Skip to main content
    • Aa
    • Aa

Cyclic multicategories, multivariable adjunctions and mates


A multivariable adjunction is the generalisation of the notion of a 2-variable adjunction, the classical example being the hom/tensor/cotensor trio of functors, to n + 1 functors of n variables. In the presence of multivariable adjunctions, natural transformations between certain composites built from multivariable functors have “dual” forms. We refer to corresponding natural transformations as multivariable or parametrised mates, generalising the mates correspondence for ordinary adjunctions, which enables one to pass between natural transformations involving left adjoints to those involving right adjoints. A central problem is how to express the naturality (or functoriality) of the parametrised mates, giving a precise characterization of the dualities so-encoded.

We present the notion of “cyclic double multicategory” as a structure in which to organise multivariable adjunctions and mates. While the standard mates correspondence is described using an isomorphism of double categories, the multivariable version requires the framework of “double multicategories”. Moreover, we show that the analogous isomorphisms of double multicategories give a cyclic action on the multimaps, yielding the notion of “cyclic double multicategory”. The work is motivated by and applied to Riehl's approach to algebraic monoidal model categories.

Corresponding author
Department of Mathematics, University of Sheffield,
Department of Mathematics, University of Sheffield,
Department of Mathematics, Harvard University,
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

4. Richard Garner . Understanding the small object argument. Applied Categorical Structures 17(3) (2009), 247285.

6. Paul G. Goerss and John F. Jardine . Simplicial homotopy theory, Progress in Mathematics 174, Birkhäuser Verlag, Basel, 1999.

13. André Joyal and Joachim Kock . Feynman graphs, and nerve theorem for compact symmetric multicategories (extended abstract). Electronic Notes in Theoretical Computer Science 270 (2011), 105113.

16. G. M. Kelly and S. Mac Lane . Coherence in closed categories. Journal of Pure and Applied Algebra 1(1) (1971), 97140.

17. G. M. Kelly and Ross Street . Review of the elements of 2-categories. In Category Seminar, Springer Lecture Notes in Mathematics 420, pages 75103, 1974.

18. Stephen Lack . A Quillen model structure for 2-categories. K-Theory 26(2) (2002), 171205.

19. Stephen Lack . A Quillen model structure for bicategories. K-Theory 33(3) (2004), 185197.

21. Tom Leinster . Higher operads, higher categories. London Mathematical Society Lecture Note Series 298. Cambridge University Press, 2004.

25. Emily Riehl . Monoidal algebraic model structures. J. Pure Appl. Algebra 217(6) (2013), 10691104.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of K-Theory
  • ISSN: 1865-2433
  • EISSN: 1865-5394
  • URL: /core/journals/journal-of-k-theory
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 3 *
Loading metrics...

Abstract views

Total abstract views: 109 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 22nd September 2017. This data will be updated every 24 hours.