Adjoints, multi-adjoints, pluri-adjoints (Q1901502)

Let \((G_p: {\mathcal A}_p\to {\mathcal X})_{p\in P}\) be a small family of functors having left adjoints, between small categories. If \({\mathcal A}\) is the coproduct of \(({\mathcal A}_p )_{p\in P}\) in \({\mathcal C}at\), the functor \(G: {\mathcal A}\to {\mathcal X}\) defined by \((G_p )_{p\in P}\) is proved to have a left multiadjoint in the sense of the reviewer. With some additional assumptions on \({\mathcal A}_p\) and \({\mathcal X}\), the authors build up a small category \(\overline {\mathcal A}\) and a functor \(\overline {G}: \overline {\mathcal A}\to {\mathcal X}\) which has a left pluriadjoint in the sense of the authors. These constructions show how the transitions from adjoint to multiadjoint, and from multiadjoint to pluriadjoint can be performed.