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From: Alex Simpson <als@inf.ed.ac.uk>
To: categories@mta.ca
Subject: Re: Inductive datatypes in toposes
Date: Tue, 04 Mar 2003 01:49:36 +0000 (GMT)	[thread overview]
Message-ID: <1046742576.3e640630a95fb@mail.inf.ed.ac.uk> (raw)
In-Reply-To: <200303031730.SAA17354@fb04209.mathematik.tu-darmstadt.de>


Lutz Schroeder asked:

> > is there a good reference for the construction of inductive
> datatypes
> > (lists, trees etc.) in a topos with nno (assuming that my guess that
> > such a construction is indeed possible is correct)?

Thomas Streicher replied:

> It is not clear to me to which extent one can characterise those
> inductive
> types that can be reduced to inductively defined predicates in HAH.
> However, one knows that assuming W-types (`a la Martin-Loef) one can
> reduce
> most inductive types to W-types in extensional type theory. As far as
> I
> understand that's the reason why Moerdijk and Palmgren introduced lccc's
> with
> W-types as sort of ``predicative toposes''.

Just to point out that "assuming W-types" here is no extra assumption.
Moerdijk and Palmgren observe that every elementary topos with nno
has W-types. This observation more than answers Schroeder's
original question affirmatively. See Moerdijk and Palmgren,
"Well-founded trees in categories", APAL 104, 2000, for the
observation. I'm not sure whether they include the details (I
don't have the paper to hand), but it's not a hard exercise.

Alex Simpson

Alex Simpson, LFCS, Division of Informatics, Univ. of Edinburgh
Email: Alex.Simpson@ed.ac.uk           Tel: +44 (0)131 650 5113
Web: http://www.dcs.ed.ac.uk/home/als  Fax: +44 (0)131 667 7209






      reply	other threads:[~2003-03-04  1:49 UTC|newest]

Thread overview: 6+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2003-02-25 19:55 projective algebras Andrei Popescu
2003-02-26 10:24 ` Mamuka Jibladze
2003-03-03  9:43   ` Inductive datatypes in toposes Lutz Schroeder
2003-03-03 16:23     ` Luigi Santocanale
2003-03-03 17:30     ` Thomas Streicher
2003-03-04  1:49       ` Alex Simpson [this message]

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