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Serge Bouc - One of the best experts on this subject based on the ideXlab platform.
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Diagonal $p$-permutation Functors
2019Co-Authors: Serge Bouc, Deniz YılmazAbstract:Let $k$ be an algebraically closed field of positive characteristic $p$, and $\mathbb{F}$ be an algebraically closed field of characteristic 0. We consider the $\mathbb{F}$-linear category $\mathbb{F} pp_k^\Delta$ of finite groups, in which the set of morphisms from $G$ to $H$ is the $\mathbb{F}$-linear extension $\mathbb{F} T^\Delta(H,G)$ of the Grothendieck group $T^\Delta(H,G)$ of $p$-permutation $(kH,kG)$-bimodules with (twisted) diagonal vertices. The $\mathbb{F}$-linear Functors from $\mathbb{F} pp_k^\Delta$ to $\mathbb{F}\hbox{\rm-Mod}$ are called {\em diagonal $p$-permutation Functors}. They form an abelian category $\mathcal{F}_{pp_k}^\Delta$. We study in particular the Functor $\mathbb{F}T^\Delta$ sending a finite group $G$ to the Grothendieck group $\mathbb{F}T(G)$ of $p$-permutation $kG$-modules, and show that $\mathbb{F}T^\Delta$ is a semisimple object of $\mathcal{F}_{pp_k}^\Delta$, equal to the direct sum of specific simple Functors parametrized by isomorphism classes of pairs $(P,s)$ of a finite $p$-group $P$ and a generator $s$ of a $p'$-subgroup acting faithfully on $P$. This leads to a precise description of the evaluations of these simple Functors. In particular, we show that the simple Functor indexed by the trivial pair $(1,1)$ is isomorphic to the Functor sending a finite group $G$ to $\mathbb{F} K_0(kG)$, where $K_0(kG)$ is the group of projective $kG$-modules.
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Tensor product of correspondence Functors
2019Co-Authors: Serge Bouc, Jacques ThévenazAbstract:As part of the study of correspondence Functors, the present paper investigates their tensor product and proves some of its main properties. In particular, the correspondence Functor associated to a finite lattice has the structure of a commutative algebra in the tensor category of all correspondence Functors.
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Simple and projective correspondence Functors
arXiv: Representation Theory, 2019Co-Authors: Serge Bouc, Jacques ThévenazAbstract:A correspondence Functor is a Functor from the category of finite sets and correspondences to the category of $k$-modules, where $k$ is a commutative ring. We determine exactly which simple correspondence Functors are projective. Moreover, we analyze the occurrence of such simple projective Functors inside the correspondence Functor $F$ associated with a finite lattice and we deduce a direct sum decomposition of $F$.
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Correspondence Functors and lattices
arXiv: Representation Theory, 2019Co-Authors: Serge Bouc, Jacques ThévenazAbstract:A correspondence Functor is a Functor from the category of finite sets and correspondences to the category of k-modules, where k is a commu-tative ring. A main tool for this study is the construction of a correspondence Functor associated to any finite lattice T. We prove for instance that this Functor is projective if and only if the lattice T is distributive. Moreover, it has quotients which play a crucial role in the analysis of simple Functors. The special case of total orders yields some more specific and complete results.
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Correspondence Functors and finiteness conditions
Journal of Algebra, 2018Co-Authors: Serge Bouc, Jacques ThévenazAbstract:We investigate the representation theory of finite sets. The correspondence Functors are the Functors from the category of finite sets and correspondences to the category of k-modules, where k is a commutative ring. They have various specific properties which do not hold for other types of func-tors. In particular, if k is a field and if F is a correspondence Functor, then F is finitely generated if and only if the dimension of F (X) grows exponentially in terms of the cardinality of the finite set X. Moreover, in such a case, F has actually finite length. Also, if k is noetherian, then any subFunctor of a finitely generated Functor is finitely generated.
Hiroyuki Nakaoka - One of the best experts on this subject based on the ideXlab platform.
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Biset Functors as module Mackey Functors, and its relation to derivators
arXiv: Category Theory, 2016Co-Authors: Hiroyuki NakaokaAbstract:In this article, we will show that the category of biset Functors can be regarded as a reflective monoidal subcategory of the category of Mackey Functors on the 2-category of finite groupoids. This reflective subcategory is equivalent to the category of modules over the Burnside Functor. As a consequence of the reflectivity, we can associate a biset Functor to any derivator on the 2-category of finite categories.
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Biset transformations of Tambara Functors
Journal of Algebra, 2014Co-Authors: Hiroyuki NakaokaAbstract:Abstract If we are given an H - G -biset U for finite groups G and H , then any Mackey Functor on G can be transformed by U into a Mackey Functor on H . In this article, we show that the biset transformation is also applicable to Tambara Functors when U is right-free, and in fact forms a Functor between the category of Tambara Functors on G and H . This biset transformation Functor is compatible with some algebraic operations on Tambara Functors, such as ideal quotients or fractions. In the latter part, we also construct the left adjoint of the biset transformation.
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A Mackey-Functor theoretic interpretation of biset Functors
arXiv: Category Theory, 2013Co-Authors: Hiroyuki NakaokaAbstract:In this article, we consider a formulation of biset Functors using the 2-category of finite sets with variable finite group actions. We introduce a 2-category $\mathbb{S}$, on which a biset Functor can be regarded as a special kind of Mackey Functors. This gives an analog of Dress' definition of a Mackey Functor, in the context of biset Functors.
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A generalization of The Dress construction for a Tambara Functor, and polynomial Tambara Functors
arXiv: Category Theory, 2010Co-Authors: Hiroyuki NakaokaAbstract:For a finite group $G$, (semi-)Mackey Functors and (semi-)Tambara Functors are regarded as $G$-bivariant analogs of (semi-)groups and (semi-)rings respectively. In fact if $G$ is trivial, they agree with the ordinary (semi-)groups and (semi-)rings, and many naive algebraic properties concerning rings and groups have been extended to these $G$-bivariant analogous notions. In this article, we investigate a $G$-bivariant analog of the semi-group rings with coefficients. Just as a coefficient ring $R$ and a monoid $Q$ yield the semi-group ring $R[Q]$, our constrcution enables us to make a Tambara Functor $T[M]$ out of a semi-Mackey Functor $M$, and a coefficient Tambara Functor $T$. This construction is a composant of the Tambarization and the Dress construction. As expected, this construction is the one uniquely determined by the righteous adjoint property. Besides in analogy with the trivial group case, if $M$ is a Mackey Functor, then $T[M]$ is equipped with a natural Hopf structure. Moreover, as an application of the above construction, we also obtain some $G$-bivariant analogs of the polynomial rings.
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tambarization of a mackey Functor and its application to the witt burnside construction
arXiv: Category Theory, 2010Co-Authors: Hiroyuki NakaokaAbstract:For an arbitrary group $G$, a (semi-)Mackey Functor is a pair of covariant and contravariant Functors from the category of $G$-sets, and is regarded as a $G$-bivariant analog of a commutative (semi-)group. In this view, a $G$-bivariant analog of a (semi-)ring should be a (semi-)Tambara Functor. A Tambara Functor is firstly defined by Tambara, which he called a TNR-Functor, when $G$ is finite. As shown by Brun, a Tambara Functor plays a natural role in the Witt-Burnside construction. It will be a natural question if there exist sufficiently many examples of Tambara Functors, compared to the wide range of Mackey Functors. In the first part of this article, we give a general construction of a Tambara Functor from any Mackey Functor, on an arbitrary group $G$. In fact, we construct a Functor from the category of semi-Mackey Functors to the category of Tambara Functors. This Functor gives a left adjoint to the forgetful Functor, and can be regarded as a $G$-bivariant analog of the monoid-ring Functor. In the latter part, when $G$ is finite, we invsetigate relations with other Mackey-Functorial constructions ---crossed Burnside ring, Elliott's ring of $G$-strings, Jacobson's $F$-Burnside ring--- all these lead to the study of the Witt-Burnside construction.
Arthur Soulié - One of the best experts on this subject based on the ideXlab platform.
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Generalized Long-Moody Functors
2018Co-Authors: Arthur SouliéAbstract:In this paper, we generalize the Long-Moody construction for representations of braid groups to other groups, such as mapping class groups of surfaces. Moreover, we introduce Long-Moody endoFunctors over a Functor category that encodes representations of a family of groups. In this context, notions of polynomial Functor are defined; these play an important role in the study of homological stability. We prove that, under some additional assumptions, a Long-Moody Functor increases the (very) strong (respectively weak) polynomial degree of Functors by one.
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The Long-Moody construction and polynomial Functors
2018Co-Authors: Arthur SouliéAbstract:In 1994, Long and Moody gave a construction on representations of braid groups which associates a representation of Bn with a representation of Bn+1. In this paper, we prove that this construction is Functorial: it gives an endoFunctor, called the Long-Moody Functor, between the category of Functors from the homogeneous category associated with the braid groupoid to a module category. Then we study the effect of the Long-Moody Functor on strong polynomial Functors: we prove that it increases by one the degree of strong polynomiality.
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Long-Moody Functors and stable homology of mapping class groups
2018Co-Authors: Arthur SouliéAbstract:Among the linear representations of braid groups, Burau representations are recovered from a trivial representation using a construction introduced by Long in 1994, following a collaboration with Moody. This construction, called the Long-Moody construction, thus allows to construct more and more complex representations of braid groups. In this thesis, we have a Functorial point of view on this construction, which allows find more easily some variants. Moreover, the degree of polynomiality of a Functor measures its complexity. We thus show that the Long-Moody construction defines a Functor LM, which increases the degree of polynomiality. Furthermore, we define analogous Functors for other families of groups such as mapping class groups of surfaces and 3-manifolds, symmetric groups or automorphism groups of free groups. They satisfy similar properties on the polynomiality. Hence, Long-Moody Functors provide twisted coefficients fitting into the framework of the homological stability results of Randal-Williams and Wahl for the afore mentioned families of groups. Finally, we give a comparison result for the stable homology with coefficient given by a Functor F and the one with coefficient given by the Functor LM(F), obtained applying a Long-Moody Functor. This thesis has three chapters. The first one introduces Long-Moody Functors for braid groups and deals with their effect on the polynomiality. The first one deals with the generalisation of Long-Moody Functors for other families of groups. The last chapter touches on stable homology computations for mapping class group.
Jacques Thévenaz - One of the best experts on this subject based on the ideXlab platform.
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Tensor product of correspondence Functors
2019Co-Authors: Serge Bouc, Jacques ThévenazAbstract:As part of the study of correspondence Functors, the present paper investigates their tensor product and proves some of its main properties. In particular, the correspondence Functor associated to a finite lattice has the structure of a commutative algebra in the tensor category of all correspondence Functors.
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Simple and projective correspondence Functors
arXiv: Representation Theory, 2019Co-Authors: Serge Bouc, Jacques ThévenazAbstract:A correspondence Functor is a Functor from the category of finite sets and correspondences to the category of $k$-modules, where $k$ is a commutative ring. We determine exactly which simple correspondence Functors are projective. Moreover, we analyze the occurrence of such simple projective Functors inside the correspondence Functor $F$ associated with a finite lattice and we deduce a direct sum decomposition of $F$.
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Correspondence Functors and lattices
arXiv: Representation Theory, 2019Co-Authors: Serge Bouc, Jacques ThévenazAbstract:A correspondence Functor is a Functor from the category of finite sets and correspondences to the category of k-modules, where k is a commu-tative ring. A main tool for this study is the construction of a correspondence Functor associated to any finite lattice T. We prove for instance that this Functor is projective if and only if the lattice T is distributive. Moreover, it has quotients which play a crucial role in the analysis of simple Functors. The special case of total orders yields some more specific and complete results.
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Correspondence Functors and finiteness conditions
Journal of Algebra, 2018Co-Authors: Serge Bouc, Jacques ThévenazAbstract:We investigate the representation theory of finite sets. The correspondence Functors are the Functors from the category of finite sets and correspondences to the category of k-modules, where k is a commutative ring. They have various specific properties which do not hold for other types of func-tors. In particular, if k is a field and if F is a correspondence Functor, then F is finitely generated if and only if the dimension of F (X) grows exponentially in terms of the cardinality of the finite set X. Moreover, in such a case, F has actually finite length. Also, if k is noetherian, then any subFunctor of a finitely generated Functor is finitely generated.
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The representation theory of finite sets and correspondences
2015Co-Authors: Serge Bouc, Jacques ThévenazAbstract:We investigate correspondence Functors, namely the Functors from the category of finite sets and correspondences to the category of $k$-modules, where $k$ is a commutative ring. They have various specific properties which do not hold for other types of Functors. In particular, if $k$ is a field and if $F$ is a correspondence Functor, then $F$ is finitely generated if and only if the dimension of $F(X)$ grows exponentially in terms of the cardinality of the finite set $X$. In such a case, $F$ has finite length. Also, if $k$ is noetherian, then any subFunctor of a finitely generated Functor is finitely generated. When $k$ is a field, we give a description of all the simple Functors and we determine the dimension of their evaluations at any finite set. A main tool is the construction of a Functor associated to any finite lattice $T$. We prove for instance that this Functor is projective if and only if the lattice $T$ is distributive. Moreover, it has quotients which play a crucial role in the analysis of simple Functors. The special case of total orders yields some more specific results. Several other properties are also discussed, such as projectivity, duality, and symmetry. In an appendix, all the lattices associated to a given poset are described.
Pedro Nora - One of the best experts on this subject based on the ideXlab platform.
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Hausdorff Coalgebras
Applied Categorical Structures, 2020Co-Authors: Dirk Hofmann, Pedro NoraAbstract:As composites of constant, finite (co)product, identity, and powerset Functors, Kripke polynomial Functors form a relevant class of $$\textsf {Set}$$ Set -Functors in the theory of coalgebras. The main goal of this paper is to expand the theory of limits in categories of coalgebras of Kripke polynomial Functors to the context of quantale-enriched categories. To assume the role of the powerset Functor we consider “powerset-like” Functors based on the Hausdorff $${\mathcal {V}}$$ V -category structure. As a starting point, we show that for a lifting of a $$\textsf {Set}$$ Set -Functor to a topological category $$\textsf {X}$$ X over $$\textsf {Set}$$ Set that commutes with the forgetful Functor, the corresponding category of coalgebras over $$\textsf {X}$$ X is topological over the category of coalgebras over $$\textsf {Set}$$ Set and, therefore, it is “as complete” but cannot be “more complete”. Secondly, based on a Cantor-like argument, we observe that Hausdorff Functors on categories of quantale-enriched categories do not admit a terminal coalgebra. Finally, in order to overcome these “negative” results, we combine quantale-enriched categories and topology à la Nachbin. Besides studying some basic properties of these categories, we investigate “powerset-like” Functors which simultaneously encode the classical Hausdorff metric and Vietoris topology and show that the corresponding categories of coalgebras of “Kripke polynomial” Functors are (co)complete.
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Hausdorff coalgebras.
arXiv: Category Theory, 2019Co-Authors: Dirk Hofmann, Pedro NoraAbstract:As composites of constant, (co)product, identity, and powerset Functors, Kripke polynomial Functors form a relevant class of $\mathsf{Set}$-Functors in the theory of coalgebras. The main goal of this paper is to expand the theory of limits in categories of coalgebras of Kripke polynomial Functors to the context of quantale-enriched categories. To assume the role of the powerset Functor we consider "powerset-like" Functors based on the Hausdorff $\mathsf{V}$-category structure. As a starting point, we show that for a lifting of a $\mathsf{SET}$-Functor to a topological category $\mathsf{X}$ over $\mathsf{Set}$ that commutes with the forgetful Functor, the corresponding category of coalgebras over $\mathsf{X}$ is topological over the category of coalgebras over $\mathsf{Set}$ and, therefore, it is "as complete" but cannot be "more complete". Secondly, based on a Cantor-like argument, we observe that Hausdorff Functors on categories of quantale-enriched categories do not admit a terminal coalgebra. Finally, in order to overcome these "negative" results, we combine quantale-enriched categories and topology \emph{\`a la} Nachbin. Besides studying some basic properties of these categories, we investigate "powerset-like" Functors which simultaneously encode the classical Hausdorff metric and Vietoris topology and show that the corresponding categories of coalgebras of "Kripke polynomial" Functors are (co)complete.
