Subject: categories: Naturality Squares and Pullbacks

From: Neil Ghani(1998-03) <[email-masked]>
Date: Tue, 24 Mar 1998 11:15:42 +0900

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

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From: "Dr. P.T. Johnstone" <[email-masked]>
Date: Wed, 25 Mar 1998 10:04:10 +0000 (GMT)

Natural transformations for which the naturality square is a pullback
are commonly called cartesian: it's not an ideal name for them, but
it's quite well established. For the question of when the unit and
multiplication of a monad on Sets are cartesian, see section 3 of
"Connected limits, familial representability and Artin glueing"
by A. Carboni and P.T. Johnstone (Math. Struct. Comp. Sci. 5 (1995),
441--459).

Peter Johnstone