  We axiomatically define (pre-)Hilbert categories. The axioms resemble those
for monoidal Abelian categories with the addition of an involutive functor. We
then prove embedding theorems: any locally small pre-Hilbert category whose
monoidal unit is a simple generator embeds (weakly) monoidally into the
category of pre-Hilbert spaces and adjointable maps, preserving adjoint
morphisms and all finite (co)limits. An intermediate result that is important
in its own right is that the scalars in such a category necessarily form an
involutive field. In case of a Hilbert category, the embedding extends to the
category of Hilbert spaces and continuous linear maps. The axioms for
(pre-)Hilbert categories are weaker than the axioms found in other approaches
to axiomatizing 2-Hilbert spaces. Neither enrichment nor a complex base field
is presupposed. A comparison to other approaches will be made in the
introduction.
