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The motive behind this paper is to generalize the concept of approach groups. In so doing, we introduce various categories, specifically, the categories of quantale-valued convergence groups, quantale-valued approach groups, quantale-valued gauge groups and quantale-valued approach system groups, and study their functorial relationships including the fact that the category of quantale-valued gauge groups as well as the category of quantale-valued approach system groups is topological over the category of groups. Besides obtaining a variety of categorical connections, we note that if the quantale is linearly ordered satisfying certain conditions, then the categories of quantale-valued approach system groups and quantale-valued gauge groups are isomorphic. Finally, we look into the embeddings of the category of quantale-valued metric groups into the categories of quantale-valued generalizations of approach groups including commutativity of some diagrams. [ABSTRACT FROM AUTHOR]

Abstract Let G be the F -points of a connected reductive group over a non-archimedean local field F of residue characteristic p and R be a commutative ring. Let P = L U be a parabolic subgroup of G and Q be a parabolic subgroup of G containing P. We study the functor St Q G taking a smooth R -representation σ of L which extends to a representation e G (σ) of G trivial on U to the smooth R -representation e G (σ) ⊗ R St Q G (R) of G where St Q G (R) is the generalised Steinberg representation. [ABSTRACT FROM AUTHOR]

The notion of an exponential R-group, where R is an arbitrary associative ring with unity, was introduced by R. Lyndon. A. G. Myasnikov and V. N. Remeslennikov refined this notion by adding an extra axiom. In particular, the new notion of an exponential MR-group is an immediate generalization of the notion of an R-module to the case of noncommutative groups. Basic concepts in the theory of exponential MR-groups are presented, and we propose a particular method for constructing tensor completion—the key construction in the category of MR-groups. As a consequence, free MR-groups and free MR-products are described using the language of group constructions. [ABSTRACT FROM AUTHOR]

The Ehresmann-Schein-Nambooripad (ESN) Theorem asserts an equivalence between the category of inverse semigroups and the category of inductive groupoids. In this paper, we consider the category of inverse categories and functors - a natural generalization of inverse semigroups and semigroup homomorphisms - and extend the ESN Theorem to an equivalence between this category and the category of locally complete inductive groupoids and locally inductive functors. From the proof of this extension, we also generalize the ESN Theorem to an equivalence between the category of inverse semicategories and the category of locally inductive groupoids and to an equivalence between the category of inverse categories with oplax functors and the category of locally complete inductive groupoids and ordered functors. [ABSTRACT FROM AUTHOR]

In this paper, we give some categorical objects of racks such as product, pullback and equalizer objects. [ABSTRACT FROM AUTHOR]

The category of strict polynomial functors inherits an internal tensor product from the category of divided powers. To investigate this monoidal structure, we consider the category of representations of the symmetric group S d which admits a tensor product coming from its Hopf algebra structure. It is classical that there exists a functor F from the category of strict polynomial functors to the category of representations of the symmetric group. Our main result is that this functor F is monoidal. In addition we study the relations under F between projective strict polynomial functors and permutation modules and the link to symmetric functions. [ABSTRACT FROM AUTHOR]

We define the partial group cohomology as the right derived functor of the functor of partial invariants, we relate this cohomology with partial derivations and with the partial augmentation ideal and we show that there exists a Grothendieck spectral sequence relating cohomology of partial smash products with partial group cohomology and algebra cohomology. [ABSTRACT FROM AUTHOR]

The construction and study of the orthogonal completion functor is an important step in the orthogonal completeness theory developed by K. I. Beidar and A. V. Mikhalev. The research of the graded orthogonal completion begun by the author is continued in this work. We consider associative rings graded by a group and modules over such rings graded by a polygon over the same group. Note that the graduation of a module by a group is a partial case of a more general and natural construction. For any topology ℱ of a graded ring R consisting of graded right dense ideals and containing all two-sided graded dense ideals, the functor O of the graded orthogonal completion is constructed and studied in this paper. This functor maps the category of right graded R-modules into the category of right graded O ( R)-modules. The important feature of the graded case is that the graded modules O ( M) and O ( M) (where M is a right graded R-module) may not be orthogonal complete. A criterion for the orthogonal completeness is proved. As a corollary we get that these modules are orthogonal complete in the case of a finite polygon. The properties of the functor O and a criterion of its exactness are also established. [ABSTRACT FROM AUTHOR]

Derived functors (or Zuckerman functors) play a very important role in the study of unitary representations of real reductive groups. These functors are usually applied on highest weight modules in the so-called good range and the theory is well-understood. On the other hand, there were several studies on the irreducibility and unitarizability, in which derived functors are applied to singular modules. See Enright et al. (Acta Math. 1985), for example. In this article, we apply derived functors to certain modules arising from the formalism of local theta lifting, and investigate the irreducible constituent of resulting modules. The key technique is to understand U ( g ) K -actions in the setting of a see-saw pair. Our results suggest that derived functor constructions are compatible with local theta liftings. [ABSTRACT FROM AUTHOR]

Let G be a compact metrizable group and H its closed subgroup. We prove that every compact metrizable G -space over G / H admits a G -fibrant extension over G / H . This implies that the functor of twisted product G × H − takes H -fibrant extensions to G -fibrant extensions. In particular, the twisted product G × H S is a G -fibrant space when S is a compact H -fibrant space. [ABSTRACT FROM AUTHOR]

We construct twisting functors for quantum group modules. First over the field Q ( v ) but later over any Z [ v , v − 1 ] -algebra. The main results in this paper are a rigorous definition of these functors, a proof that they satisfy braid relations and applications to Verma modules. [ABSTRACT FROM AUTHOR]

We interpret several constructions with C*-algebras as colimits in the bicategory of correspondences. This includes crossed products for actions of groups and crossed modules, Cuntz-Pimsner algebras of proper product systems, direct sums and inductive limits, and certain amalgamated free products. [ABSTRACT FROM AUTHOR]

In the first part of this article we formalize the concepts of terminal and initial object, categorical product [4] and natural transformation within a free-object category [1]. In particular, we show that this definition of natural transformation is equivalent to the standard definition [13]. Then we introduce the exponential object using its universal property and we show the isomorphism between the exponential object of categories and the functor category [12]. [ABSTRACT FROM AUTHOR]

For groups with a uniform bound on the length of chains of finite subgroups, we study the relationship between the Bredon cohomological dimension for proper actions and the notions of cohomological dimension one obtains by restricting the coefficients of Bredon cohomology to (cohomological) Mackey functors or fixed point functors. We also investigate the closure properties of the class of groups with finite Bredon cohomological dimension for Mackey functors. [ABSTRACT FROM AUTHOR]

Under study is the inclusion regular transitive subgroup functors. We prove that each of these functors is X-subnormal for some class X of primitive groups. We also inspect the structure of a total eigenclass of an inclusion regular transitive functor and the properties of special subgroup lattices. [ABSTRACT FROM AUTHOR]

In this sequel to [3] we try to give a comprehensive account of the “connected components” G 00 and G 000 as well as the various quotients G / G 00 , G / G 000 , G 00 / G 000 , for G a group definable in a (saturated) o -minimal expansion of a real closed field. Key themes are the structure of G 00 / G 000 and the problem of “exactness” of the G ↦ G 00 functor. We prove that the examples produced in [3] are typical, and that for any G , G 00 / G 000 is naturally the quotient of a connected compact commutative Lie group by a dense finitely generated subgroup (where we allow the trivial Lie group). [ABSTRACT FROM AUTHOR]

We associate with every group G a set τ ( G ) of subgroups of G with 1 ∈ τ ( G ) . If H ∈ τ ( G ) , then we say that H is a τ-subgroup of G . If θ ( τ ( G ) ) ⊆ τ ( θ ( G ) ) for each epimorphism θ : G → G ⁎ , then we say that τ is a subgroup functor . We say also that a subgroup functor τ is: hereditary provided H ∈ τ ( E ) whenever H ≤ E ≤ G and H ∈ τ ( G ) ; regular provided for any group G , whenever H ∈ τ ( G ) is a p -group and N is a minimal normal subgroup of G , then | G : N G ( H ∩ N ) | is a power of p ; Φ -regular (respectively Φ -quasiregular ) provided for any primitive group G , whenever H ∈ τ ( G ) is a p -group and N is a (respectively abelian) minimal normal subgroup of G , then | G : N G ( H ∩ N ) | is a power of p . Let K ≤ H be subgroups of G and τ a subgroup functor. Then we say that: the pair ( K , H ) satisfies the F -supplement condition in G if G has a subgroup T such that H T = G and H ∩ T ⊆ K Z F ( T ) ; H is F τ -supplemented in G if for some τ -subgroup S ¯ of G ¯ contained in H ¯ the pair ( S ¯ , H ¯ ) satisfies the F -supplement condition in G ¯ , where G ¯ = G / H G and H ¯ = H / H G . In this paper we study the structure of a group G under the condition that some primary subgroups of G are F τ -supplemented in G . In particular, we prove the following result. Theorem A. Let F be a saturated formation containing the class U of all supersoluble groups, E a normal subgroup of G with G / E ∈ F , X = E or X = F ⁎ ( E ) , and τ a regular or hereditary Φ -regular subgroup functor. Suppose that every τ-subgroup of G contained in X is subnormally embedded in G. If every maximal subgroup of every non-cyclic Sylow subgroup of X is U τ -supplemented in G, then G ∈ F . Moreover, in the case when τ is regular, then every chief factor of G below E is cyclic. The results in this paper not only cover and unify a long list of some known results but also cause a wide series of new results. [ABSTRACT FROM AUTHOR]

Given a point p of the topos Δ ˆ of simplicial sets and the corresponding flat covariant functor F : Δ ⟶ Sets , we determine the extensions of F to the cyclic category Λ ⊃ Δ . We show that to each such cyclic structure on a point p of Δ ˆ corresponds a group G p , that such groups can be noncommutative and that each G p is described as the quotient of a left-ordered group by the subgroup generated by a central element. Moreover for any cyclic set X the fiber (or geometric realization) of the underlying simplicial set of X at p inherits canonically the structure of a G p -space. This gives a far reaching generalization of the well-known circle action on the geometric realization of cyclic sets. [ABSTRACT FROM AUTHOR]

Symmetric cohomology of groups, defined by M. Staic in [2], is similar to the way one defines the cyclic cohomology for algebras. We show that there is a welldefined restriction, conjugation and transfer map in symmetric cohomology, which form a Mackey functor under a restriction. Some new properties for the symmetric cohomology group using normalized cochains are also given. [ABSTRACT FROM AUTHOR]

We study the local descent from irreducible, supercuspidal, self-conjugate representations of GL 2 n ( E ) to irreducible, supercuspidal and generic representations of the quasi-split unitary group U 2 n ( F ) in 2 n variables, corresponding to a quadratic extension E / F of p -adic fields. We construct the descent and prove that it is nontrivial, supercuspidal, generic and irreducible. We write the relations with poles of local gamma factors and functoriality. We also consider representations as above of GL 2 n + 1 ( E ) . [ABSTRACT FROM AUTHOR]

We study Gorenstein right derived functors of − ⊗ −with respect to semidualizing modules. As applications, some new criteria for a semidualizing module to be dualizing are given provided thatRis a ring with a dualizing complex. [ABSTRACT FROM AUTHOR]

Abstract: We consider Frobenius algebras in the monoidal category of right comodules over a Hopf algebra H. If H is a group Hopf algebra, we study a more general Frobenius type property, uncover the structure of graded Frobenius algebras, and investigate graded symmetric algebras. The graded Frobenius concept is related to Frobenius functors. [Copyright &y& Elsevier]

The aim of this paper is to investigate the relationship between the Abelian groups G with and those satisfying for various cardinals κ. In particular, we will give the structure of self-Ext-small groups. Our results answer several open problems, and give new proofs of known theorems concerning -splitters, tilting groups and finitely faithful S-groups. [ABSTRACT FROM AUTHOR]

In this paper, we focus on enriched cl-premonoid-valued topological groups, and their so-called change-of-basis lattice. In so doing, we take L as an enriched cl-premonoid and present a category SL-NTopGrp, of stratified enriched cl-premonoid-valued neighborhood topological groups. We produce some characterization theorems, and prove that every stratified L-neighborhood topological group is uniformizable. Finally, we look at the enriched lattice-valued neighborhood topological group when the underlying basis is changed under certain functorial mechanism. [ABSTRACT FROM AUTHOR]

In the paper we describe the subcategory of the category of ℤ-graded Lie algebras which is equivalent to the category of Jordan pairs via a functorial modification of the Tits, Kantor, and Koecher TKK construction. For instance, we prove thatL = L−1⊕L0⊕L1can be constructed from a Jordan pair if and only ifL0 = [L−1,L1] and the second graded homology groupis trivial. Similar descriptions are obtained for Jordan triple systems and Jordan algebras. New functorial versions of the TKK construction are given for pairs and algebras. [ABSTRACT FROM AUTHOR]

Abstract: We study internal profunctors and their normalization under various conditions on the base category. In the Mal’tsev case we give an easy characterization of profunctors. Moreover, when the base category is regular with any regular epimorphism effective for descent, we characterize those profunctors which are fractions of internal functors with respect to weak equivalences. [Copyright &y& Elsevier]

Consider the Mackey functor that assigns to each finite group G the Green ring of finitely generated kG-modules, where k is a field of characteristic p > 0. Thévenaz foresaw in 1988 that the class of primordial groups for this functor is the family of k-Dress groups. In this paper we prove that this is true for the subfunctor defined by the Green ring of finitely generated kG-modules of trivial source. [ABSTRACT FROM PUBLISHER]

In this research, by using filter regular sequences, we obtain some exact sequences of right or left derived functors of local cohomology modules. Then we use them to gain some conditions under which a right or left derived functor of some special functors over local cohomology modules belongs to a Serre subcategory. These results can conclude some generalizations of previous results in this context or regain some of them. [ABSTRACT FROM AUTHOR]

Abstract: We state a conjecture on the geometric local Arthur–Langlands functoriality at the Iwahori level. Given a morphism of Langlands dual groups over , we construct a bimodule over the affine extended Hecke algebras of G and H which should realize the geometric local Langlands functoriality at the Iwahori level. We state a second conjecture relating this bimodule and the bimodule arising in the geometric Howe correspondence at the Iwahori level. This conjecture is verified for all dual reductive pairs . [Copyright &y& Elsevier]

We present an introduction to the equivariant slice filtration. After reviewing the definitions and basic properties, we determine the slice-connectivity of various families of naturally arising spectra. This leads to an analysis of pullbacks of slices defined on quotient groups, producing new collections of slices. Building on this, we determine the slice tower for the Eilenberg-Mac Lane spectrum associated to a Mackey functor for a cyclic p-group. We then relate the Postnikov tower to the slice tower for various spectra. Finally, we pose a few conjectures about the nature of slices and pullbacks. [ABSTRACT FROM AUTHOR]

We define quasi-locally presentable categories as big unions of a chain of coreflective subcategories that are locally presentable. Under appropriate hypotheses we prove a representability theorem for exact contravariant functors defined on a quasi-locally presentable category taking values in abelian groups. We show that the abelianization of a well generated triangulated category is quasi-locally presentable, and we obtain a new proof of the Brown representability theorem. Examples of functors that are not representable are also given. [ABSTRACT FROM AUTHOR]

We relate the construction of groups which realize saturated fusion systems and signaliser functors with homology decompositions of p-local finite groups. We prove that the cohomology ring of Robinson's construction is in some precise sense very close to the cohomology ring of the fusion system it realizes. [ABSTRACT FROM AUTHOR]

Abstract: Let be an Artinian local ring with algebraically closed residue field , and let be an affine smooth group scheme over . The Greenberg functor associates to a linear algebraic group over , such that . We prove that if is a reductive group scheme over , and is a maximal torus of , then is a Cartan subgroup of , and every Cartan subgroup of is obtained uniquely in this way. Moreover, we prove that if is reductive and is a parabolic subgroup of , then is a self-normalising subgroup of , and if and are two Borel subgroups of , then the corresponding subgroups and are conjugate in . [Copyright &y& Elsevier]

Let V be a braided vector space, i.e., a vector space together with a solution $${\hat{R}\in {{End}}(V\otimes V)}$$ of the Yang-Baxter equation. Denote $${T(V):=\bigoplus_k V^{\otimes k}}$$ . We associate to $${\hat{R}}$$ a one-parameter family of solutions $${T(\hat{R})\in {\rm End}(T(V)\otimes T(V))}$$ of the Yang-Baxter equation on the tensor space T ( V). Main ingredients of the solution are braid analogues of the binomial coefficients and of the Pochhammer symbols. The association $${\hat{R}\rightsquigarrow T(\hat{R})}$$ is functorial with respect to V. [ABSTRACT FROM AUTHOR]

Let G be a profinite group and q be an integer. In this paper, we investigate the structure of the Witt–Burnside ring functor qG and the generalized Burnside ring functor BGq, which are introduced in Oh [‘q-deformation of Witt–Burnside rings’, Math. Z. 257 (2007) 151–191]. More precisely, we prove a classification theorem as q ranges over the set of integers and a decomposition theorem over the category of binomial rings when G is given by a direct sum of finitely many finite groups whose orders are mutually relatively prime. Also connections between the ring of truncated Witt vectors and Witt–Burnside rings will be dealt with. [ABSTRACT FROM PUBLISHER]

Abstract: For a finite group G, a semi-Mackey (resp. Tambara) functor is regarded as a G-bivariant analog of a commutative monoid (resp. ring). As such, some naive algebraic constructions are generalized to this G-bivariant setting. In this article, as a G-bivariant analog of the fraction of a ring, we consider fraction of a Tambara (and a semi-Mackey) functor, by a multiplicative semi-Mackey subfunctor. [Copyright &y& Elsevier]

Abstract: We prove some finiteness theorems for the Picard functor of an algebraic stack, in the spirit of SGA 6, exp. XII and XIII. In particular, we give a stacky version of Raynaudʼs relative representability theorem, we give sufficient conditions for the existence of the torsion component of the Picard functor, and for the finite generation of the Néron–Severi groups or of the Picard group itself. We give some examples and applications. In Appendix A, we prove the semicontinuity theorem for a (non-necessarily tame) algebraic stack. [Copyright &y& Elsevier]

The parameterization process used in the symbolic computation systems Kenzo and EAT is studied here as a general construction in a categorical framework. This parameterization process starts from a given specification and builds a parameterized specification by adding a parameter as a new variable to some operations. Given a model of the parameterized specification, each interpretation of the parameter, called an argument, provides a model of the given specification. Moreover, under some relevant terminality assumption, this correspondence between the arguments and the models of the given specification is a bijection. It is proved in this paper that the parameterization process is provided by a functor and the subsequent parameter passing process by a natural transformation. Various categorical notions are used, mainly adjoint functors, pushouts and lax colimits. [ABSTRACT FROM AUTHOR]

We develop methods for recognizing the Fitting classes and radicals of finite groups by Fitting functors and the prescribed properties of Hall π-subgroups. [ABSTRACT FROM AUTHOR]

Abstract: We say that a fusion system is the composition product of two subsystems if every morphism can be factored as a morphism in one fusion system followed by a morphism in the other. We establish a relationship between the characteristic idempotent of a saturated fusion system that is the composition product of saturated subsystems and the characteristic idempotents of the component systems. Consequently we obtain a compatibility result for transfer through the composition product and transfer through the component systems. [Copyright &y& Elsevier]

This paper describes a relationship between essentially finite groupoids and two-vector spaces. In particular, we show to construct two-vector spaces of Vect- valued presheaves on such groupoids. We define two-linear maps corresponding to functors between groupoids in both a covariant and contravariant way, which are ambidextrous adjoints. This is used to construct a representation-a weak functor-from Span( FinGpd) (the bicategory of essentially finite groupoids and spans of groupoids) into 2Vect. In this paper we prove this and give the construction in detail. [ABSTRACT FROM AUTHOR]

Abstract: 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 investigate relations with other Mackey-functorial constructions — crossed Burnside ring, Elliottʼs ring of G-strings, Jacobsonʼs F-Burnside ring — all these lead to the study of the Witt–Burnside construction. [Copyright &y& Elsevier]

Abstract: The coshape invariant and continuous extensions of group-valued covariant and contravariant functors, defined on the category of pairs of spaces with the homotopy type of a pair of finite CW-complexes, are constructed. [Copyright &y& Elsevier]

Abstract: A graded tensor category over a group G will be called a crossed product tensor category if every homogeneous component has at least one multiplicatively invertible object. Our main result is a description of crossed product tensor categories, graded monoidal functors, monoidal natural transformations, and braidings in terms of coherent outer G-actions over tensor categories. [Copyright &y& Elsevier]

Let XR be a geometrically irreducible smooth projective curve, defined over R, such that XR does not have any real points. Let X = XR×RC be the complex curve. We show that there is a universal real algebraic line bundle over XR × Picd (XR) if and only if the Euler characteristic ?(L) is odd for L ∈ Picd (XR). There is a universal quaternionic algebraic line bundle over X × Picd(X) if and only if the degree d is odd. (Quaternionic algebraic vector bundles are defined only on a complexification.) Take integers r and d such that ≥ 2, and d is coprime to r. Let MXR (r, d) (respectively, MX (r, d)) be the moduli space of stable vector bundles over XR (respectively, X) of rank r and degree d. We prove that there is a universal real algebraic vector bundle over XR × MXR (r, d) if and only if X(E) is odd for E ∈ MXR (r, d). There is a universal quaternionic vector bundle over X × MX (r, d) if and only if the degree d is odd. The cases where XR is geometrically reducible or XR has real points are also investigated. [ABSTRACT FROM AUTHOR]

We investigate the functors from modules to modules that occur as the summands of tensor powers and the functors from modules to Hopf algebras that occur as natural coalgebra summands of tensor algebras. The main results provide some explicit natural coalgebra summands of tensor algebras. As a consequence, we obtain some decompositions of Lie powers over the general linear groups. [ABSTRACT FROM PUBLISHER]