Re: question on finiteness in toposes

Let me expand. If one bores into just why Set^2_kf can't be boolean and looks for a minimal example of its non-booleaness one inevitably lands on the object 2 -> 1. At first blush its lattice of subobjects does look boolean. Until one notices that there's a monomorphism from 2 -> 2 to 2 -> 1 (where 2 -> 2 is the identity map). Having noticed that, one has a quicker proof that it's not a topos: not every mono-epi is an equalizer.