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Homotopy Coherent Category Theory
, 1996
"... this paper we try to lay some of the foundations of such a theory of categories `up to homotopy' or more exactly `up to coherent homotopies'. The method we use is based on earlier work on: ..."
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Cited by 22 (6 self)
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this paper we try to lay some of the foundations of such a theory of categories `up to homotopy' or more exactly `up to coherent homotopies'. The method we use is based on earlier work on:
New Model Categories From Old
 J. Pure Appl. Algebra
, 1995
"... . We review Quillen's concept of a model category as the proper setting for defining derived functors in nonabelian settings, explain how one can transport a model structure from one category to another by mean of adjoint functors (under suitable assumptions), and define such structures for categor ..."
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Cited by 13 (5 self)
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. We review Quillen's concept of a model category as the proper setting for defining derived functors in nonabelian settings, explain how one can transport a model structure from one category to another by mean of adjoint functors (under suitable assumptions), and define such structures for categories of cosimplicial coalgebras. 1. Introduction Model categories, first introduced by Quillen in [Q1], have proved useful in a number of areas  most notably in his treatment of rational homotopy in [Q2], and in defining homology and other derived functors in nonabelian categories (see [Q3]; also [BoF, BlS, DwHK, DwK, DwS, Goe, ScV]). From a homotopy theorist's point of view, one interesting example of such nonabelian derived functors is the E 2 term of the mod p unstable Adams spectral sequence of Bousfield and Kan. They identify this E 2 term as a sort of Ext in the category CA of unstable coalgebras over the mod p Steenrod algebra (see x7.4). The original purpose of this note w...
Triple cohomology of LieRinehart algebras and the canonical class of associative algebras. ArXive math.KT/0307354
"... Dedicated to the memory of Prof. A. R.Grandjeán Abstract We apply the general theory of triple cohomology to LieRinehart algebras in oreder to construct a canonical class in the third dimensional cohomology corresponding to an associative algebra. We prove that the triple cohomology is isomorphic ..."
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Dedicated to the memory of Prof. A. R.Grandjeán Abstract We apply the general theory of triple cohomology to LieRinehart algebras in oreder to construct a canonical class in the third dimensional cohomology corresponding to an associative algebra. We prove that the triple cohomology is isomorphic to the Rinehart cohomology [11] provided the LieRinehart algebra is projective over the corresponding commutative algebra. Key words: LieRinehart algebra, Hochschild cohomology, cotriple. A. M. S. Subject Class. (2000): 18G60, 16W25, 17A99.
Homological Algebra of Racks and Quandles
"... Contents Introduction 1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1 Extensions 5 1.1 Extensions and expan ..."
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Cited by 1 (1 self)
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Contents Introduction 1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1 Extensions 5 1.1 Extensions and expansions . . . . . . . . . . . . . . . . . . . . . . 8 1.2 Factor systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Abelian extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4 Quandle extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.5 Involutory extensions . . . . . . . . . . . . . . . . . . . . . . . . . 26 2 Modules 29 2.1 Rack modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2 Beck modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.3 A digression on Gmodules . . . . . . . . . . . . . . . . . . . . . 41 2.4 Free modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.5 The rack
A CARTANEILENBERG APPROACH TO HOMOTOPICAL ALGEBRA
, 707
"... Abstract. In this paper we propose an approach to homotopical algebra where the basic ingredient is a category with two classes of distinguished morphisms: strong and weak equivalences. These data determine the cofibrant objects by an extension property analogous to the classical lifting property of ..."
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Cited by 1 (0 self)
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Abstract. In this paper we propose an approach to homotopical algebra where the basic ingredient is a category with two classes of distinguished morphisms: strong and weak equivalences. These data determine the cofibrant objects by an extension property analogous to the classical lifting property of projective modules. We define a CartanEilenberg category as a category with strong and weak equivalences such that there is an equivalence of categories between its localisation with respect to weak equivalences and the relative localisation of the subcategory of cofibrant objets with respect to strong equivalences. This equivalence of categories allows us to extend the classical theory of derived additive functors to this non additive setting. The main examples include Quillen model categories and categories of functors defined on a category endowed with a cotriple (comonad) and taking values
Postnikov Invariants of Crossed Complexes
, 2004
"... We determine the Postnikov Tower and Postnikov Invariants of a Crossed Complex in a purely algebraic way. Using the fact that Crossed Complexes are homotopy types for filtered spaces, we use the above “algebraically defined ” Postnikov tower and Postnikov invariants to obtain from them those of filt ..."
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We determine the Postnikov Tower and Postnikov Invariants of a Crossed Complex in a purely algebraic way. Using the fact that Crossed Complexes are homotopy types for filtered spaces, we use the above “algebraically defined ” Postnikov tower and Postnikov invariants to obtain from them those of filtered spaces. We argue that a similar “purely algebraic ” approach to Postnikov invariants may also be used in other categories of spaces. 1
ON THE NON–BALANCED PROPERTY OF THE CATEGORY OF CROSSED MODULES IN GROUPS.
, 2003
"... Abstract. An algebraic category C is called balanced if the cotriple cohomology of any object of C vanishes in positive dimensions on injective coefficient modules. Important examples of balanced and of nonbalanced categories occur in the literature. In this paper we prove that the category of cros ..."
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Abstract. An algebraic category C is called balanced if the cotriple cohomology of any object of C vanishes in positive dimensions on injective coefficient modules. Important examples of balanced and of nonbalanced categories occur in the literature. In this paper we prove that the category of crossed modules in groups is nonbalanced.
G G GGG
, 2003
"... Cosimplicial resolutions and homotopy spectral sequences in model categories ..."
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Cosimplicial resolutions and homotopy spectral sequences in model categories
G G GGG
, 2003
"... Cosimplicial resolutions and homotopy spectral sequences in model categories ..."
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Cosimplicial resolutions and homotopy spectral sequences in model categories