2,016 research outputs found
Brane structures in microlocal sheaf theory
Let be an exact Lagrangian submanifold of a cotangent bundle ,
asymptotic to a Legendrian submanifold . We study
a locally constant sheaf of -categories on , called the sheaf of
brane structures or . Its fiber is the -category of
spectra, and we construct a Hamiltonian invariant, fully faithful functor from
to the -category of sheaves of spectra on
with singular support in .Comment: 35 pages, 13 figure
Sheaf Theory through Examples
An approachable introduction to elementary sheaf theory and its applications beyond pure math. Sheaves are mathematical constructions concerned with passages from local properties to global ones. They have played a fundamental role in the development of many areas of modern mathematics, yet the broad conceptual power of sheaf theory and its wide applicability to areas beyond pure math have only recently begun to be appreciated. Taking an applied category theory perspective, Sheaf Theory through Examples provides an approachable introduction to elementary sheaf theory and examines applications including n-colorings of graphs, satellite data, chess problems, Bayesian networks, self-similar groups, musical performance, complexes, and much more. With an emphasis on developing the theory via a wealth of well-motivated and vividly illustrated examples, Sheaf Theory through Examples supplements the formal development of concepts with philosophical reflections on topology, category theory, and sheaf theory, alongside a selection of advanced topics and examples that illustrate ideas like cellular sheaf cohomology, toposes, and geometric morphisms. Sheaf Theory through Examples seeks to bridge the powerful results of sheaf theory as used by mathematicians and real-world applications, while also supplementing the technical matters with a unique philosophical perspective attuned to the broader development of ideas
Quantitative sheaf theory
We introduce a notion of complexity of a complex of ell-adic sheaves on a
quasi-projective variety and prove that the six operations are "continuous", in
the sense that the complexity of the output sheaves is bounded solely in terms
of the complexity of the input sheaves. A key feature of complexity is that it
provides bounds for the sum of Betti numbers that, in many interesting cases,
can be made uniform in the characteristic of the base field. As an
illustration, we discuss a few simple applications to horizontal
equidistribution results for exponential sums over finite fields.Comment: v3, 68 pages; the key ideas of this paper are due to W. Sawin; A.
Forey, J. Fres\'an and E. Kowalski drafted the current version of the text;
revised after referee report
Periodic twisted cohomology and T-duality
The initial motivation of this work was to give a topological interpretation
of two-periodic twisted de-Rham cohomology which is generalizable to arbitrary
coefficients. To this end we develop a sheaf theory in the context of locally
compact topological stacks with emphasis on the construction of the sheaf
theory operations in unbounded derived categories, elements of Verdier duality
and integration. The main result is the construction of a functorial
periodization functor associated to a U(1)-gerbe. As applications we verify the
-duality isomorphism in periodic twisted cohomology and in periodic twisted
orbispace cohomology.Comment: 128 pages; v2: small corrections (e.g. of typos), version to appear
in Asterisqu
Sheaf theory for stacks in manifolds and twisted cohomology for S^1-gerbes
This is the first of a series of papers on sheaf theory on smooth and
topological stacks and its applications. The main result of the present paper
is the characterization of the twisted (by a closed integral three-form) de
Rham complex on a manifold. As an object in the derived category it will be
related with the push-forward of the constant sheaf from a S^1-gerbe with
Dixmier-Douady class represented by the three-form. In order to formulate and
prove this result we develop in detail the foundations of sheaf theory for
smooth stacks.Comment: 39 pages, v2 typos corrected and references added. v3 confusion in
2.2.5 cleaned u
Semantic Unification A sheaf theoretic approach to natural language
Language is contextual and sheaf theory provides a high level mathematical
framework to model contextuality. We show how sheaf theory can model the
contextual nature of natural language and how gluing can be used to provide a
global semantics for a discourse by putting together the local logical
semantics of each sentence within the discourse. We introduce a presheaf
structure corresponding to a basic form of Discourse Representation Structures.
Within this setting, we formulate a notion of semantic unification --- gluing
meanings of parts of a discourse into a coherent whole --- as a form of
sheaf-theoretic gluing. We illustrate this idea with a number of examples where
it can used to represent resolutions of anaphoric references. We also discuss
multivalued gluing, described using a distributions functor, which can be used
to represent situations where multiple gluings are possible, and where we may
need to rank them using quantitative measures.
Dedicated to Jim Lambek on the occasion of his 90th birthday.Comment: 12 page
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