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From: "Dr. P.T. Johnstone"
Newsgroups: gmane.science.mathematics.categories
Subject: Re: Neil Ghani's question
Date: Sat, 4 Apr 1998 15:57:18 +0100 (BST)
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To: barr@triples.math.mcgill.ca (Michael Barr)
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Mike Barr's answer to Neil Ghani's question is very pretty, but
unfortunately wrong. There are many examples of cartesian natural
transformations (e.g. between functors Set --> Set) which are neither
epic nor monic: for example, the natural transformation
(-) x A --> (-) x B induced by an arbitrary map A --> B.
The mistake in Mike's proof occurs when he says
> Let E be the equalizer of u and v
> and let h^B ---> E be any arrow.
The trouble is that E could be the zero functor, so that there may
not be any arrows h^B --> E.
Peter Johnstone