Geodesic
Coexponentiability and Projectivity: Rigs, Rings, and Quantales
Theory and applications of categories, Tome 32 (2017), pp. 1222-1228.

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We show that a commutative monoid A is coexponentiable in CMon(V) if and only if $-\otimes A : V \to V$ has a left adjoint, when V is a cocomplete symmetric monoidal closed category with finite biproducts and in which every object is a quotient of a free. Using a general characterization of the latter, we show that an algebra over a rig or ring R is coexponentiable if and only if it is finitely generated and projective as an R-module. Omitting the finiteness condition, the same result (and proof) is obtained for algebras over a quantale.
Publié le :
Classification : 18A40, 18D15, 13C10, 16Y60
Keywords: monoidal category, projective module, rig 
@article{TAC_2017_32_a35,
     author = {S.B. Niefield and R.J. Wood},
     title = {Coexponentiability and  {Projectivity:} {Rigs,} {Rings,} and {Quantales}},
     journal = {Theory and applications of categories},
     pages = {1222--1228},
     publisher = {mathdoc},
     volume = {32},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2017_32_a35/}
}
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S.B. Niefield; R.J. Wood. Coexponentiability and  Projectivity: Rigs, Rings, and Quantales. Theory and applications of categories, Tome 32 (2017), pp. 1222-1228. http://geodesic.mathdoc.fr/item/TAC_2017_32_a35/