  The notion of multiplier Hopf monoid in any braided monoidal category is
introduced as a multiplier bimonoid whose constituent fusion morphisms are
isomorphisms. In the category of vector spaces over the complex numbers, Van
Daele's definition of multiplier Hopf algebra is re-obtained. It is shown that
the key features of multiplier Hopf algebras (over fields) remain valid in this
more general context. Namely, for a multiplier Hopf monoid A, the existence of
a unique antipode is proved --- in an appropriate, multiplier-valued sense ---
which is shown to be a morphism of multiplier bimonoids from a twisted version
of A to A. For a regular multiplier Hopf monoid (whose twisted versions are
multiplier Hopf monoids as well) the antipode is proved to factorize through a
proper automorphism of the object A. Under mild further assumptions, duals in
the base category are shown to lift to the monoidal categories of modules and
of comodules over a regular multiplier Hopf monoid. Finally, the so-called
Fundamental Theorem of Hopf modules is proved --- which states an equivalence
between the base category and the category of Hopf modules over a multiplier
Hopf monoid.
