13,039 research outputs found
Levels in the toposes of simplicial sets and cubical sets
The essential subtoposes of a fixed topos form a complete lattice, which
gives rise to the notion of a level in a topos. In the familiar example of
simplicial sets, levels coincide with dimensions and give rise to the usual
notions of n-skeletal and n-coskeletal simplicial sets. In addition to the
obvious ordering, the levels provide a stricter means of comparing the
complexity of objects, which is determined by the answer to the following
question posed by Bill Lawvere: when does n-skeletal imply k-coskeletal? This
paper answers this question for several toposes of interest to homotopy theory
and higher category theory: simplicial sets, cubical sets, and reflexive
globular sets. For the latter, n-skeletal implies (n+1)-coskeletal but for the
other two examples the situation is considerably more complicated: n-skeletal
implies (2n-1)-coskeletal for simplicial sets and 2n-coskeletal for cubical
sets, but nothing stronger. In a discussion of further applications, we prove
that n-skeletal cyclic sets are necessarily (2n+1)-coskeletal.Comment: This paper subsumes earlier work of the first, third, and fourth
authors. 19 page
Cubic complexes and finite type invariants
Cubic complexes appear in the theory of finite type invariants so often that
one can ascribe them to basic notions of the theory. In this paper we begin the
exposition of finite type invariants from the `cubic' point of view. Finite
type invariants of knots and homology 3-spheres fit perfectly into this
conception. In particular, we get a natural explanation why they behave like
polynomials.Comment: Published by Geometry and Topology Monographs at
http://www.maths.warwick.ac.uk/gt/GTMon4/paper14.abs.htm
- …