Subject: categories: Artin glueing et cetera; comments of Barr, Jay, Crole, and Johnstone

From: [email-masked] (Max Kelly)
Date: Mon, 1 Feb 1999 17:55:01 +1100 (EST)

Barry Jay recently sought comments on A^2 is cartesian closed when A is so,
where 2 is the arrow-category. Roy Crole and Peter Johnstone both pointed
out that this was a special case of Artin glueing.

Is there not a more general fact? The forgetful 2-functor from cartesian-
closed categories to categories is fairly easily seen to be monadic - the
monadicity is well-known for categories with finite products or finite
limits, and adding the right adjoint to (- x A) is purely equational. But
for any 2-monad T on K, T-Alg has all lax and pseudo limits - indeed all
flexible limits - formed as in K; see [Blackwell, Kelly, and Power, Two-
-dimensional monad theory, J. Pure Appl. Algebra 50 (1989), 1-41]. Artin
glueing is the case of the lax limit of a map, if I remember correctly its
meaning. But (-)^X for any category X is also a flexible limit, so that
Jay's observation is included directly without recourse to Artin glueing.

Then, again, Michael Barr on 30 Jan asserts that A^X is cartesian closed if
A is so when X is finite and A has finite limits. Surely such conditions are
not needed - A^X is cartesian closed when A is so.

Am I missing something?

Max Kelly