MONIC SKELETA, BOUNDARIES, AUFHEBUNG, AND THE MEANING OF 'ONE-DIMENSIONALITY'

Let epsilon be a topos. If l is a level of E with monic skeleta then it makes sense to consider the objects in epsilon that have l-skeletal boundaries. In particular, if p : epsilon -> S is a pre-cohesive geometric morphism then its centre (that may be called level 0) has monic skeleta. Let level 1 be the Aufhebung of level 0. We show that if level 1 has monic skeleta then the quotients of 0-separated objects with 0-skeletal boundaries are 1-skeletal. We also prove that in several examples (such as the classifier of non-trivial Boolean algebras, simplicial sets and the classifier of strictly bipointed objects) every 1-skeletal object is of that form.