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Trialgebras and families of polytopes
 in “Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic Ktheory” Contemporary Mathematics
, 2004
"... Abstract. We show that the family of standard simplices and the family of Stasheff polytopes are dual to each other in the following sense. The chain modules of the standard simplices, resp. the Stasheff polytopes, assemble to give an operad. We show that these operads are dual of each other in the ..."
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Abstract. We show that the family of standard simplices and the family of Stasheff polytopes are dual to each other in the following sense. The chain modules of the standard simplices, resp. the Stasheff polytopes, assemble to give an operad. We show that these operads are dual of each other in the operadic sense. The main result of this paper is to show that they are both Koszul operads. As a consequence the generating series of the standard simplices and the generating series of the Stasheff polytopes are inverse to each other. The two operads give rise to new types of algebras with 3 generating operations, 11 relations, respectively 7 relations, that we call associative trialgebras and dendriform trialgebras respectively. The free dendriform trialgebra, which is based on planar trees, has an interesting Hopf algebra structure, which will be dealt with in another paper. Similarly the family of cubes gives rise to an operad which happens to be selfdual for Koszul duality.
Weak complicial sets, a simplicial weak ωcategory theory. Part II: nerves of complicial Graycategories. Available as arXiv:math/0604416
"... To Ross Street on the occasion of his 60 th birthday. Abstract. This paper develops the foundations of a simplicial theory of weak ωcategories, which builds upon the insights originally expounded by Ross Street in his 1987 paper on oriented simplices. The resulting theory of weak complicial sets pr ..."
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To Ross Street on the occasion of his 60 th birthday. Abstract. This paper develops the foundations of a simplicial theory of weak ωcategories, which builds upon the insights originally expounded by Ross Street in his 1987 paper on oriented simplices. The resulting theory of weak complicial sets provides a common generalisation of the theories of (strict) ωcategories, Kan complexes and Joyal’s quasicategories. We generalise a number of results due to the current author with regard to complicial sets and strict ωcategories to provide an armoury of well behaved technical devices, such as joins and Gray tensor products, which will be used to study these the weak ωcategory theory of these structures in a series of companion papers. In particular, we establish their basic homotopy theory by constructing a Quillen model structure on the category of stratified simplicial sets whose fibrant objects are the weak complicial sets. As a simple corollary of this work we provide an independent construction of Joyal’s model structure on simplicial sets for
Ordinal subdivision and special pasting in quasicategories
"... Quasicategories are simplicial sets with properties generalising those of the nerve of a category. They model weak∞categories. Using a combinatorially defined ordinal subdivision, we examine composition rules for certain special pasting diagrams in quasicategories. The subdivision is of combinatori ..."
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Quasicategories are simplicial sets with properties generalising those of the nerve of a category. They model weak∞categories. Using a combinatorially defined ordinal subdivision, we examine composition rules for certain special pasting diagrams in quasicategories. The subdivision is of combinatorial interest in its own right and is linked with various combinatorial constructions. 1 Introduction. The most usual method of subdivision for a simplicial complex used in elementary algebraic and geometric topology is the barycentric subdivision. There is however another very well structured subdivision construction encountered from time to time. The basic geometric construction involves chopping up a
ON THE FOUNDATION OF ALGEBRAIC TOPOLOGY
, 2005
"... Abstract. In this paper we add “a real unit with respect to topological join ” to the classical ..."
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Abstract. In this paper we add “a real unit with respect to topological join ” to the classical
Email: rgbea@algebra.us.esTo my grandfather Mariano and my grandmother Carmen. Acknowledgements
, 804
"... Luis Narváez Macarro ..."
Iterated Chromatic Subdivisions are Collapsible ∗
, 2014
"... The standard chromatic subdivision of the standard simplex is a combinatorial algebraic construction, which was introduced in theoretical distributed computing, motivated by the study of the view complex of layered immediate snapshot protocols. A most important property of this construction is the f ..."
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The standard chromatic subdivision of the standard simplex is a combinatorial algebraic construction, which was introduced in theoretical distributed computing, motivated by the study of the view complex of layered immediate snapshot protocols. A most important property of this construction is the fact that the iterated subdivision of the standard simplex is contractible, implying impossibility results in faulttolerant distributed computing. Here, we prove this result in a purely combinatorial way, by showing that it is collapsible, studying along the way fundamental combinatorial