Subject: categories: Barry Jay's question

From: Michael Barr <[email-masked]>
Date: Fri, 29 Jan 1999 08:08:26 -0500 (EST)

Unless I am missing something, it seems to me that for any CCC C with
finite limits and any finite category I, the functor category C^I is a
CCC.  You can also replace finite by any cardinal if you do it in both
places.  The argument is roughly this.  Let |I| denote the discrete
category with the objects of I.  The inclusion |I| --> I induces U: C^I
--> C^{|I|} and the limits imply the existence of a right adjoint R of U.
Since U also preserves limits, the cotriple (UR,e,d) preserves finite
limits.  It is easy to see that U is cotripleable and that C^{|I|} is a
CCC and, hence, so is C^I.  For define A -o RC = R(UA -o C).  For a
general object B, the line B --> URB ==> URURB is an equalizer and you can
define A -o B as the equalizer of A -o URB ==> A -o URURB.  

Michael

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