From: "Prof. Peter Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>
To: categories@mta.ca
Subject: Re: Can we construct free semi-lattice from free dist. lattice?
Date: Tue, 27 May 2003 21:52:21 +0100 (BST) [thread overview]
Message-ID: <Pine.LNX.3.96.1030527214517.27931B-100000@siskin.dpmms.cam.ac.uk> (raw)
In-Reply-To: <Law10-F112duSIDtBSD0002e6f0@hotmail.com>
On Tue, 27 May 2003, Christopher Townsend wrote:
> Dear All,
> I have a problem which I had thought very specialised, but actually can be
> stated very generally: -
>
> I have a category C with finite limits, and so I also have a category
> DLat(C) of distributive lattices which, lets say, has coequalizers. If free
> distibrutive lattices can be constructed (i.e. if there exists F:C->DLat(C)
> left adjoint to the forgetful functor) then do free semilattices exist?
>
I suspect the answer is no, for irritatingly trivial reasons.
If C is a pointed category (i.e. has a zero object), then any internal
distributive lattice in C has its top and bottom elements equal, and
so is degenerate (i.e. isomorphic to the terminal object 1). Hence
the free-distributive-lattice functor exists, and is the constant
functor with value 1. But I'm sure there must be examples of pointed,
finitely complete categories which don't have a free-semilattice
functor (though I have to admit I don't have one at my fingertips).
Peter Johnstone
prev parent reply other threads:[~2003-05-27 20:52 UTC|newest]
Thread overview: 2+ messages / expand[flat|nested] mbox.gz Atom feed top
2003-05-27 8:52 Christopher Townsend
2003-05-27 20:52 ` Prof. Peter Johnstone [this message]
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