Monoidal functors generated by adjunctions, with applications to transport of structure

G.M. Kelly and Stephen Lack

Bénabou pointed out in 1963 that a functor f:B-->A with a right adjoint u induces a monoidal functor [f,u]:[A,A]-->[B,B] between the (strict) monoidal categories of endofunctors. We show that this result about adjunctions in the monoidal 2-category Cat extends to adjunctions in any right-closed monoidal 2-category V, or more generally in any 2-category A with an action * of a monoidal 2-category V admitting an adjunction A(T*A,B)~V(T,[A,B]); certainly such an adjunction exists when * is the canonical action of [A,A] on A, provided that A is complete and locally small. This result allows a concise and general treatment of the transport of algebraic structure along an equivalence.

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Steve Lack
Last modified: Mon Dec 23 17:53:13 EST 2002