## Full completeness for categories enriched in commutative monoids

### Robin Cockett – 29 April 1998

Work with S. Soloviev and M. Hyland: Often full-completeness, that is establishing that a representation functor is full, is regarded as something which is more difficult to obtain than faithfulness. This has been particulaly the case for linear logics. Blute and Scott in their paper on Lauchli semantics used a representation in a functor category so that here fullness reduced to the problem of establishing the form of the natural transformations between various functors. The main result of this talk can be stated as: Prop: In a CM-enriched monoidal closed category in which the unit generates every natural transformation between tensor powers is a linear combination of formal transformations. The result covers SETS, Mod_R (R commutative ring), Span(X) (X etxtensive).

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