# Covariant presheaves and subalgebras

## Ulrich Höhle

For small involutive and integral quantaloids ${\cal Q}$ it is shown that covariant presheaves on symmetric ${\cal Q}$-categories are equivalent to certain subalgebras of a specific monad on the category of symmetric ${\cal Q}$-categories. This construction is related to a weakening of the subobject classifier axiom which does not require the classification of all subalgebras, but only guarantees that classifiable subalgebras are uniquely classifiable. As an application the identification of closed left ideals of non-commutative $C^*$-algebras with certain "open", subalgebras of freely generated algebras is given.

Keywords: Involutive quantale, involutive quantaloid, symmetric ${\cal Q}$-category, covariant presheaf, monad of weak singletons, classifiable subalgebra, closed left ideal

2000 MSC: 06F07, 18C15, 18D20, 18F20

Theory and Applications of Categories, Vol. 25, 2011, No. 13, pp 342-367.

Published 2011-07-14.

TAC Home