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Coexponentiability and Projectivity: Rigs, Rings, and Quantales

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S.B. Niefield and R.J. Wood

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.

Keywords:
monoidal category, projective module, rig

2010 MSC:
18A40, 18D15, 13C10, 16Y60

*Theory and Applications of Categories,*
Vol. 32, 2017,
No. 36, pp 1222-1228.

Published 2017-09-11.

http://www.tac.mta.ca/tac/volumes/32/36/32-36.pdf

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