Multivariable adjunctions and mates
Abstract
We present the notion of "cyclic double multicategory", as a structure in which to organise multivariable adjunctions and mates. The classic example of a 2variable adjunction is the hom/tensor/cotensor trio of functors; we generalise this situation to n+1 functors of n variables. Furthermore, we generalise the mates correspondence, which enables us to pass between natural transformations involving left adjoints to those involving right adjoints. 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2012
 arXiv:
 arXiv:1208.4520
 Bibcode:
 2012arXiv1208.4520C
 Keywords:

 Mathematics  Category Theory;
 Mathematics  Algebraic Topology;
 18A40;
 18D05;
 18D50