Discussion of Homotopy Type Theory and Univalent Foundations
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From: Thomas Streicher <streicher@mathematik.tu-darmstadt.de>
To: homotopytypetheory@googlegroups.com
Subject: [HoTT] "type-theoretic model structures"
Date: Mon, 18 Feb 2019 11:25:42 +0100	[thread overview]
Message-ID: <20190218102542.GC28450@mathematik.tu-darmstadt.de> (raw)

I was a bit imprecise in my mail about "type-theoretic model structures".
I think there are (at least) 2 different uses of the word. The first
is as certain model structures whose fibrations give rise to a model
of type theory. In the old days these were called "categories with
display maps" which have got rebaptized by Joyal as "tribes" which is
a nice name since it's about families which interact in a certain way.

Another use seems to be for particular model structures on categories
(of presheaves) whose fibrations provide a model of type theory. Sometimes,
e.g. for simplicial and cubical sets these are minimal Cisinski model
structures where "minimal" means "fewest anodyne cofibrations", typically
generated by open box inclusions.

But not every (minimal) Cisinski model structure provides a model of
type theory and, thus, it is not at all a good idea to call them
"type-theoretic model structues".

Thomas

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             reply	other threads:[~2019-02-18 10:25 UTC|newest]

Thread overview: 6+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2019-02-18 10:25 Thomas Streicher [this message]
2019-02-18 14:32 ` Michael Shulman
2019-02-18 20:30   ` Thomas Streicher
2019-02-18 20:44     ` Michael Shulman
2019-02-18 20:57       ` Thomas Streicher
2019-02-18 21:07         ` Michael Shulman

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