When I am not doing applied mathematics in business (fashionably called data science these days) I do basic work towards self-dual topos theory. My name is Phillip-Jan van Zyl.
Hi Phillip-Jan. What does "self-dual topos theory" consist of/look like?
In my setting, I work in a category together with a functor into that category.
I define duality as taking the op-functor. Thus, self-dual means that when you formulate the property (and you will end up formulating it in relation to the chosen functor) then the dual property, attained by reversing F: D -> C to F^op: D^op -> C^op, also has to hold.
My supervisor for my MSc was Zurab Janelidze, and this idea comes from him in his formulation of group theory via a self-dual functor (of subobjects). This has been extended to set theory by constructing a higher order pullback of the subjobject and quotient functors. In turn, his formulation is related to Grothendieck's idea of fibrations (since you are also working with a category and a functor into it, moreover, often in my setting it is often a fibration, opfibration or bifibration); in topology it is related to Guillaume Brümmer's topological functor; also, in lattice theory it is inspired by work by Marco Grandis (his subobject lattice conditions do hold for groups in Zurab Janelidze's setting).
You can formulate a self-dual subobject/quotient classifier in this way, and you can also construct a generalised adjunction (using the category of families of sets) that generalises a adjunction chain property of groups. In terms of Zurab's group theory self-dual axioms, you can readily generalise them to sets (my MSc) or alternatively, you can use that pullback of the subobject functor and quotient functor and his axioms directly hold then for Set.
This shows, in more informal language, that Set is Noetherian (Z. Janelidze and F. Koch van Niekerk), which essentially means that Set can be viewed as more "group-like" than it seems under the more traditional group theory axioms. Pointed sets, or Set^op can already be viewed as algebraic (in loose terms), but in fact Set itself is more like algebraic structure than we would expect from our intuition around the difference between algebra and set theory.
Sorry for taking a while to see your reply. I've just been reading Z. Janelidze and A. Goswami's self-dual axiomatization, and I'm a little confused about the claims you made in the post above. They seem to have the goal of extracting the strongest categorical axioms A of the category of groups C such that both A and the dual of A hold once we equip C with the data of subobjects [this part is strange to me, since subobjects are normally understood as monomorphisms with a given object as codomain, and this is far from being invariant under dualization, but let's ignore that for now].
Certainly when it comes to toposes there aren't many self-dual axioms which apply; there are some finitary completeness/cocompleteness axioms, but the way in which limits and colimits interact is not self-dual. There is no quotient classifier (in the sense of a direct dual to the subobject classifier), either. So I'm curious how your self-dual subobject/quotient classifier works. Similarly, the "subobject functor" and "quotient functor" have various choices available regarding what they do to morphisms. Which do you choose, and what does taking the pullback of these achieve, exactly?