Consider a symmetric monoidal closed category C with finite coproducts. Write -o for the exponentiation. Are there non-trivial examples of objects A of C such that the functor _-oA maps every tensor product X \tensor Y to (the object part of) a coproduct X-oA + Y-oA (in C)? Such an A would be a nice candidate for an "answers object" for "continuation semantics" in computer science. -- J"urgen -- J"urgen Koslowski | If I don't see you no more in this world | I meet you in the next world | and don't be late! koslowj@math.ksu.edu | Jimi Hendrix (Voodoo Chile) +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
From: koslowj@math.ksu.edu (Juergen Koslowski) Date: Thu, 29 Jul 93 9:53:18 CDT Consider a symmetric monoidal closed category C with finite coproducts. Write -o for the exponentiation. Are there non-trivial examples of objects A of C such that the functor _-oA maps every tensor product X \tensor Y to (the object part of) a coproduct X-oA + Y-oA (in C)? Such an A would be a nice candidate for an "answers object" for "continuation semantics" in computer science. X@Y -o A consists of the bilinear maps to A, those maps that can be viewed, loosely speaking, BOTH as an X-indexed FAMILY of maps of Y-oA AND a Y-indexed family of maps of X-oA. In contrast X-oA + Y-oA consists of those maps to A coming EITHER from X OR Y alone. I can't imagine how continuation semantics could usefully connect up either BOTH-AND or FAMILY with EITHER-OR. -- Vaughan Pratt
Vaughan Pratt's reply confirms my suspicion that for a symmetric monoidal closed category with tensor @ and exponential -o the Keisli category for the (strong) monad induced by the functor T that maps X to (X -o A) -o A (A fixed) is highly unlikely to be cartesian closed AND non-trivial. This probably is a well-known result. I'd appreciate pointers to the literature. -- J"urgen Koslowski | If I don't see you no more in this world | I meet you in the next world | and don't be late! koslowj@math.ksu.edu | Jimi Hendrix (Voodoo Chile) +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
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