Copower Objects and Their Applications to Finiteness in Topoi

In this paper, we examine a new approach to topos theory – rather than considering subobjects, look at quotients. This leads to the notion of a copower object, which is the object of quotients of a given object. We study some properties of copower objects, many of which are similar to the properties of a power objects. Given enough categorical structure (i.e. in a pretopos) it is possible to get power objects from copower objects, and vice versa. We then examine some new definitions of finiteness arising from the notion of a copower object. We will see that the most naturally occurring such notions are equivalent to the standard notions, K-finiteness (at least for well-pointed objects) and K̃-finiteness, but that this new way of looking at them gives new information, and in fact gives rise to another notion of finiteness, which is related to the classical notion of an amorphous set — i.e. an infinite set that is not the disjoint union of two infinite sets. Finally, We look briefly at two similar notions: potency objects and per objects.