Dedekind Finiteness in Topoi
A Dedekind finite object in a topos is an object such that any monic endomorphism is an epimorphism. This paper proves the basic properties of Dedekind finiteness and then gives examples which show that the class of Dedekind finite objects is not closed under quotients, subobjects, exponentiation, or finite powerobjects. Examples also show that having no nontrivial epic endomorphisms is distinct from Dedekind finiteness.
Stout, Lawrence, "Dedekind Finiteness in Topoi" (1987). Scholarship. 82.