39,596 research outputs found

    A Model Structure for Enriched Coloured Operads

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    We prove that, under certain conditions, the model structure on a monoidal model category V\mathcal{V} can be transferred to a model structure on the category of V\mathcal{V}-enriched coloured (symmetric) operads. As a particular case we recover the known model structure on simplicial operads.Comment: 44 pages, Preliminary version, comments are welcom

    Operads, configuration spaces and quantization

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    We review several well-known operads of compactified configuration spaces and construct several new such operads, C, in the category of smooth manifolds with corners whose complexes of fundamental chains give us (i) the 2-coloured operad of A-infinity algebras and their homotopy morphisms, (ii) the 2-coloured operad of L-infinity algebras and their homotopy morphisms, and (iii) the 4-coloured operad of open-closed homotopy algebras and their homotopy morphisms. Two gadgets - a (coloured) operad of Feynman graphs and a de Rham field theory on C - are introduced and used to construct quantized representations of the (fundamental) chain operad of C which are given by Feynman type sums over graphs and depend on choices of propagators.Comment: 58 page

    C∗C^*-algebraic drawings of dendroidal sets

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    In recent years the theory of dendroidal sets has emerged as an important framework for higher algebra. In this article we introduce the concept of a C∗C^*-algebraic drawing of a dendroidal set. It depicts a dendroidal set as an object in the category of presheaves on C∗C^*-algebras. We show that the construction is functorial and, in fact, it is the left adjoint of a Quillen adjunction between combinatorial model categories. We use this construction to produce a bridge between the two prominent paradigms of noncommutative geometry via adjunctions of presentable ∞\infty-categories, which is the primary motivation behind this article. As a consequence we obtain a single mechanism to construct bivariant homology theories in both paradigms. We propose a (conjectural) roadmap to harmonize algebraic and analytic (or topological) bivariant K-theory. Finally, a method to analyse graph algebras in terms of trees is sketched.Comment: 28 pages; v2 expanded version with some improvements; v3 revised and added references; v4 some changes according to the suggestions of the referees (to appear in Algebr. Geom. Topol.

    A tale of three homotopies

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    For a Koszul operad P\mathcal{P}, there are several existing approaches to the notion of a homotopy between homotopy morphisms of homotopy P\mathcal{P}-algebras. Some of those approaches are known to give rise to the same notions. We exhibit the missing links between those notions, thus putting them all into the same framework. The main nontrivial ingredient in establishing this relationship is the homotopy transfer theorem for homotopy cooperads due to Drummond-Cole and Vallette.Comment: 22 pages, final versio

    Random tree recursions: which fixed points correspond to tangible sets of trees?

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    Let B\mathcal{B} be the set of rooted trees containing an infinite binary subtree starting at the root. This set satisfies the metaproperty that a tree belongs to it if and only if its root has children uu and vv such that the subtrees rooted at uu and vv belong to it. Let pp be the probability that a Galton-Watson tree falls in B\mathcal{B}. The metaproperty makes pp satisfy a fixed-point equation, which can have multiple solutions. One of these solutions is pp, but what is the meaning of the others? In particular, are they probabilities of the Galton-Watson tree falling into other sets satisfying the same metaproperty? We create a framework for posing questions of this sort, and we classify solutions to fixed-point equations according to whether they admit probabilistic interpretations. Our proofs use spine decompositions of Galton-Watson trees and the analysis of Boolean functions.Comment: 41 pages; small changes in response to referees' comments; to appear in Random Structures & Algorithm

    Geometric presentations for Thompson's groups

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    We prove that Thompson's groups FF and VV are the geometry groups of associativity, and of associativity together with commutativity, respectively. We deduce new presentations of FF and VV. These presentations lead to considering a certain subgroup of VV and an extension of this subgroup. We prove that the latter are the geometry groups of associativity together with the law x(yz)=y(xz)x(yz) = y(xz), and of associativity together with a twisted version of this law involving self-distributivity, respectively
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