(0,1)-category theory: logic, order theory
proset, partially ordered set (directed set, total order, linear order)
distributive lattice, completely distributive lattice, canonical extension
The notion of $(0,1)$-topos is that of topos in the context of (0,1)-category theory.
The notion of $(0,1)$-topos is essentially equivalent to that of Heyting algebra; similarly, a Grothendieck $(0,1)$-topos is a locale.
Notice that every $(1,1)$-Grothendieck topos comes from a localic groupoid, i.e. a groupoid internal to locales, hence a groupoid internal to $(0,1)$-toposes. See classifying topos of a localic groupoid for more.
$(0,1)$-topos
J. C. Baez, M. Shulman, Lectures on n-categories and cohomology , pp.1-68 in J. C. Baez, P. May (eds.), Towards Higher Categories, Springer Heidelberg 2010. (preprint; section 5.3)
Jacob Lurie, Higher Topos Theory , Princeton UP 2009. (section 6.4.2)
.