From: Thomas Streicher <streicher@mathematik.tu-darmstadt.de>
To: "Eduardo J. Dubuc" <edubuc@dm.uba.ar>
Cc: Paul Blain Levy <P.B.Levy@cs.bham.ac.uk>,
"Categories list\" <Categories list>" <categories@mta.ca>
Subject: Re: V-included categories
Date: Thu, 4 Jan 2018 22:20:47 +0100 [thread overview]
Message-ID: <E1eXUg3-0006gH-2B@mlist.mta.ca> (raw)
In-Reply-To: <5c1ec079-335b-5609-9cb7-ae4e519f6716@dm.uba.ar>
Dear Eduardo,
thanks for your attempts to clarify the situation!
> Aparently the universes are not closed in the sense that:
>
> 1) X ~ Y, Y belongs U ===> X belongs U
>
> which is currently accepted (as we see in Thomas posting above) in the naive
> practice of category theory with universes.
I certainly did not incline this (despite my partial involvement into
HoTT :-))
I rather tacitly assumed that a locally U-small means that all homsets
are elements of U. This I find natural though I read that Grothendieck
and Verdier formulated it much more liberally.
However, using AC every locally U-small category is isomorphic to some
category where all hom-sets are elements of U. Moreover, every category
equivalent to such a category is locally U-small in the liberal sense
of Grothendieck and Verdier.
Now, for a category C which is locally U-small in the restricted sense
[C^op,U] is locally U-small in the restricted sense as shown by Paul's
argument.
But if C' is locally U-small in the liberal sense then [C'^op,U] \cong
[C^op,U] and the latter is locally U-small in the restricted sense and
thus [C'^op,U] is locally U-small in the liberal sense.
Thus, I think in SGA4 they made a mistake. No question that both guys
were great mathematicians but that doesn't prevent them from making
little mistakes in logic.
Thomas
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
next prev parent reply other threads:[~2018-01-04 21:20 UTC|newest]
Thread overview: 11+ messages / expand[flat|nested] mbox.gz Atom feed top
2018-01-01 13:10 Paul Blain Levy
2018-01-01 18:28 ` Paul Blain Levy
2018-01-01 21:14 ` Eduardo Julio Dubuc
[not found] ` <e962e844-fa2d-5a56-e3e7-be308a483c12@dm.uba.ar>
2018-01-01 22:46 ` Paul Blain Levy
2018-01-02 18:40 ` rosicky
2018-01-02 19:15 ` Michael Shulman
[not found] ` <03876a66-f7ee-a161-091c-32944a0d8556@dm.uba.ar>
2018-01-03 6:44 ` Paul Blain Levy
[not found] ` <E1eWtjA-0006aj-48@mlist.mta.ca>
[not found] ` <918B0A9E-DFD0-4033-AB7A-1A8A364DB8A9@cs.bham.ac.uk>
[not found] ` <20180104110046.GA24344@mathematik.tu-darmstadt.de>
2018-01-04 20:47 ` Steve Vickers
[not found] ` <5c1ec079-335b-5609-9cb7-ae4e519f6716@dm.uba.ar>
2018-01-04 21:20 ` Thomas Streicher [this message]
[not found] ` <C16D1DBE-89F3-42EC-86A0-B69B5DBF5713@cs.bham.ac.uk>
2018-01-04 21:30 ` Thomas Streicher
[not found] ` <E1eWtkf-0006d1-O1@mlist.mta.ca>
2018-01-04 23:28 ` Richard Williamson
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