In Logics of Worlds, an “object” is the “objectivation” - the projection into some world - of a multiple-being (a “thing”). What’s interesting about this projection is that it conserves the multiple composition of the thing projected: there is a correspondence between the “elements” of the multiple and the “atoms” of the object, such that every atom is a “real” atom. Another way of saying this is to say that every difference (in the projected components of the enworlded object) indexes a difference (in the multiple composition of the thing in itself): this is the import of Badiou’s “materialist postulate”, which we might translate colloquially as “there’s no (phenomenal) smoke without (ontological) fire”. Objects are not phantoms, not subjective illusions in some free-floating phenomenal realm, but the worldly-being of real beings. Yet another way of saying this is to say that the “chairness of a chair” (for example) does not consist entirely of such worldly attributes as woodenness or four-leggedness or ergonomic suitability for sitting on: the “chair-thing” of which the chair-object is the worldly projection forms the indispensable support for the chair-object’s discernible components.