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.

The published version in TAC has been updated so as to fix the numbering of subsections and of definitions/theorems/etc. in it (but is otherwise unchanged). This arXiv revision updates the numbering so as to agree with that in the updated published version (and is likewise otherwise unchanged)