#
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.

http://www.tac.mta.ca/tac/volumes/34/25/34-25.pdf

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