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**1 - 2**of**2**### Strong Categorical Datatypes I

, 1991

"... An endofunctor of a cartesian closed category is often called strong if it is enriched over the exponential. Equivalently this strength can be provided as a natural transformation ` A;X : F (A) \Theta X \Gamma! F (A \Theta X) satisfying some elementary coherence conditions. This latter formulation ..."

Abstract
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An endofunctor of a cartesian closed category is often called strong if it is enriched over the exponential. Equivalently this strength can be provided as a natural transformation ` A;X : F (A) \Theta X \Gamma! F (A \Theta X) satisfying some elementary coherence conditions. This latter formulation does not require exponentials, relies only on the presence of an X--action over an X--strong category, and thereby provides a first--order viewpoint of strength. The 2-category of X--strong categories is not finitely complete. It particularly lacks many standard constructions including the Eilenberg--Moore construction. Thankfully, the suggestion --- attributed to Plotkin by Moggi --- that strength can also be equivalently framed in terms of fibrations using projections to X--objects as display maps can be fully realized. The equivalence can be portrayed as an embedding of X--strong categories into the 2--category of X--indexed categories or split fibrations over X. This embedding can be u...