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18
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...
Operads, homotopy algebra, and iterated integrals for double loop spaces
 15 T. KASHIWABARA – ON THE HOMOTOPY TYPE OF CONFIGURATION COMPLEXES, CONTEMP. MATH. 146
, 1995
"... Chen's theory of iterated integrals provides a remarkable model for the di erential forms on the based loop space M of a di erentiable manifold M (Chen [10]; see also HainTondeur [23] and GetzlerJonesPetrack [21]). This article began as an attempt to nd an analogous model for 2 the complex of di ..."
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Chen's theory of iterated integrals provides a remarkable model for the di erential forms on the based loop space M of a di erentiable manifold M (Chen [10]; see also HainTondeur [23] and GetzlerJonesPetrack [21]). This article began as an attempt to nd an analogous model for 2 the complex of di erentiable forms on the double loop space M, motivated in part by the hope that this might provide an algebraic framework for understanding twodimensional topological eld theories. Our approach is to use the formalism of operads. Operads can be de ned in any symmetric monoidal category, although we will mainly be concerned with dgoperads (di erential graded operads), that is, operads in the category of chain complexes with monoidal structure de ned by the graded tensor product. An operad is a sequence of objects a(k), k 0, carrying an action of the symmetric group Sk, with products a(k) a(j1) : : : a(jk) �! a(j1 + + jk) which are equivariant and associative  we give a precise de nition in Section 1.2. An operad such that a(k) = 0 for k 6 = 1 is a monoid: in this sense, operads are a nonlinear generalization of monoids. If V is a chain complex, we may de ne an operad with EV (k) = Hom(V (k) ; V); where V (k) is the kth tensor power of V. The symmetric group Sk acts on EV (k) through its action on V (k) , and the structure maps of EV are the obvious ones. This operad plays the same role in the theory of operads that the algebra End(V) does in the theory of associative algebras. An algebra over an operad a (or aalgebra) is a chain complex A together with a morphism of operads: a �! EA. In other words, A is equipped with structure maps k: a(k)
Homological Perturbation Theory and Associativity
, 2000
"... In this paper, we prove various results concerning DGAalgebras in the context of the Homological Perturbation Theory. We distinguish two class of contractions for algebras: full algebra contractions and semifull algebra contractions. A full algebra contraction is, in particular, a semifull algebr ..."
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In this paper, we prove various results concerning DGAalgebras in the context of the Homological Perturbation Theory. We distinguish two class of contractions for algebras: full algebra contractions and semifull algebra contractions. A full algebra contraction is, in particular, a semifull algebra contraction. Taking a full algebra contraction and an "algebra perturbation" as data of the Basic Perturbation Lemma, the Algebra Perturbation Lemma (or simply, FAPL) of [20] and [29] appears in a natural way. We establish here a perturbation machinery, the SemiFull Algebra Perturbation Lemma (or, simply, SFAPL) that is a generalization of the previous one in the sense that the application range of SFAPL is wider than that of FAPL. We show four important applications in which this result is essential for the construction of algebra or coalgebra structures in various chain complexes.
A combinatorial method for computing Steenrod squares
, 1999
"... We present here a combinatorial method for computing Steenrod squares of a simplicial set X . This method is essentially based on the determination of explicit formulae for the component morphisms of a higher diagonal approximation (i.e., a family of morphisms measuring the lack of commutativity of ..."
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We present here a combinatorial method for computing Steenrod squares of a simplicial set X . This method is essentially based on the determination of explicit formulae for the component morphisms of a higher diagonal approximation (i.e., a family of morphisms measuring the lack of commutativity of the cup product on the cochain level) in terms of face operators of X.A generalization of this method to Steenrod reduced powers is sketched. c 1999 Elsevier Science B.V. All rights reserved.
Spaces of maps into classifying spaces for equivariant crossed complexes
 Indag. Math. (N.S
, 1997
"... Abstract. The results of a previous paper on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for crossed complexes, using ..."
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Abstract. The results of a previous paper on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for crossed complexes, using detailed results on the appropriate Eilenberg–Zilber theory, and of its relation to simplicial homotopy coherence. Again, our results give information not just on the homotopy classification of certain equivariant maps, but also on the weak equivariant homotopy type of the corresponding equivariant function spaces. Mathematics Subject Classifications (2001): 55P91, 55U10, 18G55. Key words: equivariant homotopy theory, classifying space, function space, crossed complex.
Homological Perturbation Theory And Computability Of Hochschild And Cyclic Homologies Of Cdgas
, 1997
"... . We establish an algorithm computing the homology of commutative dierential graded algebras (briey, CDGAs). The main tool in this approach is given by the Homological Perturbation Theory particularized for the algebra category (see [21]). Taking into account these results, we develop and rene some ..."
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. We establish an algorithm computing the homology of commutative dierential graded algebras (briey, CDGAs). The main tool in this approach is given by the Homological Perturbation Theory particularized for the algebra category (see [21]). Taking into account these results, we develop and rene some methods already known about the computation of the Hochschild and cyclic homologies of CDGAs. In the last section of the paper, we analyze the plocal homology of the iterated bar construction of a CDGA (p prime). 1. Introduction. The description of eÆcient algorithms of homological computation might be considered as a very important question in Homological Algebra, in order to use those processes mainly in the resolution of problems on algebraic topology; but this subject also inuence directly on the development of non so closedareas as Cohomological Physics (in this sense, we nd useful references in [12], [24], [25]) and Secondary Calculus ([14], [27], [28]). Working in the context ...
Rational cohomology and cohomological stability in generic representation theory
"... whose objects are functors from finite dimensional Fq–vector spaces to ¯ Fp– vector spaces. Extension groups in F(q) can be interpreted as MacLane (or Topological Hochschild) cohomology with twisted coefficients. Furthermore, evaluation on an m dimensional vector space Vm induces a homorphism from E ..."
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whose objects are functors from finite dimensional Fq–vector spaces to ¯ Fp– vector spaces. Extension groups in F(q) can be interpreted as MacLane (or Topological Hochschild) cohomology with twisted coefficients. Furthermore, evaluation on an m dimensional vector space Vm induces a homorphism from Ext ∗ (F, G) to finite group cohomology Ext ∗ (F (Vm), G(Vm)). F(q) GLm(Fq)
Simplicial Degrees Of Functors
"... this paper is to show that if G is a simplicial group of finite length, then H n G also has finite length. Here the length of a simplicial group means the length of the corresponding Moore normalization and H n G is a simplicial abelian group given by [k] 7! H n G k . A similar fact is true if we ..."
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this paper is to show that if G is a simplicial group of finite length, then H n G also has finite length. Here the length of a simplicial group means the length of the corresponding Moore normalization and H n G is a simplicial abelian group given by [k] 7! H n G k . A similar fact is true if we replace G by a simplicial ring and we take the algebraic Kfunctors instead of group homology. The origin of such results goes back to the classical paper of Dold and Puppe (see Hilfsatz 4.23 of [DP]), where the following was proved: let
On the computability of the plocal homology of twisted cartesian products of EilenbergMac Lane spaces
, 1999
"... Working in the framework of the Simplicial Topology, a method for calculating the plocal homology of a twisted cartesian product X(#, m, #, # # , n) = K(#, m) # K(# # , n) of EilenbergMac Lane spaces is given. The chief technique is the construction of an explicit homotopy equivalence between the ..."
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Working in the framework of the Simplicial Topology, a method for calculating the plocal homology of a twisted cartesian product X(#, m, #, # # , n) = K(#, m) # K(# # , n) of EilenbergMac Lane spaces is given. The chief technique is the construction of an explicit homotopy equivalence between the normalized chain complex of X and a free DGAmodule of finite type M , via homological perturbation. If X is a commutative simplicial group (being its inner product the natural one of the cartesian product of K(#, m) and K(# # , n)), then M is a DGAalgebra. Finally, in the special case K(#, 1) ## X K(# # , n), we prove that M can be a small twisted tensor product. 1