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

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

B ́enabou pointed out in 1963 that a pair f ⊣ u : A → B of adjoint functors 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 gen- erally 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, pro- vided that A is complete and locally small. This result allows a concise and general treatment of the transport of algebraic structure along an equivalence.
Original languageEnglish
Title of host publicationGalois theory, Hopf algebras, and semiabelian categories
EditorsGeorge Janelidze, Bodo Pareigis, Walter Tholen
Place of PublicationProvidence, R.I.
PublisherAmerican Mathematical Society
Pages319-340
ISBN (Print)9780821832905 , 9780821871478
Publication statusPublished - 2004
Externally publishedYes

Publication series

NameFields Institute Communications
PublisherAmerican Mathematical Society
Volume43

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  • Cite this

    Lack, S. (2004). Monoidal functors generated by adjunctions, with applications to transport of structure. In G. Janelidze, B. Pareigis, & W. Tholen (Eds.), Galois theory, Hopf algebras, and semiabelian categories (pp. 319-340). (Fields Institute Communications; Vol. 43). Providence, R.I.: American Mathematical Society.