J.R.B. Cockett and R.A.G. Seely \Weakly distributive categories", Journal of Pure and Applied Algebra 114 (1997) 133-173. (Updated version available on http://www.math.mcgill.ca/~rags.)  Home/Search   Document Not in Database   Summary   Related Articles

....preferring for the coproduct sum . We wish to thank the anonymous referee for several helpful remarks and suggestions. 1 The core of a MIX category 1.1 Preliminaries 1.1. 1 Linearly distributive categories For the full de nition of a linearly distributive category, we refer the reader to [CS92,CS92j,BCST96] (where the term weakly distributive category is used) Brie y, a linearly distributive category is a category with two tensors ; and two strength natural transformations, making each tensor strong (respectively costrong) with respect to the other. These strength transformations will be ....

.... D A (B (C D) R R 1 D = 1 A L L (A (B C) D a A ( B C) D) The third diagram above is the most controversial, as it fails to be true in distributive categories, and is why distributive categories cannot be linearly distributive (unless they are posetal, see [CS92j]) However, it is a direct consequence of the logical interpretation of linearly distributive categories: it corresponds to a natural (and necessary) cut elimination step. For in essence, ignoring associativity, L L ; R R 1 = B; C B C A B A; B A B; C A; B C C D C; D A ....

J.R.B. Cockett and R.A.G. Seely \Weakly distributive categories", Journal of Pure and Applied Algebra 114 (1997) 133-173. (Updated version available on http://www.math.mcgill.ca/~rags.)

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