Group Functor Research Articles

A note on a free group. The decomposition of a free group functor through the category of heaps

This note aims to introduce a left adjoint functor to the functor which assigns a heap to a group. The adjunction is monadic. It is explained how one can decompose a free group functor through the previously introduced adjoint and employ it to describe a slightly different construction of free groups.

Rewriting the elements in the intersection of the kernels of two morphisms between free groups

Let $F_g$ be the free group functor, left adjoint to the forgetful functor between the category of groups $\mathsf{Grp}$ and the category of sets $\mathsf{Set}$. Let $f\colon A \to B$ and $h\colon A \to C$ be two functions in $\mathsf{Set}$ and let $\mathrm{Ker}(\mathrm{F}_g(f))$ and $\mathrm{Ker}(\mathrm{F}_g(h))$ be the kernels of the induced morphisms between free groups. Provided that the kernel pairs $Eq(f)$ and $Eq(h)$ of $f$ and $h$ permute (such as it is the case when the pushout of $f$ and $h$ is a double extension in $\mathsf{Set}$), this short article describes a method to rewrite a general element in the intersection $\mathrm{Ker}(\mathrm{F}_g(f)) \cap \mathrm{Ker}(\mathrm{F}_g(g))$ as a product of generators in $A$ which is $\langle f,h \rangle$-symmetric in the sense of the higher covering theory of racks and quandles.

Right-cancellable protomodular algebras

A new protomodular analog of the classical criterion for the existence of a group term in the algebraic theory of a variety of universal algebras is given. To this end, the notion of a right-cancellable protomodular algebra is introduced. The translation group functor from the category of right-cancellable algebras of a protomodular variety to the category of groups is constructed. It is proved that the algebraic theory of a variety of universal algebras contains a group term if and only if it contains protomodular terms with respect to which all algebras from the variety are right-cancellable. Moreover, the right-cancellable algebras from the simplest protomodular varieties are characterized as sets with principal group actions as well as groups with simple additional structures.

On Vlasenko’s formal group laws

Given a Laurent polynomial over a flat \(\mathbf {Z}\)-algebra, Vlasenko defines a formal group law. We identify this formal group law with a coordinate system of a formal group functor. When the “Hasse–Witt matrix” of the Laurent polynomial is invertible, Vlasenko defines a matrix by taking a certain \(p\)-adic limit. We show that this matrix is the Frobenius of the Dieudonné module of this formal group modulo \(p\).

Density of the topological fundamental groups of quotient spaces

Let p:X→X/A be a quotient map where A is a subspace of X. Applying the fundamental group functor π1qtop on p, we obtain the map p⁎:π1qtop(X,a)→π1qtop(X/A,⁎) where these two spaces have their topological structure inherited from the loop spaces. In this paper, we show that under some conditions the map p⁎ has dense image. Moreover, so does p⁎:π1qtop(X,a)→π1qtop(X/(A1,A2,…,An),⁎), where A1,A2,…,An are pairwise separated subspaces of X.

On Two Group Functors Extending Schur Multipliers

Liedtke has introduced group functors K and , which are used in the context of describing certain invariants for complex algebraic surfaces. He proved that these functors are connected to the theory of central extensions and Schur multipliers. In this work, we relate K and to a group functor τ arising in the construction of the non-abelian exterior square of a group. In contrast to , there exist efficient algorithms for constructing τ, especially for polycyclic groups. Supported by computations with the computer algebra system GAP, we investigate when is a quotient of , and when and are isomorphic.

Categories and Weak Equivalences of Graded Algebras

In the study of the structure of graded algebras (such as graded ideals, graded subspaces, and radicals) or graded polynomial identities, the grading group can be replaced by any other group that realizes the same grading. Here we come to the notion of weak equivalence of gradings: two gradings are weakly equivalent if there exists an isomorphism between the graded algebras that maps each graded component onto a graded component. Each group grading on an algebra can be weakly equivalent to G-gradings for many different groups G; however, it turns out that there is one distinguished group among them, called the universal group of the grading. In this paper we study categories and functors related to the notion of weak equivalence of gradings. In particular, we introduce an oplax 2-functor that assigns to each grading its support, and show that the universal grading group functor has neither left nor right adjoint.

The functor of singular chains detects weak homotopy equivalences

The normalized singular chains of a path connected pointed space X X may be considered as a connected E ∞ E_{\infty } -coalgebra C ∗ ( X ) \mathbf {C}_*(X) with the property that the 0 0 th homology of its cobar construction, which is naturally a cocommutative bialgebra, has an antipode; i.e., it is a cocommutative Hopf algebra. We prove that a continuous map of path connected pointed spaces f : X → Y f: X\to Y is a weak homotopy equivalence if and only if C ∗ ( f ) : C ∗ ( X ) → C ∗ ( Y ) \mathbf {C}_*(f): \mathbf {C}_*(X)\to \mathbf {C}_*(Y) is an Ω \mathbf {\Omega } -quasi-isomorphism, i.e., a quasi-isomorphism of dg algebras after applying the cobar functor Ω \mathbf {\Omega } to the underlying dg coassociative coalgebras. The proof is based on combining a classical theorem of Whitehead together with the observation that the fundamental group functor and the data of a local system over a space may be described functorially from the algebraic structure of the singular chains.

Obstructions for gluing biset functors

We develop an obstruction theory for the existence and uniqueness of a solution to the gluing problem for a biset functor defined on the subquotients of a finite group G. The obstruction groups for this theory are the reduced cohomology groups of a category DG⁎ whose objects are the sections (U,V) of G, where 1≠V⊴U≤G, and whose morphisms are defined as a generalization of morphisms in the orbit category. Using this obstruction theory, we calculate the obstruction group for some well-known p-biset functors, such as the Dade group functor defined on p-groups with p odd.

1-Dimensional intrinsic persistence of geodesic spaces

Given a compact geodesic space [Formula: see text], we apply the fundamental group and alternatively the first homology group functor to the corresponding Rips or Čech filtration of [Formula: see text] to obtain what we call a persistence object. This paper contains the theory describing such persistence: properties of the set of critical points, their precise relationship to the size of holes, the structure of persistence and the relationship between open and closed, Rips and Čech induced persistences. Amongst other results, we prove that a Rips critical point [Formula: see text] corresponds to an isometrically embedded circle of length [Formula: see text], that a homology persistence of a locally contractible space with coefficients in a field encodes the lengths of the lexicographically smallest base, and that Rips and Čech induced persistences are isomorphic up to a factor [Formula: see text]. The theory describes geometric properties of the underlying space encoded and extractable from persistence.

On functor points of affine supergroups

To construct an affine supergroup from a Harish-Chandra pair, Gavarini [2] invented a natural method, which first constructs a group functor and then proves that it is representable. We give a simpler and more conceptual presentation of his construction in a generalized situation, using Hopf superalgebras over a superalgebra. As an application of the construction, given a closed super-subgroup of an algebraic supergroup, we describe the normalizer and the centralizer, using Harish-Chandra pairs. We also prove a tensor product decomposition theorem for Hopf superalgebras, and describe explicitly by cocycle deformation, the difference which results from the two choices of dualities found in literature.

Categorical Aspects of Quantum Groups: Multipliers and Intrinsic Groups

AbstractWe show that the assignment of the (left) completely bounded multiplier algebra Mlcb(L1()) to a locally compact quantum group , and the assignment of the intrinsic group, form functors between appropriate categories. Morphisms of locally compact quantum groups can be described by Hopf *-homomorphisms between universal C*-algebras, by bicharacters, or by special sorts of coactions. We show that the whole theory of completely bounded multipliers can be lifted to the universal C*-algebra level, and that the different pictures of both multipliers (reduced, universal, and as centralisers) and morphisms then interact in extremely natural ways. The intrinsic group of a quantum group can be realised as a class of multipliers, and so our techniques immediately apply. We also show how to think of the intrinsic group using the universal C*-algebra picture, and then, again, how the differing views on the intrinsic group interact naturally with morphisms. We show that the intrinsic group is the “maximal classical” quantum subgroup of a locally compact quantum group, that it is even closed in the strong Vaes sense, and that the intrinsic group functor is an adjoint to the inclusion functor from locally compact groups to quantum groups.

Continuity of coarse shape groups

We show that every coarse shape group can be obtained as the inverse limit of an inverse system of the groups coming from an HPol_{; ; ⋆}; ; -expansion. This provides a way of computing of these interesting topological invariants (whose algebraic structure is significantly richer than those of the homotpy and shape groups) in an easier manner. It is proven that, for inverse systems of compact polyhedra, the coarse shape group functor commutes with the inverse limit.

The Automorphism Group Functor of the N = 4 Lie Conformal Superalgebra

In this article, we study the structure and representability of the automorphism group functor of the N = 4 Lie conformal superalgebra over an algebraically closed field k of characteristic zero.

Jacobienne locale d'une courbe formelle relative

This article is devoted to the proof of a relative duality formula on a noetherian scheme S , giving rise on the spectrum of a field S=\operatorname{Spec}\,k to local symbols of class field theory. Relative local symbols are obtained in terms of the universal property of a couple (\Im,f) , of a S -group functor \Im , associated to a S -formal curve \mathfrak X locally of the form \mathfrak X=\operatorname{ Spf}\, A[[T]] ( S=\operatorname{Spec}\,A) . \Im is a S -group extension of the completion \check{W} of the universal S -Witt vectors group W , by the group of units {\cal O}_{S}[[T]]^{*} . We associate an S -functor \Im_{\text omb} to \Im , and we define an Abel–Jacobi morphism f:\mathfrak A=\operatorname{Spec}\ A[[T]][T^{-1}],\longrightarrow \,\Im_{\text omb} , setting up a group isomorphism: \operatorname{Hom}_{S-gr}(\Im,G)\simeq G(\mathfrak A), where G denotes a commutative smooth S -group scheme. We define an S -bihomomorphism \Im \times \Im \longrightarrow \mathbb G_m which is a local symbol (The Tame Symbol), identifying \Im to its own Cartier dual group \check\Im=\underline{\operatorname{Hom}} (\Im;\mathbb G_m) , and inducing the above isomorphism for G=\mathbb G_m . It follows that \Im may be interpreted as the relative Loop Group: \mathbb G_m(\mathfrak A) : S' \longrightarrow \mathbb G_m(\mathfrak A_{\{S'\}}) , S'=\operatorname{Spec} A' denotes a S -scheme, and we write \mathfrak A_{\{S'\}}=\operatorname{Spec} A'[[T]][T^{-1}] , and as the A -universal group of Witt-Bivectors. The couple (\Im,f) may be seen as the local analogue of the relative Rosenlicht Jacobian (Generalized Jacobian) defined by a S -smooth curve X .

Morita homotopy theory of [formula omitted]-categories

In this article we establish the foundations of the Morita homotopy theory of C⁎-categories. Concretely, we construct a cofibrantly generated simplicial symmetric monoidal Quillen model structure (denoted by MMor) on the category C1⁎cat of small unital C⁎-categories. The weak equivalences are the Morita equivalences and the cofibrations are the ⁎-functors which are injective on objects. As an application, we obtain an elegant description of Brown–Green–Rieffelʼs Picard group in the associated homotopy category Ho(MMor). We then prove that Ho(MMor) is semi-additive. By group completing the induced abelian monoid structure at each Hom-set we obtain an additive category Ho(MMor)−1 and a composite functor C1⁎cat→Ho(MMor)→Ho(MMor)−1 which is characterized by two simple properties: inversion of Morita equivalences and preservation of all finite products. Finally, we prove that the classical Grothendieck group functor becomes co-represented in Ho(MMor)−1 by the tensor unit object.

Differential conformal superalgebras and their forms

We introduce the formalism of differential conformal superalgebras, which we show leads to the “correct” automorphism group functor and accompanying descent theory in the conformal setting. As an application, we classify forms of N = 2 and N = 4 conformal superalgebras by means of Galois cohomology.

Probabilistic zeta functions of subgroup functors

A probabilistic zeta function of a finite group is generalized by using the notion of a group functor. Factorization properties of this function as a Dirichlet polynomial are studied and an example of the simplest subgroup functor associating a group with the set of its normal subgroups is considered.

Representability of Hom implies flatness

LetX be a projective scheme over a noetherian base schemeS, and letF be a coherent sheaf onX. For any coherent sheaf e onX, consider the set-valued contravariant functor Hom(e,F)S-schemes, defined by Hom(e,F) (T)= Hom(e T ,F T) where e T andF T are the pull-backs of e andF toX T =X x S T. A basic result of Grothendieck ([EGA], III 7.7.8, 7.7.9) says that ifF is flat over S then Kome,F) is representable for all e. We prove the converse of the above, in fact, we show that ifL is a relatively ample line bundle onX over S such that the functor Hom(L -n ,F) is representable for infinitely many positive integersn, thenF is flat overS. As a corollary, takingX =S, it follows that ifF is a coherent sheaf on S then the functorT ↦H°(T, F t) on the category ofS-schemes is representable if and only ifF is locally free onS. This answers a question posed by Angelo Vistoli. The techniques we use involve the proof of flattening stratification, together with the methods used in proving the author’s earlier result (see [N1]) that the automorphism group functor of a coherent sheaf onS is representable if and only if the sheaf is locally free.

EXISTENCE OF LATTICES IN KAC–MOODY GROUPS OVER FINITE FIELDS

Let [Formula: see text] be a Kac–Moody Lie algebra. We give an interpretation of Tits' associated group functor using representation theory of [Formula: see text] and we construct a locally compact "Kac–Moody group" G over a finite field k. Using (twin) BN-pairs (G,B,N) and (G,B-,N) for G we show that if k is "sufficiently large", then the subgroup B- is a non-uniform lattice in G. We have also constructed an uncountably infinite family of both uniform and non-uniform lattices in rank 2. We conjecture that these form uncountably many distinct conjugacy classes in G. The basic tool for the construction of non-uniform lattices in rank 2 is a spherical Tits system for G which we also construct.