Discussion of Homotopy Type Theory and Univalent Foundations
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From: Richard Williamson <rwilliamson62@gmail.com>
To: Thomas Streicher <streicher@mathematik.tu-darmstadt.de>
Cc: "Steve Awodey" <awodey@cmu.edu>,
	"Michael Shulman" <shulman@sandiego.edu>,
	"\"Anders Mörtberg\"" <andersmortberg@gmail.com>,
	"Homotopy Type Theory" <homotopytypetheory@googlegroups.com>
Subject: Re: [HoTT] A unifying cartesian cubical type theory
Date: Sun, 17 Feb 2019 14:49:41 +0100	[thread overview]
Message-ID: <20190217134941.GA2100@richard.richard> (raw)
In-Reply-To: <20190217091522.GA3415@mathematik.tu-darmstadt.de>

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 donnée 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

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  reply	other threads:[~2019-02-17 13:49 UTC|newest]

Thread overview: 18+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2019-02-14 19:04 Anders Mortberg
2019-02-14 20:06 ` Andrew Pitts
2019-02-15 15:38   ` Anders Mörtberg
2019-02-15  8:16 ` Bas Spitters
2019-02-15 16:32   ` Anders Mörtberg
2019-02-16  0:01     ` Michael Shulman
2019-02-16  0:14       ` Steve Awodey
2019-02-16 12:30         ` streicher
2019-02-16 19:51           ` Thomas Streicher
2019-02-16 22:27             ` Steve Awodey
2019-02-17  9:43               ` Thomas Streicher
2019-02-17 14:14                 ` Licata, Dan
2019-02-16 21:58           ` Richard Williamson
2019-02-17  9:15             ` Thomas Streicher
2019-02-17 13:49               ` Richard Williamson [this message]
2019-02-18 14:05 ` [HoTT] " Andrew Swan
2019-02-18 15:31   ` Anders Mörtberg
2019-06-16 16:04     ` Anders Mörtberg

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