1,790 research outputs found
On Structure and Organization: An Organizing Principle
We discuss the nature of structure and organization, and the process of
making new Things. Hyperstructures are introduced as binding and organizing
principles, and we show how they can transfer from one situation to another. A
guiding example is the hyperstructure of higher order Brunnian rings and
similarly structured many-body systems.Comment: Minor revision of section
Contractible stability spaces and faithful braid group actions
We prove that any `finite-type' component of a stability space of a
triangulated category is contractible. The motivating example of such a
component is the stability space of the Calabi--Yau- category
associated to an ADE Dynkin quiver. In addition to
showing that this is contractible we prove that the braid group
acts freely upon it by spherical twists, in particular
that the spherical twist group is isomorphic to
. This generalises Brav-Thomas' result for the
case. Other classes of triangulated categories with finite-type components in
their stability spaces include locally-finite triangulated categories with
finite rank Grothendieck group and discrete derived categories of finite global
dimension.Comment: Final version, to appear in Geom. Topo
Crossed simplicial groups and structured surfaces
We propose a generalization of the concept of a Ribbon graph suitable to
provide combinatorial models for marked surfaces equipped with a G-structure.
Our main insight is that the necessary combinatorics is neatly captured in the
concept of a crossed simplicial group as introduced, independently, by
Krasauskas and Fiedorowicz-Loday. In this context, Connes' cyclic category
leads to Ribbon graphs while other crossed simplicial groups naturally yield
different notions of structured graphs which model unoriented, N-spin, framed,
etc, surfaces. Our main result is that structured graphs provide orbicell
decompositions of the respective G-structured moduli spaces. As an application,
we show how, building on our theory of 2-Segal spaces, the resulting theory can
be used to construct categorified state sum invariants of G-structured
surfaces.Comment: 86 pages, v2: revised versio
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