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From: maxk@maths.usyd.edu.au (Max Kelly)
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Subject: Re: Naturality Squares and Pullbacks
Date: Wed, 25 Mar 1998 18:23:46 +1100 (EST)
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In response to the question of Neil Ghani, namely
	
	
	A natural transformation is an indexed family of arrows such that a
	certain diagram commutes. One could require a stronger condition,
	namely that the said diagram is a pullback. What would such a
	transformation be called? I'm sure I've seen this in the literature
	before but I cant remember where. Pointers?
	
	This problem arose in the context of finitary monads where 
	T(X) is the derived operations over a set X for some signature. 
	The naturality square for the unit turns out to be a pullback. 
	This then implies that the unit of the monad is a monic - 
	presumably this is a result in the literature somewhere. 
	Again, pointers?
	
	Neil Ghani
	
	

this phenomenon is now quite well recognised. some call such natural
transformations "cartesian", while others use Robin Cockett's term "shapely".
For my own contribution to the subject, see [G.M. Kelly, On clubs and 
data-type constructors, in applications of Categories to Computer Science
(Proc. LMS Symposium, Durham 1991), Cambridge Univ. Press 1992, 163-190].

Max Kelly.