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.
Keywords: Topoi, Foundations, 2-categories
2020 MSC: 18B25, 03G30, 18N10, 18N25
Theory and Applications of Categories, Vol. 45, 2026, No. 21, pp 779-862.
Published 2026-04-13.
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