Lingyuan Ye and I have put on the arXiv a new version of our paper, Domains and Classifying Topoi, which is currently under review. We have made an important improvement, correcting an terminological error that I deeply regret.
The error I am referring to is not a mathematical or technical one, but rather an issue of clarity and terminological coherence. This work is about transplanting Ingo Blechschmidt’s notion of synthetic quasi-coherence from algebraic geometry to domain theory. There is a very general way to state the synthetic quasi-coherence principle in the context of any algebraic theory \mathbb {T} that has a notion of module, which is Definition 18.18 of Bleschmidt’s PhD thesis.
We now consider a special case of Definition [0LVX].
The definition of quasi-coherence that we inherit from Ingo Blechschmidt is fixed specifically to finitely presented algebras. The idea of some recent works, including A Foundation for Synthetic Stone Duality and our own paper, is to consider versions of synthetic quasi-coherence with respect to larger classes of algebras (particularly, the countably presented ones).
Our own ideological contribution was to make the trivial observation that synthetic quasi-coherence of the line object is really about characterising the fixed points of the adjunction \mathcal {O}\dashv \operatorname {Spec}. This reflects our perspective that the fundamental property of concern is whether something is a fixed point of this adjunction, not whether it is (e.g.) finitely presented, countably presented, etc. In specific models, there is a direct correspondence between the two notions, but it is another matter entirely to design a civilised axiomatics.