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.
Keywords: (pre-)Hilbert category, embedding, monoidal Abelian category
2000 MSC: 16B50,16D90,46C50
Theory and Applications of Categories,
Vol. 22, 2009,
No. 13, pp 321-344.
http://www.tac.mta.ca/tac/volumes/22/13/22-13.dvi
http://www.tac.mta.ca/tac/volumes/22/13/22-13.ps
http://www.tac.mta.ca/tac/volumes/22/13/22-13.pdf
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/22/13/22-13.dvi
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/22/13/22-13.ps
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/22/13/22-13.pdf