39,596 research outputs found
A Model Structure for Enriched Coloured Operads
We prove that, under certain conditions, the model structure on a monoidal
model category can be transferred to a model structure on the
category of -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
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
-algebraic drawings of dendroidal sets
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
-algebraic drawing of a dendroidal set. It depicts a dendroidal set as an
object in the category of presheaves on -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 -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
For a Koszul operad , there are several existing approaches to
the notion of a homotopy between homotopy morphisms of homotopy
-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?
Let 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 and such that the
subtrees rooted at and belong to it. Let be the probability that a
Galton-Watson tree falls in . The metaproperty makes satisfy a
fixed-point equation, which can have multiple solutions. One of these solutions
is , 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
We prove that Thompson's groups and are the geometry groups of
associativity, and of associativity together with commutativity, respectively.
We deduce new presentations of and . These presentations lead to
considering a certain subgroup of and an extension of this subgroup. We
prove that the latter are the geometry groups of associativity together with
the law , and of associativity together with a twisted version
of this law involving self-distributivity, respectively
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