From: "Jon Sterling" <jon@jonmsterling.com>
To: "Thomas Streicher" <streicher@mathematik.tu-darmstadt.de>,
"Christian Sattler" <sattler.christian@gmail.com>
Cc: "Richard Garner" <richard.garner@mq.edu.au>,
"David Roberts" <droberts.65537@gmail.com>,
"categories@mq.edu.au" <categories@mq.edu.au>
Subject: Re: Fibrewise opposite fibration
Date: Thu, 01 Feb 2024 09:43:27 +0000 [thread overview]
Message-ID: <d767a725-8c2b-46ac-8575-a77e3abe607e@app.fastmail.com> (raw)
In-Reply-To: <5aca1a1590406e68498c51bb858d89b5.squirrel@webmail.mathematik.tu-darmstadt.de>
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Hi Thomas et al,
I believe the proposed non-quotiented construction works even with a cleaving that is not split. Does anyone know if this is true? (I thought I had checked this a while ago, but now I cannot find my notes and am less confident; however, it is claimed by Von Glehn in Example 3.9 here: http://www.tac.mta.ca/tac/volumes/33/36/33-36.pdf<https://protect-au.mimecast.com/s/bbuoCwV1jpS1nNELuq9eMh?domain=tac.mta.ca>.) Glancing at it briefly, it looks rather like the identity and associativity laws follow from the higher identities governing associators and unitors in a pseudofunctor.
I agree it is inelegant to assume a splitting, but I think it is rather elegant to assume a (non-split) cleaving as property-like structure (so we do not ask that it be preserved) — Indeed, this is precisely what you get from the Chevalley criterion for fibrations in a 2-category. I think this is analogous to the way that in an internal setting we must assume chosen structures but not typically ask to preserve the choices; when we do not wish to assume chosen structures at all, we can either pass to stacks or work in a univalent / fully saturated setting.
Best,
Jon
On Wed, Jan 31, 2024, at 11:41 PM, streicher@mathematik.tu-darmstadt.de wrote:
> If we work with split fibrations and arbitrary cartesian functors between
> them we can construct the opposite of a fibration without quotienting.
> That is possible but in my eyes less elegant than the usual approach where
> one assumes that one can factorize modulo equivalence relations even if
> they are big.
>
> Thomas
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next prev parent reply other threads:[~2024-02-01 20:08 UTC|newest]
Thread overview: 33+ messages / expand[flat|nested] mbox.gz Atom feed top
2024-01-28 0:51 David Roberts
2024-01-28 11:54 ` Jon Sterling
2024-01-28 20:03 ` Thomas Streicher
2024-01-30 6:42 ` David Roberts
2024-01-31 0:35 ` Richard Garner
2024-01-31 19:31 ` Christian Sattler
2024-01-31 23:41 ` streicher
2024-02-01 4:48 ` Martin Bidlingmaier
2024-02-01 9:43 ` Jon Sterling [this message]
2024-02-01 11:06 ` Thomas Streicher
2024-02-01 11:18 ` Jon Sterling
2024-02-01 11:46 ` Thomas Streicher
[not found] ` <ZbuFZoT9b9K8o7zi@mathematik.tu-darmstadt.de>
2024-02-02 10:11 ` Thomas Streicher
2024-02-01 11:26 ` Christian Sattler
2024-02-09 0:02 ` Dusko Pavlovic
2024-02-09 1:48 ` Michael Barr, Prof.
2024-02-09 19:55 ` Dusko Pavlovic
2024-02-10 6:28 ` David Roberts
2024-02-10 8:42 ` Jon Sterling
2024-02-09 11:25 ` Fibrewise opposite fibration + computers Sergei Soloviev
2024-02-09 20:25 ` Dusko Pavlovic
2024-02-12 13:20 ` Fibrewise opposite fibration Nath Rao
2024-02-13 8:16 ` Jon Sterling
2024-02-13 10:04 ` Thomas Streicher
2024-02-13 10:56 ` Jon Sterling
2024-02-13 11:38 ` Thomas Streicher
2024-02-13 11:53 ` Jon Sterling
2024-02-13 12:18 ` Thomas Streicher
2024-02-13 16:35 ` Thomas Streicher
2024-02-23 1:50 ` Dusko Pavlovic
2024-02-23 1:52 ` Dusko Pavlovic
2024-02-23 1:42 ` Dusko Pavlovic
2024-02-26 7:31 ` Dusko Pavlovic
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