Perfect toposes and infinitesimal weak generation

Peter Johnstone – 6 December 2017

In his seminal 2007 paper `Axiomatic cohesion', Bill Lawvere introduced a notion of `weak generation' of a topos by a family of objects, and showed that at least one simple model of `sufficient cohesion' is weakly generated by an object which is infinitesimal in the sense of Penon (i.e., its associated not-not-sheaf is 1). Recently, Matias Menni produced an ingenious argument which shows that this condition holds in all models of sufficient cohesion; but he failed to notice a much stronger result, namely that it holds in all toposes which are perfect in the sense of Freyd. Recently I noticed that the converse also holds: that is, weak generation by infinitesimals is equivalent to perfectness.