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Ernst-erich Doberkat - One of the best experts on this subject based on the ideXlab platform.

Vladimir Voevodsky - One of the best experts on this subject based on the ideXlab platform.

  • Motivic Eilenberg-MacLane spaces
    Publications mathématiques de l'IHÉS, 2010
    Co-Authors: Vladimir Voevodsky
    Abstract:

    In this paper we construct symmetric powers in the motivic homotopy categories of morphisms and finite correspondences associated with f -admissible subcategories in the categories of schemes of finite type over a field. Using this construction we provide a description of the motivic Eilenberg-MacLane spaces representing motivic cohomology on some f -admissible categories including the category of semi-normal quasi-projective schemes and, over fields which admit resolution of singularities, on some admissible subcategories including the category of smooth schemes. This description is then used to give a complete computation of the algebra of bistable motivic cohomological operations on smooth schemes over fields of characteristic zero and to obtain partial results on unstable operations which are required for the proof of the Bloch-Kato conjecture.

  • motivic Eilenberg maclane spaces
    arXiv: Algebraic Geometry, 2008
    Co-Authors: Vladimir Voevodsky
    Abstract:

    This paper is the second one in a series of papers about operations in motivic cohomology. Here we show that in the context of smooth schemes over a field of characteristic zero all the bi-stable operations can be obtained in the usual way from the motivic reduced powers and the Bockstein homomorphism.

Markus Szymik - One of the best experts on this subject based on the ideXlab platform.

  • preludes to the Eilenberg moore and the leray serre spectral sequences
    arXiv: Algebraic Topology, 2021
    Co-Authors: Frank Neumann, Markus Szymik
    Abstract:

    The Leray-Serre and the Eilenberg-Moore spectral sequence are fundamental tools for computing the cohomology of a group or, more generally, of a space. We describe the relationship between these two spectral sequences when both of them share the same abutment. There exists a joint tri-graded refinement of the Leray-Serre and the Eilenberg-Moore spectral sequence. This refinement involves two more spectral sequences, the preludes from the title, which abut to the initial terms of the Leray-Serre and the Eilenberg-Moore spectral sequence, respectively. We show that one of these always degenerates from its second page on and that the other one satisfies a local-to-global property: It degenerates for all possible base spaces if and only if it does so when the base space is contractible. When the preludes degenerate early enough, they appear to echo Deligne's decalage, but in general, this is an illusion. We discuss several principal fibrations to illustrate the possible cases and give applications, in particular, to Lie groups, torus bundles, and generalizations.

  • a new kenzo module for computing the Eilenberg moore spectral sequence
    ACM Communications in Computer Algebra, 2020
    Co-Authors: Ana Romero, Julio Rubio, Francis Sergeraert, Markus Szymik
    Abstract:

    In this work we present a new module for the computer algebraic topology system Kenzo computing the Eilenberg-Moore spectral sequence of fibrations between spaces with effective homology. These programs can be applied to determine the Eilenberg-Moore spectral sequence of extensions of finitely generated abelian groups.

Benoit Razet - One of the best experts on this subject based on the ideXlab platform.

  • machines d Eilenberg effectives
    2009
    Co-Authors: Benoit Razet
    Abstract:

    La theorie des automates est apparue pour resoudre des problemes aussi bien pratiques que theoriques, et ceci des le debut de l'informatique. Desormais, les automates font partie des notions fondamentales de l'informatique, et se retrouvent dans la plupart des logiciels. En 1974, Samuel Eilenberg proposa un modele de calcul qui unifie la plupart des automates (transducteurs, automates a pile et machines de Turing) et qui a une propriete de modularite interessante au vu d'applications reposant sur differentes couches d'automates ; comme cela peut etre le cas en linguistique computationnelle. Nous proposons d'etudier les techniques permettant d'avoir des machines d'Eilenberg effectives. Cette etude commence par la modelisation de relations calculables a base de flux, puis continue avec l'etude de la simulation des machines d'Eilenberg definies avec ces relations. Le simulateur est un programme fonctionnel enumerant progressivement les solutions, en explorant un espace de recherche selon differentes strategies. Nous introduisons, en particulier, la notion de machine d'Eilenberg finie pour laquelle nous fournissons une preuve formelle de correction de la simulation. Les relations sont une premiere composante des machines d'Eilenberg, la deuxieme composante etant son controle, qui est defini par un automate fini. Dans ce contexte, on peut utiliser une expression reguliere comme syntaxe pour decrire la composante de controle d'une machine d'Eilenberg. Recemment, un ensemble de travaux exploitant la notion de derivees de Brzozowski, a ete la source d'algorithmes efficaces de synthese d'automates non-deterministes a partir d'expressions regulieres. Nous faisons l'etat de l'art de ces algorithmes, tout en donnant une implementation efficace en OCaml permettant de les comparer les uns aux autres.

  • finite Eilenberg machines
    International Conference on Implementation and application of automata, 2008
    Co-Authors: Benoit Razet
    Abstract:

    Eilenberg machines define a general computational model. They are well suited to the simulation of problems specified using finite state formalisms such as formal languages and automata theory. This paper introduces a subclass of them called finite Eilenberg machines. We give a formal description of complete and efficient algorithms which permit the simulation of such machines. We show that our finiteness condition ensures a correct behavior of the simulation. Interpretations of this condition are studied for the cases of non-deterministic finite automata (NFA) and transducers, leading to applications to computational linguistics. The given implementation provides a generic simulation procedure for any problem encoded as a composition of finite Eilenberg machines.

  • simulating Eilenberg machines with a reactive engine formal specification proof and program extraction
    2008
    Co-Authors: Benoit Razet
    Abstract:

    Eilenberg machines have been introduced in 1974 in the field of formal language theory. They are finite automata for which the alphabet is interpreted by mathematical relations over an abstract set. They generalize many finite state machines. We consider in the present work a class of Eilenberg machines for which we provide an executable complete simulator. This program is described using the Coq specification language. The correction of the algorithm is also proved formally and mechanically verified using the Coq proof assistant. The Coq extraction code technology allows to translate the specification into executable OCaml code. The algorithm and proofs are inspired from the reactive engine of Gerard Huet.

Joost Vercruysse - One of the best experts on this subject based on the ideXlab platform.

  • the Eilenberg moore category and a beck type theorem for a morita context
    Applied Categorical Structures, 2011
    Co-Authors: Tomasz Brzezinski, Adrian Vazquez Marquez, Joost Vercruysse
    Abstract:

    The Eilenberg-Moore constructions and a Beck-type theorem for pairs of monads are described. More specifically, a notion of a Morita context comprising of two monads, two bialgebra functors and two connecting maps is introduced. It is shown that in many cases equivalences between categories of algebras are induced by such Morita contexts. The Eilenberg-Moore category of representations of a Morita context is constructed. This construction allows one to associate two pairs of adjoint functors with right adjoint functors having a common domain or a double adjunction to a Morita context. It is shown that, conversely, every Morita context arises from a double adjunction. The comparison functor between the domain of right adjoint functors in a double adjunction and the Eilenberg-Moore category of the associated Morita context is defined. The sufficient and necessary conditions for this comparison functor to be an equivalence (or for the moritability of a pair of functors with a common domain) are derived.

  • the Eilenberg moore category and a beck type theorem for a morita context
    arXiv: Category Theory, 2008
    Co-Authors: Tomasz Brzezinski, Adrian Vazquez Marquez, Joost Vercruysse
    Abstract:

    The Eilenberg-Moore constructions and a Beck-type theorem for pairs of monads are described. More specifically, a notion of a {\em Morita context} comprising of two monads, two bialgebra functors and two connecting maps is introduced. It is shown that in many cases equivalences between categories of algebras are induced by such Morita contexts. The Eilenberg-Moore category of representations of a Morita context is constructed. This construction allows one to associate two pairs of adjoint functors with right adjoint functors having a common domain or a {\em double adjunction} to a Morita context. It is shown that, conversely, every Morita context arises from a double adjunction. The comparison functor between the domain of right adjoint functors in a double adjunction and the Eilenberg-Moore category of the associated Morita context is defined. The sufficient and necessary conditions for this comparison functor to be an equivalence (or for the {\em moritability} of a pair of functors with a common domain) are derived.