Why we should pay more attention to Blass’ theorem

It is well known that a natural numbers object is needed in our base topos if we want to have classifying toposes for geometric theories. But even back in 1978 the question was posed by Johnstone and Wraith as to whether the condition was necessary; that is, does the existence of classifying toposes for geometric theories imply that our base topos must have a natural numbers object? (See the remark before Prop 4.9 in [JW78].) Since having a natural numbers object is essentially the same thing as satisfying the Axiom of Infinity, answering the question positively allows us to consider infinity in a way that is very much removed from its usual statement.

About a decade later Blass [B89] answered the question positively and B4.2.11 of the Elephant [J02] contains an excellent account showing that this observation is easy to derive using the direct image (back to the base topos) of the generic object in the classifying topos of the relevant theory.

In this talk I will outline the above in more detail and then provide some comments as to how it might be possible to understand infinity using these methods as a purely topological construct, removing it from its usual ‘discrete’ setting. The key is to re-interpret Blass’s theorem as a statement about localic groupoids, using the Joyal and Tierney result, [JT84]. I will explain that there are two different ways of viewing infinity in this manner; they are dual to one another in the sense that discrete can be considered to be dual to compact Hausdorff. Time permitting I will explain a conjectured third way of isolating infinity topologically that is self-dual.

The talk will lean on joint work with Henry, [HT23].