu-darmstadt.de> <20190216215813.GA1514@richard.richard> <20190217091522.GA3415@mathematik.tu-darmstadt.de> MIME-Version: 1.0 Content-Type: text/plain; charset="UTF-8" Content-Disposition: inline Content-Transfer-Encoding: quoted-printable In-Reply-To: <20190217091522.GA3415@mathematik.tu-darmstadt.de> User-Agent: Mutt/1.11.2 (2019-01-07) X-Original-Sender: rwilliamson62@gmail.com X-Original-Authentication-Results: gmr-mx.google.com; dkim=pass header.i=@gmail.com header.s=20161025 header.b=FBP0KB5x; spf=pass (google.com: domain of rwilliamson62@gmail.com designates 2a00:1450:4864:20::12e as permitted sender) smtp.mailfrom=rwilliamson62@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: , Dear Thomas, > but in simplicial sets the minimal and the test model structure > do coincide I don't think this is correct if by 'minimal' one means choosing the set S of monomorphisms in the donn=C3=A9e homotopique to be the empty set. Every Cisinski model structure is a Bousfield localisation of the minimal one in this sense. In particular, the quasi-categorical is, and the usual one whose fibrant objects are Kan complexes is a Bousfield localisation of that again. None of them are Quillen equivalent. One reference where this is touched on slightly is towards the beginning of Joyal's notes on quasi-categories. http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf I expect the same pattern for all cube-like test categories. Best wishes, Richard --=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. For more options, visit https://groups.google.com/d/optout.