Applied Categorical Structures

, Volume 26, Issue 3, pp 559–596 | Cite as

Commutants for Enriched Algebraic Theories and Monads

  • Rory B. B. Lucyshyn-Wright


We define and study a notion of commutant for \(\mathscr {V}\)-enriched \({\mathscr {J}}\)-algebraic theories for a system of arities \({\mathscr {J}}\), recovering the usual notion of commutant or centralizer of a subring as a special case alongside Wraith’s notion of commutant for Lawvere theories as well as a notion of commutant for \(\mathscr {V}\)-monads on a symmetric monoidal closed category \(\mathscr {V}\). This entails a thorough study of commutation and Kronecker products of operations in \({\mathscr {J}}\)-theories. In view of the equivalence between \({\mathscr {J}}\)-theories and \({\mathscr {J}}\)-ary monads we reconcile this notion of commutation with Kock’s notion of commutation of cospans of monads and, in particular, the notion of commutative monad. We obtain notions of \({\mathscr {J}}\)-ary commutant and absolute commutant for \({\mathscr {J}}\)-ary monads, and we show that for finitary monads on \(\text {Set}\) the resulting notions of finitary commutant and absolute commutant coincide. We examine the relation of the notion of commutant to both the notion of codensity monad and the notion of algebraic structure in the sense of Lawvere.


Commutant Commutation Commutative monad Algebraic theory Commutative algebraic theory Universal algebra Monad Enriched category theory 

Mathematics Subject Classification

18C10 18C15 18C20 18C05 18D20 18D15 08A99 08B99 08C05 08C99 03C05 18D35 18D10 18D25 18A35 


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© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Mount Allison UniversitySackvilleCanada

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