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TRIPLES, ALGEBRAS AND COHOMOLOGY
 REPRINTS IN THEORY AND APPLICATIONS OF CATEGORIES
, 2003
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Cohomology of Algebraic Theories
 J. of Algebra
, 1991
"... this paper were announced in [11], [18]. Some of them were obtained by the second author only, in particular, the main theorem of Section 2. 2. On the Mac Lane Cohomology In this section we prove Theorem A, which is concerned with the relationship between Mac Lane cohomology of rings and Ext groups ..."
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this paper were announced in [11], [18]. Some of them were obtained by the second author only, in particular, the main theorem of Section 2. 2. On the Mac Lane Cohomology In this section we prove Theorem A, which is concerned with the relationship between Mac Lane cohomology of rings and Ext groups in functor categories; related questions are discussed. Let us recall the denition of the Mac Lane cohomology from [13]. Consider the sets Cn with 2 n elements  ntuples ("1 ; :::; "n ), where " i = 0 or 1, for n > 0 and i 6 n, and the 0tuple ( ) for n = 0. For convenience Cn can be visualized as the set of vertices of an ncube, the product of n copies of the 1cube with vertices 0 and 1. Dene maps 0 i ; 1 i : Cn ! Cn+1 , 1 6 i 6 n + 1, by the equalities 0 i ("1 ; :::; "n) = ("1 ; :::; " i 1 ; 0; " i+1 ; :::; "n ); 1 i ("1 ; :::; "n) = ("1 ; :::; " i 1 ; 1; " i+1 ; :::; "n ): For an abelian group A and a set S, let A[S] denote the sum of S copies of the group A. Since the sets Cn are nite, the group A[Cn ] can be identied with the group of all maps t : Cn ! A: Let Q 0 n (A) be the free abelian group generated by the set A[Cn ], i. e., Q 0 n (A) = Z[A[Cn ]]: Following Mac Lane [13], dene for i = 1; 2; :::; n the homomorphisms R i ; S i ; P i : Q 0 n (A) ! Q 0 n 1 (A) by R i = Z[R i ]; S i = Z[S i ]; P i = Z[P i ]; COHOMOLOGY OF ALGEBRAIC THEORIES 257 where R i ; S i ; P i : A[Cn ] ! A[Cn 1 ] are homomorphisms dened for e 2 Cn 1 and t 2 A[Cn ] by (R i t)(e) = t(0 i e); (S i t)(e) = t(1 i e); (P i t)(e) = t(0 i e) + t(1 i e): In [13], Mac Lane denes the boundary homomorphism @ : Q 0 n (A) ! Q 0 n 1 (A) by the equality @ = n X i=1 ( 1) i (P i R i S i ): A generator t : Cn ! A of the group Q 0 n (A) is called a slab when t( ) = 0, for n...
The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads
, 2007
"... Lawvere theories and monads have been the two main category theoretic formulations of universal algebra, Lawvere theories arising in 1963 and the connection with monads being established a few years later. Monads, although mathematically the less direct and less malleable formulation, rapidly gained ..."
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Lawvere theories and monads have been the two main category theoretic formulations of universal algebra, Lawvere theories arising in 1963 and the connection with monads being established a few years later. Monads, although mathematically the less direct and less malleable formulation, rapidly gained precedence. A generation later, the definition of monad began to appear extensively in theoretical computer science in order to model computational effects, without reference to universal algebra. But since then, the relevance of universal algebra to computational effects has been recognised, leading to renewed prominence of the notion of Lawvere theory, now in a computational setting. This development has formed a major part of Gordon Plotkin’s mature work, and we study its history here, in particular asking why Lawvere theories were eclipsed by monads in the 1960’s, and how the renewed interest in them in a computer science setting might develop in future.
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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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
RESEARCH ARTICLE EXTENSION THEORIES FOR MONOIDS
"... Leech extensions by a centralizing functor which satisfy the other requirements given. 2. The functor called F in Theorem 8 is called F in the proof. n Y ..."
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Leech extensions by a centralizing functor which satisfy the other requirements given. 2. The functor called F in Theorem 8 is called F in the proof. n Y
NONABELIAN (CO)HOMOLOGY OF LIE ALGEBRAS
"... Abstract: Nonabelian homology of Lie algebras with coefficients in Lie algebras is constructed and studied, generalising the classical ChevalleyEilenberg homology of Lie algebras. The relation of cyclic homology with Milnor cyclic homology of noncommutative associative algebras is established in ..."
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Abstract: Nonabelian homology of Lie algebras with coefficients in Lie algebras is constructed and studied, generalising the classical ChevalleyEilenberg homology of Lie algebras. The relation of cyclic homology with Milnor cyclic homology of noncommutative associative algebras is established in terms of the long exact nonabelian homology sequence of Lie algebras. Some explicit formulas for the second and the third nonabelian homology of Lie algebras are obtained. Using the generalised notion of the Lie algebra of derivations, we introduce the second nonabelian cohomology of Lie algebras with coefficients in crossed modules and extend the seventerm exact nonabelian cohomology sequence of Guin to nineterm exact sequence.
ABELIAN EXTENSIONS OF ALGEBRAS IN CONGRUENCEMODULAR VARIETIES
, 2000
"... Abstract. We define abelian extensions of algebras in congruencemodular varieties. The theory is sufficiently general that it includes, in a natural way, extensions of Rmodules for a ring R. We also define a cohomology theory, which we call clone cohomology, such that the cohomology group in dimen ..."
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Abstract. We define abelian extensions of algebras in congruencemodular varieties. The theory is sufficiently general that it includes, in a natural way, extensions of Rmodules for a ring R. We also define a cohomology theory, which we call clone cohomology, such that the cohomology group in dimension one is the group of equivalence classes of extensions.
Journal of Pure and Applied Algebra 168 (2002) 147–176 www.elsevier.com/locate/jpaa
, 1999
"... This paper begins with the observation that the category of crossed modules is tripleable ..."
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This paper begins with the observation that the category of crossed modules is tripleable