1,790 research outputs found

    On Structure and Organization: An Organizing Principle

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    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

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    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-NN category D(Ξ“NQ)\mathcal{D}(\Gamma_N Q) associated to an ADE Dynkin quiver. In addition to showing that this is contractible we prove that the braid group Br⁑(Q)\operatorname{Br}(Q) acts freely upon it by spherical twists, in particular that the spherical twist group Br⁑(Ξ“NQ)\operatorname{Br}(\Gamma_N Q) is isomorphic to Br⁑(Q)\operatorname{Br}(Q). This generalises Brav-Thomas' result for the N=2N=2 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

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    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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