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[84.202.247.109]) by smtp.gmail.com with ESMTPSA id o11-v6sm1083072lji.0.2018.06.14.11.39.59 (version=TLS1_2 cipher=ECDHE-RSA-CHACHA20-POLY1305 bits=256/256); Thu, 14 Jun 2018 11:40:00 -0700 (PDT) Date: Thu, 14 Jun 2018 20:39:59 +0200 From: Richard Williamson To: Thierry Coquand Cc: Homotopy Theory Subject: Re: [HoTT] Quillen model structure Message-ID: <20180614183959.GA1401@richard.richard> References: MIME-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 8bit In-Reply-To: User-Agent: Mutt/1.9.5 (2018-04-13) Dear Thierry, Forgive me for not quite having an overview of exactly what has been done. My interest here is in how the model structure you have constructed compares with Cisinski's work. For example, I would expect that the category of cartesian cubical sets is a test category. Indeed, I would expect that more or less exactly the same technique as in 8.4 of Les préfaisceaux comme ... goes through to prove this. And I would expect that one can put a model structure on in the same way. Now, I believe I saw in some slides that that your model structure coincides with the Cisinski one. But this does not seem compatible with my expectations above, because then it is certainly Quillen equivalent to the Serre model structure on topological spaces (or whatever). Thus my questions: 1) Have I misinterpreted the slides or some other aspect of your work? 2) If not, where exactly do Cisinski's techniques fail? Best wishes, Richard