Subobject classifier for algebraic structures (Q1097957)

Any locally presentable category \({\mathcal C}\) can be embedded (in a reasonably ``canonical'', but non-unique, way) in a topos of presheaves \({\mathcal P}\). The authors show that there exists an object \(\Omega_{{\mathcal C}}\) in \({\mathcal P}\) which plays the role of a suboject classifier for \({\mathcal C}\) (except that it does not necessarily live in \({\mathcal C}\), and they show how localizations of \({\mathcal C}\) may be classified by Lawvere- Tierney topologies on \(\Omega_{{\mathcal C}}\), extending the well-known classification when \({\mathcal C}\) itself is a (Grothendieck) topos.