From: "Martín Hötzel Escardó" <"escardo..."@gmail.com>
To: Homotopy Type Theory <HomotopyT...@googlegroups.com>
Subject: Re: [HoTT] Yet another characterization of univalence
Date: Wed, 20 Dec 2017 12:46:39 -0800 (PST) [thread overview]
Message-ID: <7f3e4d3e-9204-4ca9-9a8c-7c8c5ed7936b@googlegroups.com> (raw)
In-Reply-To: <CAM8RHpFE+dWKLKKYf6Nb72s7epxMoR_5q_gJsAc3rioMMMzmPg@mail.gmail.com>
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On Tuesday, 19 December 2017 03:36:51 UTC, Gershom B wrote:
On December 18, 2017 at 5:58:22 PM, Martín Hötzel Escardó
(escar...@gmail.com) wrote:
> >
> > A type D is called injective if for any embedding j:X→Y, every
> > function f:X→D extends to a map f':Y→D along j.
>
> Here is a question: given a local operator (Lawvere-Tierny topology)
> j, an object D is a sheaf if for any j-dense subobject X >-> Y, every
> function f : X -> D extends to a map f’ : Y -> D. Is there a way in
> which we can view injective types as sheafs in some appropriate sense?
I don't think so, because we don't have the required uniqueness
of the sheaf condition here.
In general, if a type D is injective, many extensions of any
f:X→D along an embedding j:X→Y exist.
In the file
http://www.cs.bham.ac.uk/~mhe/agda-new/InjectiveTypes.html,
two canonical extensions are shown to exist, a minimal one f\j
using Σ, and a maximal f/j one using Π.
In fact they are left and right Kan extensions, in the sense that we
have equivalences
Nat f (g ∘ j) ≃ Nat f∖j g
and
Nat g f/j ≃ Nat (g ∘ j) f
of natural transformations involving the "presheaves" g : Y → U and
f\j, f/j : X → U.
Although these two extensions are canonical in the above sense, they are not unique.
Martin
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next prev parent reply other threads:[~2017-12-20 20:46 UTC|newest]
Thread overview: 16+ messages / expand[flat|nested] mbox.gz Atom feed top
2017-11-17 23:53 Martín Hötzel Escardó
2017-11-20 3:54 ` [HoTT] " y
2017-11-24 12:21 ` Martín Hötzel Escardó
2017-11-24 19:11 ` Martín Hötzel Escardó
2017-11-28 9:40 ` Andrej Bauer
2017-11-29 21:12 ` Evan Cavallo
[not found] ` <204e382a-efcf-cb13-006f-47fdbadd99a5@cs.bham.ac.uk>
2017-11-29 22:15 ` Evan Cavallo
2017-11-29 22:16 ` Martín Hötzel Escardó
2017-12-01 14:49 ` Martin Escardo
2017-12-01 14:53 ` Martín Hötzel Escardó
2017-12-09 0:27 ` Martín Hötzel Escardó
2017-12-18 22:58 ` Martín Hötzel Escardó
2017-12-19 3:36 ` Gershom B
2017-12-20 20:46 ` Martín Hötzel Escardó [this message]
2017-11-24 23:12 ` Martín Hötzel Escardó
2017-11-24 23:28 ` Martín Hötzel Escardó
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