Internal 1-topoi in 2-topoi

Abstract

We further develop the notion of elementary 2-topos, introduced by Weber, by proposing certain new axioms. We show that in a 2-category C satisfying these axioms, the "discrete opfibration (DOF) classifier" S is always an internal elementary 1-topos, in an appropriate sense. The axioms introduced for this purpose are closure conditions on the DOFs having "S-small fibres". Among these closure conditions, the most interesting one asserts that a certain DOF, analogous to the "subset fibration" over Set, has small fibres. The remaining new axioms concern "groupoidal" objects in a 2-category, which are seen to play a significant role in the general theory. We prove two results to the effect that a 2-category C satisfying these axioms is "determined by" its groupoidal objects: the first shows that C is equivalent to a 2-category of internal categories built out of groupoidal objects, and the second shows that the groupoidal objects are dense in C.