Cyclic multicategories, multivariable adjunctions and mates

Abstract

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.

Monoidal algebraic model structures

“…A careful statement of the "multi-functoriality" of the parameterized mates correspondence, the appropriate analog of Theorem 2.7, involves category objects in the category of multicategories equipped with certain additional structure. This result will appear in a separate paper [3], joint work with Eugenia Cheng and Nick Gurski. For present purposes, we only need a preliminary lemma in this direction.…”

Higher cyclic operads

“…This is a direct consequence of the second part of Lemma 7. 12 In other words, sSet R op * admits a cofibrantly-generated model structure with weak equivalences (resp. fibrations) those maps which are weak equivalences (resp.…”

“…For this reason, 1 We exclude the case i = 1, = 0 from (1), as the formula (g • 1 f ) · τ = (g · τ 2 ) • k f follows from the first case. Indeed, 2 We should note that our definition is not a symmetric version of the 'cyclic multicategories' of Cheng, Gurski, and Riehl [12]. A non-symmetric colored cyclic operad is a non-symmetric colored operad O together with an action of the subgroup τ n+1 ≤ Σ + n on On, so that (1) holds.…”

The additivity of traces in monoidal derivators

“…This is a two-variable version of the fact that an adjunction (F, G) : C 1 ⇄ C 2 can equally be regarded as an adjunction (G op , F op ) : C op 2 ⇄ C op 1 . (The placement of the opposites can also be made more symmetrical; see [CGR12]. )…”

“…Thus, up to isomorphism, Theorem 9.1 describes a cyclic action on two-variable adjunctions. Abstractly, we could say that derivators and two-variable adjunctions form a "pseudo cyclic double multicategory" [CGR12]. Also, by the construction in Theorem 9.1, the functor ⊲ appearing in (9.10) should be given by a canceling version of (9.9), which is to say an end in D 1 :…”