Abstract

In this talk I will present the notion of ``cyclic double multicategory’’, as a structure in which to organise multivariable adjunctions and mates. The most common example of a 2-variable 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 neatly to pass between natural transformations involving left adjoints and those involving right adjoints.

While the standard mates correspondence is elegantly described using an isomorphism of double categories, the multivariable version needs 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’’.

This is joint work with Nick Gurski and Emily Riehl, and is motivated by and applied to Riehl’s approach to algebraic monoidal model categories.