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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...
Cyclic operads and cyclic homology
 in "Geometry, Topology and Physics,"International
, 1995
"... The cyclic homology of associative algebras was introduced by Connes [4] and Tsygan [22] in order to extend the classical theory of the Chern character to the noncommutative setting. Recently, there has been increased interest in more general algebraic structures than associative algebras, characte ..."
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The cyclic homology of associative algebras was introduced by Connes [4] and Tsygan [22] in order to extend the classical theory of the Chern character to the noncommutative setting. Recently, there has been increased interest in more general algebraic structures than associative algebras, characterized by the presence of several algebraic operations. Such structures appear, for example, in homotopy theory [18], [3] and topological field theory [9]. In this paper, we extend the formalism of cyclic homology to this more general framework. This extension is only possible under certain conditions which are best explained using the concept of an operad. In this approach to universal algebra, an algebraic structure is described by giving, for each n ≥ 0, the space P(n) of all nary expressions which can be formed from the operations in the given algebraic structure, modulo the universally valid identities. Permuting the arguments of the expressions gives an action of the symmetric group Sn on P(n). The sequence P = {P(n)} of these Snmodules, together with the natural composition structure on them, is the operad describing our class of algebras. In order to define cyclic homology for algebras over an operad P, it is necessary that P is what we call a cyclic operad: this means that the action of Sn on P(n) extends to an action of Sn+1 in a way compatible with compositions (see Section 2). Cyclic
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)
Moduli Spaces of Commutative Ring Spectra
, 2003
"... Let E be a homotopy commutative ring spectrum, and suppose the ring of cooperations E#E is flat over E# . We wish to address the following question: given a commutative E# algebra A in E#Ecomodules, is there an E# ring spectrum X with E#X = A as comodule algebras? We will formulate this as ..."
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Let E be a homotopy commutative ring spectrum, and suppose the ring of cooperations E#E is flat over E# . We wish to address the following question: given a commutative E# algebra A in E#Ecomodules, is there an E# ring spectrum X with E#X = A as comodule algebras? We will formulate this as a moduli problem, and give a way  suggested by work of Dwyer, Kan, and Stover  of dissecting the resulting moduli space as a tower with layers governed by appropriate AndreQuillen cohomology groups. A special case is A = E#E itself. The final section applies this to discuss the LubinTate or Morava spectra En .
Quasismooth Derived Manifolds
"... products; for example the zeroset of a smooth function on a manifold is not necessarily a manifold, and the nontransverse intersection of submanifolds is ..."
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products; for example the zeroset of a smooth function on a manifold is not necessarily a manifold, and the nontransverse intersection of submanifolds is
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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. 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...
Derived Hilbert schemes
 J. A.M.S
"... (0.1) The Derived Deformation Theory (DDT) program (see [Kon], [CK] for more details and historical references) seeks to avoid the difficulties related to the singular nature of the moduli spaces in geometry by “passing to the derived category”, i.e., developing an appropriate version of the (nonabe ..."
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(0.1) The Derived Deformation Theory (DDT) program (see [Kon], [CK] for more details and historical references) seeks to avoid the difficulties related to the singular nature of the moduli spaces in geometry by “passing to the derived category”, i.e., developing an appropriate version of the (nonabelian) derived functor of the functor
topological AndréQuillen homology and stabilization
 Topology Appl. 121 (2002) No.3
"... The quest for an obstruction theory to E ∞ ring structures on a spectrum has led a number of authors to the investigation of homology in the category of E ∞ algebras. In this note we present three, apparently very different, constructions and show that when specialized to commutative rings they all ..."
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The quest for an obstruction theory to E ∞ ring structures on a spectrum has led a number of authors to the investigation of homology in the category of E ∞ algebras. In this note we present three, apparently very different, constructions and show that when specialized to commutative rings they all agree. (AMS subject classification 55Nxx. Key words: AndréQuillen homology, E∞homology).
Stable Homotopy of Algebraic Theories
 Topology
, 2001
"... The simplicial objects in an algebraic category admit an abstract homotopy theory via a Quillen model category structure. We show that the associated stable homotopy theory is completely determined by a ring spectrum functorially associated with the algebraic theory. For several familiar algebraic t ..."
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The simplicial objects in an algebraic category admit an abstract homotopy theory via a Quillen model category structure. We show that the associated stable homotopy theory is completely determined by a ring spectrum functorially associated with the algebraic theory. For several familiar algebraic theories we can identify the parameterizing ring spectrum; for other theories we obtain new examples of ring spectra. For the theory of commutative algebras we obtain a ring spectrum which is related to AndreH}Quillen homology via certain spectral sequences. We show that the (co)homology of an algebraic theory is isomorphic to the topological Hochschild (co)homology of the parameterizing ring spectrum. # 2000 Elsevier Science Ltd. All rights reserved. MSC: 55U35; 18C10 Keywords: Algebraic theories; Ring spectra; AndreH}Quillen homology; #spaces The original motivation for this paper came from the attempt to generalize a rational result about the homotopy theory of commutative rings. For...