A -category can be thought of as a horizontal categorification of a -algebra. Equivalently, a -algebra is thought of as a pointed one-object -category (the delooping of ). Accordingly, a more systematic name for -categories would be -algebroids.
Every arrow satisfies the -identity .
Composition satisfies for all composable pairs of arrows and . (That is, we give the projective tensor product.)
For every arrow there exists an arrow such that .
Condition (3) above is equivalent to requiring that every arrow of the form is positive in the sense of -algebras. Unlike -algebras, this does not follow automatically, as can be seen by considering the category with two objects with all morphism sets a copy of and with involution defined on by if and otherwise.
The category of Hilbert spaces and bounded linear maps is a -category.
-algebras can be represented as algebras of bounded linear operators on some choice of Hilbert space, using the G.N.S. construction. -categories have an analogue of the G.N.S. construction that allows them to represented on the category of Hilbert spaces and bounded linear maps.
For any (small) -category there exists a faithful -functor .