Which mathematician has argued that we should move from "set" to "order"?

Question

I recall reading (on this very site, in fact) that there is a mathematician whom argues that we ought to switch from "set" to "order" in the foundations, so as to "recover duality" or some such.

Does anyone know who this is? And/or where they've argued this?

Answer 1

John Baez once argued that $0$-categories should be preorders rather than sets, but the argument was based on higher category theory. Specifically, that the hierarchy of categories should start at $-2$, and goes:

• There is just one $(-2)$ category.
• $(-1)$ categories are equivalent to truth values -- a $(-1)$ category has objects, and the homsets are $(-2)$-categories. Therefore either a $(-1)$ category is empty, or it is non-empty and all objects are isomorphic.
• $0$ categories should be equivalent to preorders -- a $0$ category would have objects, and the homsets are $(-1)$-categories: i.e. truth values. Working out what the composition law should mean amounts to making the value of $\hom(X,Y)$ mean the truth of $X \leq Y$ where $\leq$ is a preorder.

Then, we recover sets as being analogous to groupoids: they are preorders whose arrows are all isomorphisms. Actually, this gives setoids -- sets with an equivalence relation -- but setoids are equivalent to sets.

(IMO, the rationale that preorders are a better basic notion that set is not implausible even without trying to imagine $(-2)$- and $(-1)$-categories. And I definitely think setoid is a better basic notion than set)

I don't recall him saying anything about duality, though, so this may have nothing to do with what you're remembering.