Covariant Presheaves and Subalgebras

For small involutive and integral quantaloids Q it is shown that covariant presheaves on symmetric Q-categories are equivalent to certain subalgebras of a speci c monad on the category of symmetric Q-categories. This construction is related to a weakening of the subobject classi er axiom which does not require the classi cation of all subalgebras, but only guarantees that classi able subalgebras are uniquely classi able. As an application the identi cation of closed left ideals of non-commutative C∗-algebras with certain open subalgebras of freely generated algebras is given. Introduction LetQ be a small involutive quantaloid. ThenQ induces an involution on the quantaloid of symmetric Q-categories and distributors. In particular the involute of every contravariant presheaf is covariant and vice versa. In this framework we ask the question whether there exists a concept of a weak subobject classi er in the sense that a subobject classi ed by a covariant presheaf is always uniquely classi ed. For this purpose we introduce a special kind of presheaves on symmetric Q-categories which we call weak singletons. As a rst property we note that weak singletons form a non-idempotent monad. Weak singletons appear already in the theory of metric spaces. If symmetric Qcategories are metric spaces, then maximal weak singletons coincide with extremal functions (cf. [11]). Further there is a close relationship between weak singletons and the type of singletons considered by H. Heymans (cf. [8]). In fact, if the Cauchy completion preserves the symmetry axiom (cf. [9], see also Proposition 3.1), then the monad associated with the Cauchy completion is a submonad of the weak singleton monad. After this brief historical digression, we return to the problem of unique classi cation of subobjects. Let W denote the weak singleton monad. For every object a of the given involutive quantaloid Q there exists a W-algebra structure on the free cocompletion of the trivial Q-category a. If Q is integral, this W-algebra serves as a weak subobject classi er in the category of W-algebras. Under the assumption of the integrality of Q we show that classi able subalgebras are uniquely classi able (cf. Theorem 4.3). Moreover, the classi able hull of every subalgebra exists (cf. Section 5). Since in general there exist more subalgebras than characteristic morphisms , this property might be of some Received by the editors 2010-04-23 and, in revised form, 2011-07-06. Transmitted by Ieke Moerdijk. Published on 2011-07-14. 2000 Mathematics Subject Classi cation: 06F07, 18C15, 18D20, 18F20.