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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.
@article{TAC_2009_22_a12,
author = {Chris Heunen},
title = {An embedding theorem for {Hilbert} categories},
journal = {Theory and applications of categories},
pages = {321--344},
publisher = {mathdoc},
volume = {22},
year = {2009},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2009_22_a12/}
}
Chris Heunen. An embedding theorem for Hilbert categories. Theory and applications of categories, Tome 22 (2009), pp. 321-344. http://geodesic.mathdoc.fr/item/TAC_2009_22_a12/