  We propose a categorical interpretation of multiplier Hopf algebras, in
analogy to usual Hopf algebras and bialgebras. Since the introduction of
multiplier Hopf algebras by Van Daele in [A. Van Daele, Multiplier Hopf
algebras, {\em Trans. Amer. Math. Soc.}, {\bf 342}(2), (1994) 917--932.] such a
categorical interpretation has been missing. We show that a multiplier Hopf
algebra can be understood as a coalgebra with antipode in a certain monoidal
category of algebras. We show that a (possibly non-unital, idempotent,
non-degenerate, $k$-projective) algebra over a commutative ring $k$ is a
multiplier bialgebra if and only if the category of its algebra extensions and
both the categories of its left and right modules are monoidal and fit,
together with the category of $k$-modules, into a diagram of strict monoidal
forgetful functors.
