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From: Michael Abbott <Michael@rcp.co.uk>
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Subject: Three questions about fibrations
Date: Thu, 15 Jun 2000 18:02:23 +0100
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I am wondering if anyone can give references for three remarks Wesley Phoa
makes in the chapter on fibrations of his paper "An introduction to
fibrations, topos theory, the effective topos and modest sets".  


1. Essentially Algebraic Theories

In the footnote on page 7 Phoa comments: 
	"[fibrations] are the models for a first-order, 'essentially
algebraic' theory".

I'm not quite sure what he means here, and this sounds like it must be a
standard and well known connection.  I'd be glad of a reference.


2. Splitting Fibrations

At the bottom of page 14 Phoa observes: 
	"Every fibration .. is equivalent to a split fibration (there is an
elegant proof due to John Power).  However, it's not clear how to extend
this result to more complicated structures.  This is the coherence problem
..."

Now I know that any fibration is equivalent to a split fibration via the
"fibred Yoneda lemma" (Borceux, "Handbook of Categorical Algebra 2", 8.2.7
and Jacobs, "Categorical Logic and Type Theory"), but I don't think that's
the only splitting available, and I'm not sure that this correspondence
helps very with coherence questions.
	I am aware of another, different, equivalent splitting, and I wonder
if there are any references.  In particular, can anyone guess what reference
by John Power Phoa was referring to?
	In particular, I'd be very interested in any other observations on
the "coherence problem".

3. Generalising the Definition

In the footnote on page 12, in reference to the definition of a fibration,
Phoa says: 
	"If one really wants to take .. 2-categorical issues seriously, one
needs .. a more sophisticated definition of 'fibration'."

I can make some promising looking guesses.  Any references?


Many thanks.
	Michael Abbott