Date: Sat, 14 Dec 1996 16:33:22 -0400 (AST) Subject: Cartesian Closed arrow categories Date: Fri, 13 Dec 96 16:01:11 EST From: Kathryn_Van_Stone@POP.CS.CMU.EDU Does anyone what conditions are necessary in a category C for its arrow category to be Cartesian Closed. I have come up with a solution, but it is rather mechanical. Thanks, Kathy Van Stone kvs@cs.cmu.edu Date: Sun, 15 Dec 1996 11:02:08 -0400 (AST) Subject: Cartesian Closed arrow categories Date: Mon, 16 Dec 1996 11:09:14 +1100 From: Ross Street Question from Kathy Van Stone: >Does anyone what conditions are necessary in a category C for its >arrow category to be Cartesian Closed. Let's agree that a category A is cartesian closed when it has finite products and each functor a x - : A --> A has a right adjoint [a,-]. By examining the various adjunctions between A and the arrow category A' of A, or otherwise, we see that: A' is cartesian closed if and only if A is cartesian closed and, for all arrows f : a --> b, g : c --> d in A, the arrows [f,1] : [b,d] --> [a,d], [1,g] : [a,c] --> [a,d] admit a pullback P(f,g) in A. The internal hom of f, g as objects of A' is then P(f,g) --> [b,d]. Perhaps this is the mechanical answer you had already. Best regards, Ross PS: Our federal minister who has slashed funding to Australian universities is Amanda Vanstone. Would that she were interested in CCCs. Date: Sun, 15 Dec 1996 11:02:58 -0400 (AST) Subject: Re: Cartesian Closed arrow categories Date: Sun, 15 Dec 96 14:53 GMT From: Dr. P.T. Johnstone Assuming that by "arrow category" you mean the functor category [2,C], this is a special case of the question of when a category obtained by Artin glueing is cartesian closed. This (and related questions) was dealt with in my joint paper with Aurelio Carboni (Math. Struct. Comp. Sci. 5 (1995), 441--459). Peter Johnstone