From: Bas Spitters <b.a.w.s...@gmail.com>
To: homotopytypetheory <homotopyt...@googlegroups.com>
Subject: Internal Universes in Models of Homotopy Type Theory
Date: Wed, 24 Jan 2018 14:10:04 +0100 [thread overview]
Message-ID: <CAOoPQuSX0m7OvZFYnvprp4V2h4HvncM3TkHK5dQV3Vez=gxaug@mail.gmail.com> (raw)
Our paper has just appeared on the arxiv:
Internal Universes in Models of Homotopy Type Theory
http://arxiv.org/abs/1801.07664
Dan, Ian, Andy, Bas
---
Internal Universes in Models of Homotopy Type Theory
Authors: Daniel R. Licata, Ian Orton, Andrew M. Pitts and Bas Spitters
We show that universes of fibrations in various models of homotopy type
theory have an essentially global character: they cannot be described in the
internal language of the presheaf topos from which the model is constructed. We
get around this problem by extending the internal language with a modal
operator for expressing properties of global elements. In this setting we show
how to construct a universe that classifies the Cohen-Coquand-Huber-M\"ortberg
(CCHM) notion of fibration from their cubical sets model, starting from the
assumption that the interval is tiny - a property that the interval in cubical
sets does indeed have. This leads to a completely internal development of
models of homotopy type theory within what we call crisp type theory.
reply other threads:[~2018-01-24 13:10 UTC|newest]
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