Résumé
Hopf monads generalise Hopf algebras. They clarify several aspects of the theory of Hopf algebras and capture several related structures such as weak Hopf algebras and Hopf al-gebroids. However, important parts of Hopf algebra theory are not reached by Hopf monads, most noticeably Drinfeld's quasi-Hopf algebras. In this paper we introduce a generalisa-tion of Hopf monads, that we call slack Hopf monads. This generalisation retains a clean theory and is flexible enough to encompass quasi-Hopf algebras as examples. A slack Hopf monad is a colax magma monad T on a magma category C such that the forgetful functor UT : CT -> C 'slackly' preserves internal Homs. We give a number of different descriptions of slack Hopf monads, and study special cases such as slack Hopf monads on cartesian categories and k-linear exact slack Hopf monads on Vectk, that is comagma algebras such that a modi-fied fusion operator is invertible. In particular, we characterise quasi-Hopf algebras in terms of slackness.(c) 2023 Elsevier Inc. All rights reserved.