Gabi-Monads

We study gabi-monads on skew-closed categories, extending the gabi-algebras of Berger, the second author, and Vercruysse beyond the linear case. Our main reconstruction theorem identifies gabi-monad structures on a monad with skew-closed structures on its Eilenberg–Moore category for which the canonical forgetful functor is strict closed. We compare this notion with closed monads in the sense of Kock, showing that in representation-theoretic cases these notions are quite different. On closed monoidal categories, every left Hopf monad is a normal gabi-monad, but the converse fails in general. We characterise when a gabi-monad is Hopf by the invertibility of the corresponding parametric mates, which recovers the ring-theoretic result that normal gabi-algebras over a commutative base ring are Hopf algebras. The theory of gabi-monads admits several natural examples, such as torsion-free modules, reflexive digraphs, and simplicial complexes, that we will explore in detail; we also study pointed sets as a quasi-example.