dual closure operators and their applications abstract departing from a suitable categorical description of closure operators this paper dualizes this notion and introduces some basic properties of dual closure operators usually these operators act on quotients rather than subobjects and much attention is being paid here to their key examples in algebra and topology which include the formation of monotone quotients eilenberg whyburn and concordant quotients collins in fair categorical generality these constructions are shown to be factors of the fundamental correspondence that relates connectednesses and disconnectednesses in topology as well as torsion classes and torsion free classes in algebra depending on a given cogenerator the paper also establishes a non trivial correspondence between closure operators and dual closure operators in the category of r modules dual closure operators must be carefully distinguished from interior operators that have been studied by other authors

quadratic functors on pointed categories we study polynomial functors of degree 2 called quadratic with values in the category of abelian groups ab and whose source category is an arbitrary category c with null object such that all objects are colimits of copies of a generating object e which is small and regular projective this includes all categories of models v of a pointed theory t more specifically we are interested in such quadratic functors f from c to ab which preserve filtered colimits and suitable coequalizers a functorial equivalence is established between such functors f c ab and certain minimal algebraic data which we call quadratic c modules these involve the values on e of the cross effects of f and certain structure maps generalizing the second hopf invariant and the whitehead product applying this general result to the case where e is a cogroup these data take a particularly simple form this application extends results of baues and pirashvili obtained for c being the category of groups or of modules over some ring here quadratic c modules are equivalent with abelian square groups or quadratic r modules respectively

the ternary commutator obstruction for internal crossed modules abstract in finitely cocomplete homological categories co smash products give rise to possibly higher order commutators of subobjects we use binary and ternary co smash products and the associated commutators to give characterisations of internal crossed modules and internal categories respectively the ternary terms are redundant if the category has the smith is huq property which means that two equivalence relations on a given object commute precisely when their normalisations do in fact we show that the difference between the smith commutator of such relations and the huq commutator of their normalisations is measured by a ternary commutator so that the smith is huq property itself can be characterised by the relation between the latter two commutators this allows us to show that the category of loops does not have the smith is huq property which also implies that ternary commutators are generally not decomposable into nested binary ones thus in contexts where smith is huq need not hold we obtain a new description of internal categories beck modules and double central extensions as well as a decomposition formula for the smith commutator the ternary commutator now also appears in the hopf formula for the third homology with coefficients in the abelianisation functor

ideals and clots in universal algebra and in semi abelian categories we clarify the relationship between basic constructions of semi abelian category theory and the theory of ideals and clots in universal algebra to name a few results in this frame which establish connections between hitherto separated subjects 0 regularity in universal algebra corresponds to the requirement that regular epimorphisms are normal we describe clots in categorical terms and show that ideals are images of clots under regular epimorphisms we show that the relationship between internal precrossed modules and internal reflexive graphs extends the relationship between compatible reflexive binary relations and clots

poncelet s theorem paraorthogonal polynomials and the numerical range of compressed multiplication operators abstract there has been considerable recent literature connecting poncelet s theorem to ellipses blaschke products and numerical ranges summarized for example in the recent book 16 we show how those results can be understood using ideas from the theory of orthogonal polynomials on the unit circle opuc and in turn can provide new insights to the theory of opuc

some perspectives on the eigenvalue problem this expository paper explores the relationships among a number of algorithms for solving eigenvalue problems including the power method subspace iteration the qr algorithm and the arnoldi and symmetric lanczos algorithms the symmetric lanczos algorithm is shown to be identical to the three term recursion stieltjes procedure for computing orthogonal polynomials with respect to a measure on the real line the connection between measures on the line and symmetric tridiagonal jacobi matrices is investigated if such a matrix is transformed by a step of the qr algorithm there is a corresponding transformation in the measure the tridiagonal matrices are also exploited for the construction of gaussian quadrature formulas for measures on the line the developments on the real line are replicated with suitable modifications on the unit circle via lanczos like procedures for unitary operators the best known procedure of this type is the recursion of szego for computing orthogonal polynomials on the

fixed point free action on groups of odd order let a be a finite abelian group that acts fixed point freely on a finite solvable group g assume that g is odd and a is of squarefree exponent coprime to 6 we show that the fitting length of g is bounded by the length of the longest chain of subgroups of a

relative group completions abstract we extend arbitrary group completions to the category of pairs g n where g is a group and n is a normal subgroup of g relative localizations are special cases our construction is a group theoretical analogue of fibrewise completion and fibrewise localization in homotopy theory and generalizes earlier work of hilton and others on relative localization at primes we use our approach to find conditions under which factoring out group radicals preserves exactness this has implications in the study of the effect of plus constructions on homotopy fibre sequences

minimal representations of unitary operators and orthogonal polynomials on the unit circle abstract in this paper we prove that the simplest band representations of unitary operators on a hilbert space are five diagonal orthogonal polynomials on the unit circle play an essential role in the development of this result and also provide a parameterization of such five diagonal representations which shows specially simple and interesting decomposition and factorization properties as an application we get the reduction of the spectral problem of any unitary hessenberg matrix to the spectral problem of a five diagonal one two applications of these results to the study of orthogonal polynomials on the unit circle are presented the first one concerns krein s theorem the second one deals with the movement of mass points of the orthogonality measure under mono parametric perturbations of the schur parameters

pure galois theory in categories abstract galois theory j pure appl algebra 19 1980 21 42 2 m barr abstract galois theory ii j pure appl algebra 25 1982 227 247 3 m barr and r diaconescu on locally simply connected toposes and their fundamen tal groups cahiers topologie gt om diffprentielle catigoriques 22 i98 i 301 3 14 4 s u chase d k harrison and a rosenberg galois theory and cohomology of commutative rings mem amer mafh sot 52 1965 15 33 5 s u chase and m e sweedler hopf algebras and galois theory in lecture notes in math vol 97 springer verlag new york berlin 1969 6 i diers categories of boolean sheaves of simple algebras in lecture notes in math vol 1187 springer verlag new york berlin 1986 7 a grothendieck revetements itales et groupe fondamental sga 1 in lecture notes in math vol 269 springer verlag new york berlin 1972 8 j f kennison galois theory and theaters of action in a topos j pure appl algebra 18 1980 149 164 9

obstruction theory in action accessible categories abstract we show that in semi abelian action accessible categories such as the categories of groups lie algebras rings associative algebras and poisson algebras the obstruction to the existence of extensions is classified by the second cohomology group in the sense of bourn moreover we describe explicitly the obstruction to the existence of extensions in the case of leibniz algebras comparing bourn cohomology with loday pirashvili cohomology of leibniz algebras

contractions with rank one defect operators and truncated cmv matrices the main issue we address in the present paper are the new models for completely nonunitary contractions with rank one defect operators acting on some hilbert space of dimension n these models complement nicely the well known models of livsic and sz nagy foias we show that each such operator acting on some finite dimensional respectively separable infinite dimensional hilbert space is unitarily equivalent to some finite respectively semi infinite truncated cmv matrix obtained from the full cmv matrix by deleting the first row and the first column and acting in cn respectively l2 n this result can be viewed as a nonunitary version of the famous characterization of unitary operators with a simple spectrum due to cantero moral and velazquez as well as an analog for contraction operators of the result from yu arlinski e tsekanovski non self adjoint jacobi matrices with a rank one imaginary part j funct anal 241 2006 383 438 concerning dissipative non self adjoint operators with a rank one imaginary part it is shown that another functional model for contractions with rank one defect operators takes the form of the compression f pk f on the hilbert space l2 t d with a probability measure onto the subspace k l2 t d c the relationship between characteristic functions of sub matrices of the truncated cmv matrix with rank one defect operators and the corresponding schur iterates is established we develop direct and inverse spectral analysis for finite and semi infinite truncated cmv matrices in particular we study the problem of reconstruction of such matrices from their spectrum or the mixed spectral data involving schur parameters it is pointed out that if the mixed spectral data contains zero eigenvalue then no solution unique solution or infinitely many solutions may occur in the inverse problem for truncated cmv matrices the uniqueness theorem for recovered truncated cmv matrix from the given mixed spectral data is established in this part the paper is closely related to the results of hochstadt and gesztesy simon obtained for finite self adjoint jacobi matrices

semi localizations of semi abelian categories abstract a semi localization of a category is a full reflective subcategory with the property that the reflector is semi left exact there are many interesting examples of semi localizations as for instance any torsion free subcategory of a semi abelian category by specializing a result due to s mantovani we first characterize the categories which are semi localizations of exact mal tsev categories we then prove a new characterization of protomodular categories in terms of binary relations allowing us to obtain an abstract characterization of the semi localizations of exact protomodular categories this result is very useful to study the hereditarily torsion free subcategories of semi abelian categories some examples are considered in detail in the categories of groups crossed modules commutative rings and topological groups we finally explain how these results extend the corresponding ones obtained in the abelian context by w rump

internal categories and groupoids in congruence modular varieties abstract it is well known that there exist several simple descriptions of the internal categories and groupoids in various classical algebraic categories such as groups rings lie algebras etc one of those descriptions is extended to arbitrary congruence modular varieties we describe all internal categories satisfying a certain commutator condition which always holds for internal groupoids and for all internal categories in mal tsev congruence permutable varieties the results of this paper show a deep connection between internal category theory and commutator theory in universal algebra the so called kiss difference operation also plays a central role

commutator theory action groupoids and an intrinsic schreier mac lane extension theorem abstract beyond groups of automorphisms in the category gp of groups and lie algebras of derivations in the category k lie of lie algebras there are structures of internal groupoids called action groupoids in both categories they allow a synthesis of the notion of obstruction to extensions this leads in any pointed protomodular category c with split extension classifiers to a general treatment of non abelian extensions which can be understood as morphisms in a certain groupoid tors c

schur parameter pencils for the solution of the unitary eigenproblem abstract let u v be an n n pencil with unitary matrices u and v an algorithm is presented which reduces u and v simultaneously to unitary block diagonal matrices g o q h up and g e q h vp with block size at most two it is an o n 3 process using householder eliminations and it is backward stable in the special case v i the block diagonal matrices g o g h e can be normalized so that their entries are just the schur parameters of the hessenberg condensed form of u we call g o g e a schur parameter pencil it can also be derived from u v by a lanczos like process for the solution of the eigenvalue problem for g o g e a qr type algorithm can be developed based on this unitary reduction of a pencil u v to a schur parameter pencil the condensed form is preserved throughout the process each iteration step needs only o n operations this method of solving the unitary eigenvalue problem seems to be the closest possible analogy to the qr method for the hermitian eigenvalue problem

higher central extensions and hopf formulae abstract higher extensions and higher central extensions which are of importance to non abelian homological algebra are studied and some fundamental properties are proven as an application a direct proof of the invariance of the higher hopf formulae is obtained

extension theory and the calculus of butterflies abstract this paper provides a unified treatment of two distinct viewpoints concerning the classification of group extensions the first uses weak monoidal functors the second classifies extensions by means of suitable h 2 actions we develop our theory formally by making explicit a connection between non abelian g torsors and fibrations then we apply our general framework to the classification of extensions in a semi abelian context by means of butterflies 1 between internal crossed modules as a main result we get an internal version of dedecker s theorem on the classification of extensions of a group by a crossed module in the semi abelian context bourn s intrinsic schreier mac lane extension theorem 13 turns out to be an instance of our theorem 6 3 actually even just in the case of groups our approach reveals a result slightly more general than classical schreier mac lane theorem

baer sums in homological categories abstract we give a unified treatment of the baer sums in the context of efficiently homological categories which on the one hand contains any category of groups with multiple operators and more generally any semi abelian variety and on the other hand the category of hausdorff groups and more generally any category of semi abelian hausdorff algebras this gives rise to a generalized euclide s postulate and a five terms exact sequence

centralizer and faithful groupoid abstract we correlate existence and aspect of some properties of centralizers with the conceptual notion of faithful objects we produce a large choice of examples from standard algebraic situations to topological models going through less classical cases as the new notions of algebras introduced by loday leibniz algebras associative dialgebras and trialgebras or the dual of any boolean topos

normalities and commutators abstract we first compare several algebraic notions of normality from a categorical viewpoint then we introduce an intrinsic description of higgins commutator for ideal determined categories and we define a new notion of normality in terms of this commutator our main result is to extend to any semi abelian category the following well known characterization of normal subgroups a subobject k is normal in a if and only if a k k

higher hopf formulae for homology via galois theory we use janelidze s categorical galois theory to extend brown and ellis s higher hopf formulae for homology of groups to arbitrary semi abelian monadic categories given such a category alpha and a chosen birkhoff subcategory beta of alpha thus we describe the barr beck derived functors of the reflector of alpha onto beta in terms of centralization of higher extensions in case alpha is the category gp of all groups and beta is the category ab of all abelian groups this yields a new proof for brown and ellis s formulae we also give explicit formulae in the cases of groups vs k nilpotent groups groups vs k solvable groups and precrossed modules vs crossed modules

protomodular aspect of the dual of a topos abstract the structure of the dual of a topos is investigated under its aspect of a barr exact and protomodular category in particular the normal monomorphisms in the fibres of the fibration of pointed objects are characterized and the change of base functors with respect to this same fibration are shown to reflect those normal monomorphisms

on the three subobjects lemma and its higher order generalisations abstract we solve a problem mentioned in the article 1 of berger and bourn we prove that in the context of an algebraically coherent semi abelian category two natural definitions of the lower central series coincide in a first standard approach nilpotency is defined as in group theory via nested binary commutators of the form x x x in a second approach higher higgins commutators of the form x x x are used to define nilpotent objects 18 16 20 the two are known to be different in general for instance in the context of loops the definition of bruck 6 is of the former kind while the commutator associator filtration of mostovoy 27 26 and his co authors is of the latter type another example in the context of moufang loops is given in 1 in this article we show that the two streams of development agree in any algebraically coherent semi abelian category such are for instance all orzech categories of interest 31 our proof of this result is based on a higher order version of the three subobjects lemma of 8 which extends the classical three subgroups lemma from group theory to categorical algebra it says that any n fold higgins commutator k 1 k n of normal subobjects k i x may be decomposed into a join of nested binary commutators

fitting height of a finite group with a frobenius group of automorphisms abstract suppose that a finite group g admits a frobenius group of automorphisms fh with kernel f and complement h such that f acts without nontrivial fixed points that is such that c g f 1 it is proved that the fitting height of g is equal to the fitting height of the fixed point subgroup c g h and the fitting series of c g h coincides with the intersections of c g h with the fitting series of g as a corollary it is also proved that for any set of primes the length of g is equal to the length of c g h

the snail lemma for internal groupoids abstract we establish a generalized form both of the gabriel zisman exact sequence associated with a pointed functor between pointed groupoids and of the brown exact sequence associated with a fibration of pointed groupoids our generalization consists in replacing pointed groupoids with groupoids internal to a pointed regular category with reflexive coequalizers

a hall higman shult type theorem for arbitrary finite groups we shall extend a fixed point theorem of shult to arbitrary finite groups this will have applications to the study of group automorphisms

torsion theories in homological categories we introduce and study the notion of torsion theory in the non abelian context of homological categories and we investigate the properties of the corresponding closure operator we then consider several new examples of torsion theories in the category of topological groups and more generally in any category of topological semi abelian algebras we finally characterize the hereditary torsion theories and we analyse a new example in the homological category of crossed modules c 2006 elsevier inc all rights reserved

torsion theories and radicals in normal categories we introduce a relativized notion of semi normalcy for categories that come equipped with a proper stable factorization system and we use radicals in the sense of module theory and normal closure operators in order to study torsion theories in such categories our results generalize and complement recent studies in the realm of semi abelian and in part homological categories in particular we characterize both torsion free and torsion classes in terms of their closure under extensions we pay particular attention to the homological and for our purposes more importantly normal categories of topological algebra such as the category of topological groups but our applications go far beyond the realm of these types of categories as they include for example the normal but non homological category of pointed topological spaces which is in fact a rich supplier for radicals of topological groups

higher central extensions and cohomology abstract we establish a galois theoretic interpretation of cohomology in semi abelian categories cohomology with trivial coefficients classifies central extensions also in arbitrarily high degrees this allows us to obtain a duality in a certain sense between internal homology and external cohomology in semi abelian categories these results depend on a geometric viewpoint of the concept of a higher central extension as well as the algebraic one in terms of commutators

precrossed modules and galois theory the adjunction between crossed modules and precrossed modules over a fixed group can be seen as a special case of a more general adjunction between internal groupoids and internal reflexive graphs in a mal tsev variety by using the categorical galois theory we characterize the central extensions with respect to this latter adjunction in terms of the universal algebraic commutator in particular we get a description of the central extensions of precrossed modules and of precrossed rings this characterization provides a natural way to define a categorical notion of peiffer commutator c 2005 elsevier inc all rights reserved

topological semi abelian algebras given an algebraic theory t whose category of models is semi abelian we study the category top t of topological models of t and generalize to it most classical results on topological groups in particular top t is homological which includes barr regularity and forces the mal cev property every open subalgebra is closed and every quotient map is open we devote special attention to the hausdorff compact locally compact connected totally disconnected and profinite t algebras c 2004 elsevier inc all rights reserved

a categorical approach to commutator theory abstract we define a notion of commutator for equivalence relations in a mal cev category and show that it verifies a universal property

weighted commutators in semi abelian categories abstract we introduce new notions of weighted centrality and weighted commutators corresponding to each other in the same way as centrality of congruences and commutators do in the smith commutator theory both the huq commutator of subobjects and pedicchio s categorical generalization of smith commutator are special cases of our weighted commutators in fact we obtain them by taking the smallest and the largest weight respectively at the end of the paper we briefly consider the universal algebraic context in connection with an older work of the third author on the ideal theory version of the commutator theory

commutator theory for loops abstract using the freese mckenzie commutator theory for congruence modular varieties as the starting point we develop commutator theory for the variety of loops the fundamental theorem of congruence commutators for loops relates generators of the congruence commutator to generators of the total inner mapping group we specialize the fundamental theorem into several varieties of loops and also discuss the commutator of two normal subloops consequently we argue that some standard definitions of loop theory such as elementwise commutators and associators should be revised and linked more closely to inner mappings using the new definitions we prove several natural properties of loops that could not be so elegantly stated with the standard definitions of loop theory for instance we show that the subloop generated by the new associators defined here is automatically normal we conclude with a preliminary discussion of abelianess and solvability in loops

opuc cmv matrices and perturbations of measures supported on the unit circle abstract let us consider a hermitian linear functional defined on the linear space of laurent polynomials with complex coefficients in the literature canonical spectral transformations of this functional are studied the aim of this research is focused on perturbations of hermitian linear functionals associated with a positive borel measure supported on the unit circle some algebraic properties of the perturbed measure are pointed out in a constructive way we discuss the corresponding sequences of orthogonal polynomials as well as the connection between the associated verblunsky coefficients then the structure of the matrices of the perturbed linear functionals which is the main tool for the comparison of their corresponding cmv matrices is deeply analyzed from the comparison between different cmv matrices other families of perturbed verblunsky coefficients will be considered we introduce a new matrix named fundamental matrix that is a tridiagonal symmetric unitary matrix containing basic information about the family of orthogonal polynomials however we show that it is connected to another family of orthogonal polynomials through the takagi decomposition

fundamental group functors in descent exact homological categories abstract we study the notion of a fundamental group in the framework of descent exact homological categories this setting is sufficiently large to include several categories of algebraic nature such as almost abelian categories semi abelian categories and categories of topological semi abelian algebras for many adjunctions in this context we describe the fundamental groups by generalised brown ellis hopf formulae for the integral homology of groups

five diagonal matrices and zeros of orthogonal polynomials on the unit circle it is shown that monic orthogonal polynomials on the unit circle are the characteristic polynomials of certain five diagonal matrices depending on the schur parameters this result is achieved through the study of orthogonal laurent polynomials on the unit circle more precisely it is a consequence of the five term recurrence relation obtained for these orthogonal laurent polynomials and the one to one correspondence established between them and the orthogonal polynomials on the unit circle as an application some results relating the behaviour of the zeros of orthogonal polynomials and the location of schur parameters are obtained

non self adjoint jacobi matrices with a rank one imaginary part we develop direct and inverse spectral analysis for finite and semi infinite non self adjoint jacobi matrices with a rank one imaginary part it is shown that given a set of n not necessarily distinct nonreal numbers in the open upper lower half plane uniquely determines an n n jacobi matrix with a rank one imaginary part having those numbers as its eigenvalues counting algebraic multiplicity algorithms of reconstruction for such finite jacobi matrices are presented a new model complementing the well known livsic triangular model for bounded linear operators with a rank one imaginary part is obtained it turns out that the model operator is a non self adjoint jacobi matrix we show that any bounded prime non self adjoint linear operator with a rank one imaginary part acting on some finite dimensional respectively separable infinite dimensional hilbert space is unitarily equivalent to a finite respectively semi infinite non self adjoint jacobi matrix this obtained theorem strengthens a classical result of stone established for self adjoint operators with simple spectrum we establish the non self adjoint analogs of the hochstadt and gesztesy simon uniqueness theorems for finite jacobi matrices with nonreal eigenvalues as well as an extension and refinement of these theorems for finite non self adjoint tri diagonal matrices to the case of mixed eigenvalues real and nonreal a unique jacobi matrix unitarily equivalent to the operator of integration ff x 2i xlf t dt in the hilbert space l2 0 l is found as well as spectral properties of its perturbations and connections with the well known bernoulli numbers we also give the analytic characterization of the weyl functions of dissipative jacobi matrices with a rank one imaginary part

higher commutator conditions for extensions in mal tsev categories abstract we define a galois structure on the category of pairs of equivalence relations in an exact mal tsev category and characterize central and double central extensions in terms of higher commutator conditions these results generalize both the ones related to the abelianization functor in exact mal tsev categories and the ones corresponding to the reflection from the category of internal reflexive graphs to the subcategory of internal groupoids some examples and applications are given in the categories of groups precrossed modules precrossed lie algebras and compact groups

on fixed point free automorphisms abstract let r be a cyclic group of prime order which acts on the extraspecial group f in such a way that f f r suppose rf acts on a group g so that c g f 1 and r g 1 it is proved that f c g r f g as corollaries to this it is shown that the fitting series of c g r coincides with the intersections of c g r with the fitting series of g and that when r is not a fermat prime the fitting heights of c g r and g are equal

hall s criterion for nilpotence in semi abelian categories abstract a well known theorem of p hall usually called hall s criterion for nilpotence states a group g is nilpotent whenever it has a normal subgroup n such that g n n and n are nilpotent we widely generalize this result replacing groups with objects in an abstract semi abelian category satisfying suitable conditions in particular these conditions are satisfied in any algebraically coherent semi abelian category and hence in many categories of classical algebraic structures including a few where hall s theorem is already known note that the categories of crossed modules crossed lie algebras n cat groups and cocommutative hopf algebras over fields are also algebraically coherent we give however a counter example showing that hall s theorem does not hold in the semi abelian variety of non associative rings

butterflies i morphisms of 2 group stacks abstract weak morphisms of non abelian complexes of length 2 or crossed modules are morphisms of the associated 2 group stacks or gr stacks we present a full description of the weak morphisms in terms of diagrams we call butterflies we give a complete description of the resulting bicategory of crossed modules which we show is fibered and biequivalent to the 2 stack of 2 group stacks as a consequence we obtain a complete characterization of the non abelian derived category of complexes of length 2 deligne s analogous theorem in the case of picard stacks and abelian sheaves becomes an immediate corollary commutativity laws on 2 group stacks are also analyzed in terms of butterflies yielding new characterizations of braided symmetric and picard 2 group stacks furthermore the description of a weak morphism in terms of the corresponding butterfly diagram allows us to obtain a long exact sequence in non abelian cohomology removing a preexisting fibration condition on the coefficients short exact sequence

butterflies in a semi abelian context it is known that monoidal functors between internal groupoids in the category grp of groups constitute the bicategory of fractions of the 2 category grpd grp of internal groupoids internal functors and internal natural transformations in grp with respect to weak equivalences that is internal functors which are internally fully faithful and essentially surjective on objects monoidal functors can be equivalently described by a kind of weak morphisms introduced by b noohi under the name of butterflies in order to internalize monoidal functors in a wide context we introduce the notion of internal butterflies between internal crossed modules in a semi abelian category c and we show that they are morphisms of a bicategory b c our main result states that when in c the notions of huq commutator and smith commutator coincide then the bicategory b c of internal butterflies is the bicategory of fractions of grpd c with respect to weak equivalences

intrinsic centrality and associated classifying properties abstract the notion of central morphism is introduced in the very soft algebraic context of unital categories it reveals inside any such category an additive core which has a strong classifying potential

3 3 lemma and protomodularity abstract the classical 3 3 lemma and snake lemma valid in any abelian category still hold in any quasi pointed the map 0 1 is a mono regular and protomodular category some applications are given in this abstract context concerning the denormalization of kernel maps and the normalization of internals groupoids i e associated crossed modules

on the regular representation of an essentially finite 2 group the regular representation of an essentially finite 2 group mathbb g in the 2 category mathbf 2vect k of kapranov and voevodsky 2 vector spaces is defined and cohomology invariants classifying it computed it is next shown that all hom categories in mathbf rep mathbf 2vect k mathbb g are 2 vector spaces under quite standard assumptions on the field k and a formula giving the corresponding intertwining numbers is obtained which proves they are symmetric finally it is shown that the forgetful 2 functor boldmath omega mathbf rep mathbf 2vect k mathbb g to mathbf 2vect k is representable with the regular representation as representing object as a consequence we obtain a k linear equivalence between the 2 vector space mathbf vect k mathcal g of functors from the underlying groupoid of mathbb g to mathbf vect k on the one hand and the k linear category mathcal e nd boldmath omega of pseudonatural endomorphisms of boldmath omega on the other hand we conclude that mathcal e nd boldmath omega is a 2 vector space and we partially describe a basis of it

resolutions higher extensions and the relative mal tsev axiom abstract we study how the concept of higher dimensional extension which comes from categorical galois theory relates to simplicial resolutions for instance an augmented simplicial object is a resolution if and only if its truncation in every dimension gives a higher extension in which sense resolutions are infinite dimensional extensions or higher extensions are finite dimensional resolutions we also relate certain stability conditions of extensions to the kan property for simplicial objects this gives a new proof of the fact that a regular category is mal tsev if and only if every simplicial object is kan using a relative setting of extensions