Handbook of Categorical Algebra
26 Aug 1994-
TL;DR: The Handbook of Categorical Algebra is intended to give, in three volumes, a rather detailed account of what, ideally, everybody working in category theory should know, whatever the specific topic of research they have chosen.
Abstract: The Handbook of Categorical Algebra is intended to give, in three volumes, a rather detailed account of what, ideally, everybody working in category theory should know, whatever the specific topic of research they have chosen. The book is planned also to serve as a reference book for both specialists in the field and all those using category theory as a tool. Volume 3 begins with the essential aspects of the theory of locales, proceeding to a study in chapter 2 of the sheaves on a locale and on a topological space, in their various equivalent presentations: functors, etale maps or W-sets. Next, this situation is generalized to the case of sheaves on a site and the corresponding notion of Grothendieck topos is introduced. Chapter 4 relates the theory of Grothendieck toposes with that of accessible categories and sketches, by proving the existence of a classifying topos for all coherent theories.
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22 Jul 2007
TL;DR: Torsion pairs in abelian and triangulated categories Torsion pair in pretriangulated classes Compactly generated torsions in triangulation categories Hereditary torsion paired in triagonality categories TORSion pairs and closed model structures (Co)torsions and generalized Tate-Vogel cohomology Nakayama categories and Cohen-Macaulay cohology Bibliography Index as mentioned in this paper.
Abstract: Introduction Torsion pairs in abelian and triangulated categories Torsion pairs in pretriangulated categories Compactly generated torsion pairs in triangulated categories Hereditary torsion pairs in triangulated categories Torsion pairs in stable categories Triangulated torsion(-free) classes in stable categories Gorenstein categories and (co)torsion pairs Torsion pairs and closed model structures (Co)torsion pairs and generalized Tate-Vogel cohomology Nakayama categories and Cohen-Macaulay cohomology Bibliography Index.
447 citations
Cites background from "Handbook of Categorical Algebra"
...Now it is easy to see that in such an adjoint triple, F is fully faithful if and only if H is fully faithful [34]....
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TL;DR: In this paper, the basics of homological algebra in exact categories in the sense of Quillen are surveyed, and Deligne's approach to derived functors is described and a comparison theorem for projective resolutions and the horseshoe lemma is given.
Abstract: We survey the basics of homological algebra in exact categories in the sense of Quillen All diagram lemmas are proved directly from the axioms, notably the five lemma, the 3 x 3-lemma and the snake lemma We briefly discuss exact functors, idempotent completion and weak idempotent completeness We then show that it is possible to construct the derived category of an exact category without any embedding into abelian categories and we sketch Deligne's approach to derived functors The construction of classical derived functors with values in an abelian category painlessly translates to exact categories, ie, we give proofs of the comparison theorem for projective resolutions and the horseshoe lemma After discussing some examples we elaborate on Thomason's proof of the Gabriel-Quillen embedding theorem in an appendix
408 citations
TL;DR: In this article, it was shown that the Kervaire invariant one elements θj ∈ π2j+1−2S exist only for j ≤ 6.
Abstract: We show that the Kervaire invariant one elements θj ∈ π2j+1−2S exist only for j ≤ 6. By Browder’s Theorem, this means that smooth framed manifolds of Kervaire invariant one exist only in dimensions 2, 6, 14, 30, 62, and possibly 126. Except for dimension 126 this resolves a longstanding problem in algebraic topology.
391 citations
Book•
20 Oct 2010
TL;DR: Benabou, Eilenberg, Kelly and Mac Lane as discussed by the authors proposed the notion of a bilax monoidal functor which plays a central role in this work and showed how ideas in Parts I and II lead to a unified approach to Hopf algebras.
Abstract: This research monograph integrates ideas from category theory, algebra and combinatorics. It is organised in three parts. Part I belongs to the realm of category theory. It reviews some of the foundational work of Benabou, Eilenberg, Kelly and Mac Lane on monoidal categories and of Joyal and Street on braided monoidal categories, and proceeds to study higher monoidal categories and higher monoidal functors. Special attention is devoted to the notion of a bilax monoidal functor which plays a central role in this work. Combinatorics and geometry are the theme of Part II. Joyal's species constitute a good framework for the study of algebraic structures associated to combinatorial objects. This part discusses the category of species focusing particularly on the Hopf monoids therein. The notion of a Hopf monoid in species parallels that of a Hopf algebra and reflects the manner in which combinatorial structures compose and decompose. Numerous examples of Hopf monoids are given in the text. These are constructed from combinatorial and geometric data and inspired by ideas of Rota and Tits' theory of Coxeter complexes. Part III is of an algebraic nature and shows how ideas in Parts I and II lead to a unified approach to Hopf algebras. The main step is the construction of Fock functors from species to graded vector spaces. These functors are bilax monoidal and thus translate Hopf monoids in species to graded Hopf algebras. This functorial construction of Hopf algebras encompasses both quantum groups and the Hopf algebras of recent prominence in the combinatorics literature. The monograph opens a vast new area of research. It is written with clarity and sufficient detail to make it accessible to advanced graduate students. Titles in this series are co-published with the Centre de Recherches Mathematiques.|This research monograph integrates ideas from category theory, algebra and combinatorics. It is organised in three parts. Part I belongs to the realm of category theory. It reviews some of the foundational work of Benabou, Eilenberg, Kelly and Mac Lane on monoidal categories and of Joyal and Street on braided monoidal categories, and proceeds to study higher monoidal categories and higher monoidal functors. Special attention is devoted to the notion of a bilax monoidal functor which plays a central role in this work. Combinatorics and geometry are the theme of Part II. Joyal's species constitute a good framework for the study of algebraic structures associated to combinatorial objects. This part discusses the category of species focusing particularly on the Hopf monoids therein. The notion of a Hopf monoid in species parallels that of a Hopf algebra and reflects the manner in which combinatorial structures compose and decompose. Numerous examples of Hopf monoids are given in the text. These are constructed from combinatorial and geometric data and inspired by ideas of Rota and Tits' theory of Coxeter complexes. Part III is of an algebraic nature and shows how ideas in Parts I and II lead to a unified approach to Hopf algebras. The main step is the construction of Fock functors from species to graded vector spaces. These functors are bilax monoidal and thus translate Hopf monoids in species to graded Hopf algebras. This functorial construction of Hopf algebras encompasses both quantum groups and the Hopf algebras of recent prominence in the combinatorics literature. The monograph opens a vast new area of research. It is written with clarity and sufficient detail to make it accessible to advanced graduate students. Titles in this series are co-published with the Centre de Recherches Mathematiques.
326 citations
Book•
01 Jan 2006TL;DR: The point-set topology of parametrized spaces change functors and compatibility relations Proper actions, equivariant bundles and fibrations, and proper actions proper actions.
Abstract: Prologue Point-set topology, change functors, and proper actions: Introduction to Part I The point-set topology of parametrized spaces Change functors and compatibility relations Proper actions, equivariant bundles and fibrations Model categories and parametrized spaces: Introduction to Part II Topologically bicomplete model categories Well-grounded topological model categories The $qf$-model structure on $\mathcal{K}_B$ Equivariant $qf$-type model structures Ex-fibrations and ex-quasifibrations The equivalence between Ho$G\mathcal{K}_B$ and $hG\mathcal{W}_B$ Parametrized equivariant stable homotopy theory: Introduction to Part III Enriched categories and $G$-categories The category of orthogonal $G$-spectra over $B$ Model structures for parametrized $G$-spectra Adjunctions and compatibility relations Module categories, change of universe, and change of groups Parametrized duality theory: Introduction to Part IV Fiberwise duality and transfer maps Closed symmetric bicategories The closed symmetric bicategory of parametrized spectra Costenoble-Waner duality Fiberwise Costenoble-Waner duality Homology and cohomology, Thom spectra, and addenda: Introduction to Part V Parametrized homology and cohomology theories Equivariant parametrized homology and cohomology Twisted theories and spectral sequences Parametrized FSP's and generalized Thom spectra Epilogue: Cellular philosophy and alternative approaches Bibliography Index Index of notation.
272 citations