52 research outputs found
Discrete Morse theory for computing cellular sheaf cohomology
Sheaves and sheaf cohomology are powerful tools in computational topology,
greatly generalizing persistent homology. We develop an algorithm for
simplifying the computation of cellular sheaf cohomology via (discrete)
Morse-theoretic techniques. As a consequence, we derive efficient techniques
for distributed computation of (ordinary) cohomology of a cell complex.Comment: 19 pages, 1 Figure. Added Section 5.
Parametrized Homology via Zigzag Persistence
This paper develops the idea of homology for 1-parameter families of
topological spaces. We express parametrized homology as a collection of real
intervals with each corresponding to a homological feature supported over that
interval or, equivalently, as a persistence diagram. By defining persistence in
terms of finite rectangle measures, we classify barcode intervals into four
classes. Each of these conveys how the homological features perish at both ends
of the interval over which they are defined
Duality and spherical adjunction from microlocalization -- An approach by contact isotopies
For a subanalytic Legendrian , we prove that when
is either swappable or a full Legendrian stop, the microlocalization
at infinity is a spherical functor, and the
spherical cotwist is the Serre functor on the subcategory
of compactly supported sheaves with perfect
stalks. In this case, when is compact the Verdier duality on
extends naturally to all compact objects
. This is a sheaf theory counterpart (with
weaker assumptions) of the results on the cap functor and cup functors between
Fukaya categories. When proving spherical adjunction, we deduce the Sato-Saboff
fiber sequence and construct the Guillermou doubling functor for any Reeb flow.
As a setup for the Verdier duality statement, we study the dualizability of
itself and obtain a classification result of
colimit-preserving functors by convolutions of sheaf kernels.Comment: 74 pages, 5 figures. A new discussion around Proposition 4.1
Wrapped sheaves
We construct a sheaf-theoretic analogue of the wrapped Fukaya category in
Lagrangian Floer theory, by localizing a category of sheaves microsupported
away from some given along continuation maps constructed
using the Guillermou-Kashiwara-Schapira sheaf quantization.
When is a subanalytic singular isotropic, we also construct a
comparison map to the category of compact objects in the category of unbounded
sheaves microsupported in , and show that it is an equivalence. The
last statement can be seen as a sheaf theoretical incarnation of the
sheaf-Fukaya comparison theorem of Ganatra-Pardon-Shende.Comment: 54 pages, 4 figure
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