Syntax and semantics of focalisation with relative monads and comonads

The logical principles of focalisation and polarisation can be used to design well-behaved term syntaxes for sequent calculus, which play a role as meta-languages for describing effectful computation. On the semantics side, this corresponds to an axiomatic and polarised notion of model of computation stated in terms of adjunctions over non-associative categories. In this paper, we study the special and delicate cases of resource and effect modalities in a general intuitionistic and linear setting: an exponential comonad $!$ (refining $\square$) and a strong monad $\lozenge$. The starting point of our contribution is noticing that the completeness for a polarised syntax for $!$ and $\lozenge$ with respect to (co)monads in linear call-by-push-value models can be achieved if we move to relative (co)monads: more precisely, comonads relative to $\downarrow$ (the positive shift functor) for $!$ and monads relative to $\uparrow$ (the negative shift functor) for $\lozenge$. These specialisations of the concept of relative (co)monad to call-by-push-value adjunctions recently appeared. Yet the syntax we present arose from proof-theoretic consideration, without the link with relative (co)monads being noticed at the time. Our first remark is thus that (co)monads relative to a call-by-push-value adjunction have been motivated previously from a proof-theoretic perspective in the context of focalisation, which also provides a meta-language for these concepts in an effectful setting. We carry out the study of these modalities from the axiomatic, non-associative point of view. We recall the notion of adjunction over non-associative categories, and establish correspondence results between this notion of adjunction and that of relative adjunction. This correspondence is then extended to linear-non-linear and strong versions of adjunctions as needed to model $!$ and $\lozenge$.