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Commutants for Enriched Algebraic Theories and Monads

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Abstract

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.

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References

  1. Birkhoff, G.: Selected Papers on Algebra and Topology. Birkhäuser Boston Inc, Boston, MA (1987)

    MATH  Google Scholar 

  2. Borceux, F., Day, B.: Universal algebra in a closed category. J. Pure Appl. Algebra 16(2), 133–147 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  3. Dlab, V., Ringel, C.M.: Rings with the double centralizer property. J. Algebra 22, 480–501 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dubuc, E.J.: Enriched semantics-structure (meta) adjointness. Rev. Un. Mat. Argentina 25, 5–26 (1970)

    MathSciNet  MATH  Google Scholar 

  5. Dubuc, E.J.: Kan Extensions in Enriched Category Theory, Lecture Notes in Mathematics, vol. 145. Springer, Berlin (1970)

    Book  MATH  Google Scholar 

  6. Eilenberg, S., Kelly, G.M.: Closed categories. In: Proceedings of Conference Categorical Algebra (La Jolla, Calif., 1965), Springer, pp. 421–562 (1966)

  7. Garner, R., López Franco, I.: Commutativity, J. Pure Appl. Algebra (2016), 1707–1751

  8. Kelly, G.M.: Monomorphisms, epimorphisms, and pull-backs. J. Austral. Math. Soc. 9, 124–142 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kelly, G.M.: Structures defined by finite limits in the enriched context. I. Cahiers Topologie Géom. Différentielle 23(1), 3–42 (1982)

    MathSciNet  MATH  Google Scholar 

  10. Kelly, G.M.: Basic concepts of enriched category theory, Repr. Theory Appl. Categ. (2005), no. 10, Reprint of the 1982 original [Cambridge Univ. Press]

  11. Kock, A.: Continuous Yoneda representation of a small category, Aarhus University Preprint, (1966)

  12. Kock, A.: Monads on symmetric monoidal closed categories. Arch. Math. (Basel) 21, 1–10 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kock, A.: On double dualization monads. Math. Scand. 27(1970), 151–165 (1971)

    MathSciNet  MATH  Google Scholar 

  14. Lawvere, F.W.: Functorial semantics of algebraic theories, Dissertation, Columbia University, New York. Available. In: Repr. Theory Appl. Categ. 5, 1963 (2004)

  15. Linton, F.E.J.: Autonomous equational categories. J. Math. Mech. 15, 637–642 (1966)

    MathSciNet  MATH  Google Scholar 

  16. Linton, F.E.J.: Some aspects of equational categories. In: Proceeding of Conference Categorical Algebra (La Jolla, Calif., 1965), Springer, pp. 84–94 (1966)

  17. Linton, F.E.J.: An outline of functorial semantics, Sem. on Triples and Categorical Homology Theory (ETH, Zürich, 1966/67), Springer, pp. 7–52 (1969)

  18. Lucyshyn-Wright, R.B.B.: Riesz-Schwartz extensive quantities and vector-valued integration in closed categories, Ph.D. thesis, York University, (2013), arXiv:1307.8088

  19. Lucyshyn-Wright, R.B.B.: A general theory of measure and distribution monads founded on the notion of commutant of a subtheory, Talk at Category Theory 2015. Aveiro, Portugal (2015)

  20. Lucyshyn-Wright, R.B.B.: Enriched algebraic theories and monads for a system of arities. Theory Appl. Categ. 31, 101–137 (2016)

    MathSciNet  MATH  Google Scholar 

  21. Lucyshyn-Wright, R.B.B.: Convex spaces, affine spaces, and commutants for algebraic theories. Appl. Categor. Struct. (2017). doi:10.1007/s10485-017-9496-9

  22. Lucyshyn-Wright, R.B.B.: Functional distribution monads in functional-analytic contexts. Adv. Math. (2017). http://dx.doi.org/10.1016/j.aim.2017.09.027

  23. Power, J.: Enriched Lawvere theories. Theory Appl. Categ. 6, 83–93 (1999)

    MathSciNet  MATH  Google Scholar 

  24. Wraith, G.C.: Algebraic theories, Lectures Autumn 1969. Lecture Notes Series, No. 22, Matematisk Institut, Aarhus Universitet, Aarhus, (1970) (Revised version 1975)

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Authors and Affiliations

  1. Mount Allison University, Sackville, New Brunswick, Canada

    Rory B. B. Lucyshyn-Wright

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  1. Rory B. B. Lucyshyn-Wright
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Correspondence to Rory B. B. Lucyshyn-Wright.

Additional information

Communicated by J. Rosický.

The author gratefully acknowledges financial support in the form of an AARMS Postdoctoral Fellowship, a Mount Allison University Research Stipend, and, earlier, an NSERC Postdoctoral Fellowship.

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Lucyshyn-Wright, R.B.B. Commutants for Enriched Algebraic Theories and Monads. Appl Categor Struct 26, 559–596 (2018). https://doi.org/10.1007/s10485-017-9503-1

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  • DOI: https://doi.org/10.1007/s10485-017-9503-1

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