# Monic skeleta, Boundaries, Aufhebung, and the meaning of `one-dimensionality'

## Matias Menni

Let E be a topos. If l is a level of E with monic skeleta then it makes sense to consider the objects in E that have l-skeletal boundaries. In particular, if p : E \to 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.

Keywords: Topos theory, Axiomatic Cohesion

2010 MSC: 18B25, 18F20

Theory and Applications of Categories, Vol. 34, 2019, No. 25, pp 714-735.

Published 2019-09-05.

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