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Topic: Category:Algebra

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h17.htm Generalizations of purity in the category of abelian groups and in module categories have many applications and are really tools of homological algebra. Finally, Buchsbaum [] and others (see []) have given axioms for a ``proper class'' of short exact sequences in any abelian category and MacLane has rewritten in his ``Homology'' [] a part of homological algebra from the point of view of relative homological algebra. Section is devoted to relative homological algebra in module categories and we discuss recent results on the classification of inductively closed proper classes which are closely related with algebraically compact modules. www.elsevier.com /homepage/saj/523281/h17.htm   (527 words)

h17.htm Generalizations of purity in the category of abelian groups and in module categories have many applications and are really tools of homological algebra. Finally, Buchsbaum [] and others (see []) have given axioms for a ``proper class'' of short exact sequences in any abelian category and MacLane has rewritten in his ``Homology'' [] a part of homological algebra from the point of view of relative homological algebra. Section is devoted to relative homological algebra in module categories and we discuss recent results on the classification of inductively closed proper classes which are closely related with algebraically compact modules. www.elsevier.com /homepage/saj/523281/h17.htm   (527 words)

IRMA Strasbourg - Publication 2001 A braided monoidal category $\Cal G_{\Lambda,\theta}$ of $\Lambda$-graded associative algebras over a field $k$ is established. The structural feature (including its PBW-basis) of the braided universal enveloping algebra $\Cal U(L)$ of a $\theta$-Lie algebra $L$ is investigated as an object in $\Cal G_{\Lambda,\theta}$ and a class of quantum groups arising from $\Cal U(L)$ is constructed. The quantum affine space $k[A_q^{n0}]$, as the braided universal enveloping algebra of an abelian $\theta$-Lie algebra, is a braided Hopf algebra. www-irma.u-strasbg.fr /irma/publications/2001/01026.shtml   (527 words)

h17.htm Generalizations of purity in the category of abelian groups and in module categories have many applications and are really tools of homological algebra. Finally, Buchsbaum [ ] and others (see [ ]) have given axioms for a ``proper class'' of short exact sequences in any abelian category and MacLane has rewritten in his ``Homology'' [ ] a part of homological algebra from the point of view of relative homological algebra. Section is devoted to relative homological algebra in module categories and we discuss recent results on the classification of inductively closed proper classes which are closely related with algebraically compact modules. www1.elsevier.com /homepage/saj/523281/h17.htm   (527 words)

h17.htm Generalizations of purity in the category of abelian groups and in module categories have many applications and are really tools of homological algebra. Section is devoted to relative homological algebra in module categories and we discuss recent results on the classification of inductively closed proper classes which are closely related with algebraically compact modules. In representation theory we have also the important notions of relative projectives and relative injectives, and the analysis of their properties has led Hochschild in 1956 to the discovery of ``relative homological algebra'' []. www.elsevier.com /homepage/saj/523281/h17.htm   (527 words)

HUEVOFRITO :: Science/Math/Algebra/Category_Theory CT Category Theory - Section of the e-print arXiv dealing with category theory, including such topics as: enriched categories, topoi, abelian categories, monoidal categories, homological algebra. Category Theory and Homological Algebra - In the "known maths" series. The Computational Category Theory Project - The aim of the project is the development of software on a wide variety of platforms for computing with mathematical categories and associated algebraic structures. www.yehoo.com.ar /dir/Science/Math/Algebra/Category_Theory   (527 words)

all An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived category of the ring R. This example shows that traditional homological algebra is encompassed by Quillen's homotopical algebra. By ``simplicial algebra'' we mean any category of algebras over a simplicial algebraic theory, which is allowed to be multi-sorted. This is equivalent to studying algebraic maps to the quotient of the infinite Grassmannians $BU(k)$ by a similar symmetric group action. claude.math.wesleyan.edu /~mhovey/archive/all   (527 words)

Algebraic structure - Pictures This category, being a concrete category, may be regarded as a category of sets with extra structure in the category-theoretic sense. For example, a topological group is a topological space with a group structure such that the operations of multiplication and taking inverses are continuous ; a topological group has both a topological and an algebraic structure. Similarly, the category of topological groups (with continuous group homomorphisms as morphisms) is a category of topological spaces with extra structure. www.greatestinfo.org /Algebraic_structure   (527 words)

mandell-taq.txt Because we are working in the category of E1 algebras instead of the category of commutative algebras, we have an artificial distinction between a commutative algebra like R or k that we can work relative to and a general E1 algebra. In the module case, similar observations apply with FA and FN(A) playing the roles of E and C. We have already shown in both the algebra and module contexts that the nor- malization functor N preserves fibrations and weak equivalences. We prove that the homotopy theory and Andre-Quillen cohomology of E1 simplicial algebras are equivalent to the homotopy theory and Andre-Quillen cohomology of E1 differential graded algebras. hopf.math.purdue.edu /Mandell/mandell-taq.txt   (12672 words)

2cats My ultimate goal is to take you to an elegant understanding of Frobenius algebras by means of a 2-category called the "walking biadjunction", but first I'll play around a bit with a simpler but more famous 2-category called the "walking adjunction". A "Frobenius algebra" is just a Frobenius object in the category of vector spaces. So: one definition of a "Frobenius object" in a monoidal category is that it's a monoid object / comonoid object satisfying the I = N equations. www.math.niu.edu /~rusin/known-math/01_incoming/2cats   (2679 words)

Citations: Coherence theorems for lax algebras and distributive laws - Kelly (ResearchIndex) In fact, just as N is a rig, satisfying all the ring axioms except those involving additive inverses, FinSet is what one might call a rig category. Just as the decategorification of a monoidal category is a monoid, the decategorification of any rig category is a rig. The analogy between the commutative rig R and the symmetric rig category Vect suggests the existence of a recursive hierarchy of n vector spaces. citeseer.ist.psu.edu /context/353441/0   (2679 words)

eg27 In order for this to make sense, the operad O has to be an operad in the category of coalgebras, in other words, a Hopf operad. Subject: Another response on Hopf algebras A Hopf algebra H over an operad O is an algebra H over the operad O in the category, not of modules, but of coalgebras. Paul Goerss and I have proved that the category of dg Hopf operads is a closed model category, and hence there is a cofibrant resolution of the associative operad here. www.lehigh.edu /~dmd1/eg27   (2679 words)

Category Theory - The Great Web Directory Category theory for conformal boundary conditions - Category theory for conformal boundary conditions Category theory for conformal boundary conditions We study properties of the category of modules of an algebra object A in a tensor category C. We show that the module category inherits... 18: Category theory, homological algebra - 18: Category theory, homological algebra. Category theory, a comparatively new field of mathematics, provides a universal framework for discussing fields of algebra and geometry. www.thegreatwebdirectory.com /Science/Math/Algebra/Category_Theory/Journals   (2679 words)

PREFACE For example, within the variety of algebras, the family of finitely presented algebras constitutes an abstract category, while the family of matrix algebras constitutes a concrete category. The major areas represented in Magma V2.12 include group theory, ring theory, commutative algebra, arithmetic fields and their completions, module theory and lattice theory, finite dimensional algebras, Lie theory, representation theory, the elements of homological algebra, general schemes and curve schemes, modular forms and modular curves, finite incidence structures, linear codes and much else. However, categories based on a concrete representation are as least as important as the abstract category in most varieties. www.math.lsu.edu /magma/preface.htm   (713 words)

Abstract Stone Duality There is a ``greatest'' construction of an adjunction out of the monad (the Eilenberg-Moore or ``monadic'' category of all algebras), and a ``least'' one (the Kleisli category of free algebras). By Stone duality, the opposite of the category C of ``spaces'' is to be a category of ``algebras'', but defined by a monad over C rather than over sets. However, an ``elementwise'' interpretation of this language is not computationally feasible, even for finite sets - nor does it agree with mathematical practice, which instead manipulates the algebra of equations and predicates. www.cs.man.ac.uk /~pt/ASD/manifesto.html   (713 words)

Cornell Math - Stephen U. Chase This work impinges upon and utilizes techniques from other areas in which I also have strong interests, such as category theory and homological algebra, group theory, group schemes and Hopf algebras, representation theory, algebraic K-theory, and algebraic number theory. Following a period in my career in which the main focus of my research was the Galois module structure of algebraic integers, I have returned to investigations in pure algebra; these involve primarily Hopf algebras (especially quantum groups and Tannakian reconstruction) and, more recently, finite groups (especially the structure of p-groups). Hopf algebras and Galois theory (with M. Sweedler), Lecture Notes in Math 97, Springer-Verlag, 1969. www.math.cornell.edu /People/Faculty/chase.html   (152 words)

The Magma Philosophy The kernel of Magma contains implementations of many of the important concrete classes of structure in five fundamental branches of algebra, namely group theory, ring theory, field theory, module theory and the theory of algebras. Magma is a Computer Algebra system designed to solve problems in algebra, number theory, geometry and combinatorics that may involve sophisticated mathematics and which are computationally hard. Most of the major algorithms currently installed in the Magma kernel are state-of-the-art and give performance similar to, or better than, specialized programs. magma.maths.usyd.edu.au /magma/Features/node2.html   (152 words)

pa_schft.html Typical examples of braided monoidal categories are the category of modules over a quasitriangular Hopf algebra and the category of comodules over a coquasitriangular Hopf algebra. We generalize this construction to the category $\Mp$ of entwined modules, that is $A$-modules and $C$-comodules over Hopf algebras $A$ and $C$ where the structures are only related by an entwining map $\psi: C \tensor A \to A \tensor C$. In the dual of the category of vector spaces this allows to work with ordinary coalgebras as if they were algebras. www.mathematik.uni-muenchen.de /~pareigis/pa_schft.html   (152 words)

Citations: Rational homotopy theory - Quillen (ResearchIndex) In the category of DGCAs over any k algebra P, a model of a DGCA (A; d) is a morphism: A; A; d) of DGCAs such that A is free as a graded.... He defined a (Quillen) model category as a category C equipped with three distinguished families of maps (cofibrations, fibrations and weak equivalences) satisfying five axioms (CM1 CM5) Waldhausen observed that, since these families of maps are closed under composition and contain all identity.... ....the corresponding Quillen differential graded Lie algebra, defined as the free Lie algebra on the desuspension of the dual of the augmentation ideal of B, with differential arising from the dual of the multiplication map. citeseer.ist.psu.edu /context/201530/0   (2432 words)

PicApril20.txt A more recent example in algebraic geometry is the A1-stable homotopy category of Morel and Voevodsky [37], which is closely analogous to the initial examples from stable homotopy theory in topology and is one of our motivating examples. When C is the stable homotopy category, the cancellation property and the struc- ture of K(C) have been studied extensively by Freyd [12, 13, 14, 15] and Margo* *lis [33]. Hu [24] has begun the study of Pic(C) when C is the A1-stable homotopy category of Morel and Voevodsky [37] by finding a surprising variety of exotic invertible elements of C. hopf.math.purdue.edu /May/PicApril20.txt   (2432 words)

Citebase - On Lie Algebras in Braided Categories The set of primitive elements of a Hopf algebra in the braided category of group graded vector spaces (with a commutative group) carry the structure of a generalized Lie algebra. We show that universal enveloping algebras in the braided category exist. In particular the graded derivations of an associative algebra carry this Lie algebra structure. citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:q-alg/9612002   (363 words)

PlanetMath: Hopf algebra The category of commutative Hopf algebras is anti-equivalent to the category of affine group schemes. Further, a commutative Hopf algebra is a cogroup object in the category of commutative algebras. The prime spectrum of a commutative Hopf algebra is an affine group scheme of multiplicative units. planetmath.org /encyclopedia/HopfAlgebra.html   (363 words)

PlanetMath: Hopf algebra The category of commutative Hopf algebras is anti-equivalent to the category of affine group schemes. Cross-references: universal enveloping algebra, Lie algebra, polynomial, Lie group, bilinear form, group algebra, complex, counit, comultiplication, finite group, functions, group, structure, coalgebra, natural transformations, algebra, opposite, group scheme of multiplicative units, prime spectrum, group schemes, commutative, category, commutative diagram, unit, map, field, bialgebra The prime spectrum of a commutative Hopf algebra is an affine group scheme of multiplicative units. planetmath.org /encyclopedia/HopfAlgebra.html   (266 words)

Triangulated Categories and Kac-Moody Algebras - Peng, Xiao (ResearchIndex) 12 the derived category of a finite-dimensional algebra (context) - Happel - 1987 Abstract: By using the Ringel-Hall algebra approach, we find a Lie algebra arising in each triangulated category with T 2 = 1, where T is the translation functor. Triangulated Categories and Kac-Moody Algebras - Peng, Xiao (ResearchIndex) citeseer.ist.psu.edu /160177.html   (266 words)

Citebase - Higher-Dimensional Algebra VI: Lie 2-Algebras We construct a 2-category of semistrict Lie 2-algebras and prove that it is 2-equivalent to the 2-category of 2-term L-infinity algebras in the sense of Stasheff. Authors: Baez, John C. Crans, Alissa S. The theory of Lie algebras can be categorified starting from a new notion of "2-vector space", which we define as an internal category in Vect. We define a "semistrict Lie 2-algebra" to be a 2-vector space L equipped with a skew-symmetric bilinear functor satisfying the Jacobi identity up to a completely antisymmetric trilinear natural transformation called the "Jacobiator", which in turn must satisfy a certain law of its own. citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0307263   (266 words)

Institut AIFB - Publikation: A categorical view on algebraic lattices in formal concept analysis To this end, we build on the the notion of approximable concept with a suitable category and show that the latter is equivalent to the category of algebraic lattices. In this paper, we explore the notion of algebraicity in formal concept analysis from a category-theoretical perspective. At the same time, the paper provides a relatively comprehensive account of the representation theory of algebraic lattices in the framework of Stone duality, relating well-known structures such as Scott information systems with further formalisms from logic, topology, domains and lattice theory. www.aifb.uni-karlsruhe.de /Publikationen/showPublikation?publ_id=786   (183 words)

relative.txt In addition to the connection between phantom maps and pure homological algebra, the authors are interested in the pure derived category as a tool for connecting the global pure dimension of a ring R to the behaviour of phantom maps in DC and DP under composition. Pure homological algebra has applications to phantom maps in the stable homo- topy category [CS98 ] and in the (usual) derived category of a ring [Chr98 ], connections to Kasparov KK-theory [Sch01 ], and is actively studied by algebraists and representation theorists. This example shows that traditional homological algebra is encompassed by Quillen's homotopical algebra, and indeed this unification was one of the main points of Quillen's influential work [Qui67 ]. jdc.math.uwo.ca /papers/relative.txt   (10317 words)

relative.txt In addition to the connection between phantom maps and pure homological algebra, the authors are interested in the pure derived category as a tool for connecting the global pure dimension of a ring R to the behaviour of phantom maps in DC and DP under composition. Pure homological algebra has applications to phantom maps in the stable homo- topy category [CS98 ] and in the (usual) derived category of a ring [Chr98 ], connections to Kasparov KK-theory [Sch01 ], and is actively studied by algebraists and representation theorists. This example shows that traditional homological algebra is encompassed by Quillen's homotopical algebra, and indeed this unification was one of the main points of Quillen's influential work [Qui67 ]. jdc.math.uwo.ca /papers/relative.txt   (10317 words)

Vahagn Minasian's Homepage We specialize to the case when B is the category of a -algebras for an operad a and F is the forgetful functor, and derive milder splitting conditions in terms of the derivative of F. In addition, we compute the differentials of the forgetful functor from the category of n-Poisson algebras in terms of the homology of configuration spaces. Abstract: In this paper we develop a topological analogue of the HKR theorem, i.e we show that for certain type of S -algebras (smooth S -algebras), the natural (derivative) map THH(A) --> \Sigma TAQ(A) has a section in the category of A -modules, which induces an equivalence of A -algebras $ P www.math.nwu.edu /~minasian   (10317 words)

all01 An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived category of the ring R. This example shows that traditional homological algebra is encompassed by Quillen's homotopical algebra. The category of simplicial modules inherits a proper closed simplicial model structure from the category of simplicial presheaves. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. claude.math.wesleyan.edu /~mhovey/archive/all01   (10317 words)

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