From: jas...@cs.washington.edu
To: Homotopy Type Theory <HomotopyT...@googlegroups.com>
Subject: Definition of semantic composition in CCHM cubical type theory
Date: Mon, 23 Oct 2017 21:10:56 -0700 (PDT) [thread overview]
Message-ID: <a4477cb1-28f0-4d7a-9931-8f8fefc3dc03@googlegroups.com> (raw)
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Hello,
I've been reading through Cubical Type Theory: a constructive
interpretation of the univalence axiom
<https://arxiv.org/pdf/1611.02108.pdf>, trying to understand the model
constructed there.
I got confused about a few details, and I hoped someone on this list could
help clarify them.
The semantic composition structure (p.19) takes as an argument ρ ∈ Γ(I, i).
For f : J → I (a map I to dM(J)) and fresh j, we have restrictions
using ρ(f, i = j), which I am reading as the restriction of ρ along g : J,
j -> I, i (a map I, i to dm(J, j)) that takes I by f (with the inclusion of
dM(J) into dM(J, j)) and takes i to j in dM(J, j). That part I think I can
follow.
Then down in the interpretation from syntax (bottom of p.22, top of p.23),
we define the interpretation of comp as
[[Γ; comp i A [ϕ → u] a0]]ρ = comp El [[Γ,i:II;A]] (I, j, ρ′ , [[Γ; ϕ]]ρ,
[[Γ, ϕ, i: II; u]]ρ′ , [[Γ; a0]]ρ)
using ρ′ = (ρ s_j , i = j), supposedly in Γ(I, j). We have ρ ∈ [[Γ]](I)
and s_j : I, j → I is the inclusion.
I don't understand how to read ρ′ = (ρ s_j , i = j). Where does i come
from, and what does it mean to set ρ′ to be a pair of ρ s_j and the
substitution i = j? I think, we have El [[Γ,i:II;A]] in Ty([[Γ,i:II]]), so
we expect ρ′ in [[Γ,i:II]](I, i), so ρ′ is a pair of [[Γ]](I, i) and II(I,
i), so i guess we have ρ s_j in [[Γ]](I, i) and i = j in II(I, i)?
As I was writing this I think I made some progress, but I still don't
understand what it means to have i = j in II(I, i).
Does it have anything to do with the other place ρ′ is used, in [[Γ, ϕ, i:
II; u]]ρ′ ?
Thanks in advance for any clarification,
- Jasper Hugunin
p.s. I've been studying the CCHM model with an eye towards giving the comp
operation a (hopefully fibrant) type. The only place I've seen (semantic)
comp used is in the translation from syntax, which would mean that ρ always
has the form (ρ s_j , i = j), and so possibly we could replace comp with
something like "a composition structure for A in Ty(Γ . II) is, for all I
and ρ in Γ(I) (which looks like a cubical type) ... (with ρ replaced with
(ρ s_j , i = j) in the body).
I'm still just taking stabs in the dark in the area of model theory, so any
thoughts on the feasibility of this would also be appreciated (if it isn't
obviously ridiculous).
The hope being that we could then define operations like fill and contr
inside the theory, and possibly making progress towards a
higher-inductive-recursive universe in the style of
https://homotopytypetheory.org/2014/06/08/hiru-tdd/.
I don't understand Glue types very well, and they seem to be the most
complicated part of the system, so I've been pondering ways to do without
them.
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next reply other threads:[~2017-10-24 4:10 UTC|newest]
Thread overview: 2+ messages / expand[flat|nested] mbox.gz Atom feed top
2017-10-24 4:10 jas... [this message]
2017-10-24 12:40 ` [HoTT] " Simon Huber
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