Fibrations of Toposes and Etale Maps
Date:
Abstract: For many special constructions of topological spaces (which for us will be point-free, and generalized in the sense of Grothendieck), a structure preserving map between the presenting structures gives a map between the corresponding spaces. Two very simple examples are: a function $f\colon X \to Y$ between sets already is a map between the corresponding discrete spaces; and a homomorphism $f\colon K \to L$ between two distributive lattices gives a map in the opposite direction between their spectra. The covariance or contravariance of this correspondence is a fundamental property of the construction.
Borrowing from work of Ross Street we define a notion of (op)fibration in the 2-category $\Con$ of contexts developed in Vickers, 2016. We establish that for every context extension (op)fibration $\thT_1 \to \thT_0$ in $\Con$ any model $M$ of $T_0$ in an elementary topos $\CS$ with natural number object gives rise to an (op)fibration for toposes with codomain $\CS$, in the sense of Johnstone, 1993.
Slides: Fibrations Of Toposes, PSSL 101, Leeds, UK
Update (April 2019): These ideas have been baked into my thesis which was submitted recently and can be download here