From: Richard Garner <rhgg2@hermes.cam.ac.uk>
To: categories@mta.ca
Subject: Re: Cartesian closed without products
Date: Mon, 7 Jan 2008 14:09:54 +0000 (GMT) [thread overview]
Message-ID: <E1JBxLm-0001wG-Eq@mailserv.mta.ca> (raw)
A minor correction:
> A small category C has finite products just when the
> representables in [C^op, Set] are closed under finite
> products. It is cartesian closed just when the
> representables are closed under finite products and
> internal hom.
>
> It seems natural, therefore, to consider a notion of
> "cartesian closedness without finite products": categories
> in which the representables are closed in [C^op, Set] under
> internal hom but not necessarily under finite products.
> This amounts to giving, for each pair of objects X and Y,
> an object [X, Y] together with a universal natural
> transformation C(-, [X, Y]) x C(-, X) -> C(-, Y). Such
> categories will be closed in the sense of Eilenberg-Kelly
> without necessarily being monoidal:
On reflection, this bit isn't necessarily true, since
EK-closed requires the unit object to be representable as
well.
Richard
next reply other threads:[~2008-01-07 14:09 UTC|newest]
Thread overview: 6+ messages / expand[flat|nested] mbox.gz Atom feed top
2008-01-07 14:09 Richard Garner [this message]
-- strict thread matches above, loose matches on Subject: below --
2008-01-09 2:29 Toby Bartels
2008-01-09 1:36 Peter Selinger
2008-01-08 21:29 Ronnie
2008-01-07 21:35 Toby Bartels
2008-01-07 11:08 Richard Garner
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