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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:
Locally Complete Intersection Homomorphisms And A Conjecture Of Quillen On The Vanishing Of Cotangent Homology
 Ann. of Math
, 1999
"... . Classical definitions of l.c.i. homomorphisms of commutative rings are limited to maps that are essentially of finite type, or flat. The concept introduced in this paper is meaningful for homomorphisms ' : R \Gamma! S of commutative noetherian rings. It is defined in terms of the structure of ' in ..."
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. Classical definitions of l.c.i. homomorphisms of commutative rings are limited to maps that are essentially of finite type, or flat. The concept introduced in this paper is meaningful for homomorphisms ' : R \Gamma! S of commutative noetherian rings. It is defined in terms of the structure of ' in a formal neighborhood of each point of Spec S. We characterize the l.c.i. property by different conditions on the vanishing of the Andr'eQuillen homology of the Ralgebra S. One of these descriptions establishes a very general form of a conjecture of Quillen that was open even for homomorphisms of finite type: If S has a finite resolution by flat Rmodules and the cotangent complex L(S jR) is quasiisomorphic to a bounded complex of flat Smodules, then ' is l.c.i. The proof uses a mixture of methods from commutative algebra, differential graded homological algebra, and homotopy theory. The l.c.i. property is shown to be stable under a variety of operations, including composition, decomp...
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).
Homological Criteria for Regular Homomorphisms and for Locally Complete Intersection Homomorphisms
, 2000
"... this paper was written. ..."
AndreQuillen Homology Of Algebra Retracts
, 2001
"... this paper. Let K k be a surjective morphism of DG algebras, where k is a eld concentrated in degree 0, and let k ,! l be a eld extension. If K ,! KhX i k is a factorization as in 1.7, then the unique DGalgebra structure on lhX i = KhX K l is admissible. Thus, Tor de nes a functor from th ..."
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Cited by 5 (3 self)
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this paper. Let K k be a surjective morphism of DG algebras, where k is a eld concentrated in degree 0, and let k ,! l be a eld extension. If K ,! KhX i k is a factorization as in 1.7, then the unique DGalgebra structure on lhX i = KhX K l is admissible. Thus, Tor de nes a functor from the category of diagrams K k ,! l, with the obvious morphisms, to the category ofalgebras and their morphisms. 2. Factorizations of local homomorphisms Let ' : (R; m; k) ! (S; n; l) be a homomorphism of local rings, which is local in the sense that '(m) n. A regular factorization of ' is a commutative diagram # # # # # # # # ;; # # # # # # # # of local homomorphisms such that the Rmodule R is at, the ring R is regular, and the map R ! S is surjective. Regular factorizations are often easily found, for instance when ' is essentially of nite type (in particular, surjective), or when ' is the canonical embedding of R in its completion with respect to the maximal ideal. In this paper they are mostly used through the following construction of Avramov, Foxby, and B. Herzog [10]
DEFORMATION THEORY (LECTURE NOTES)
"... Abstract. First three sections of this overview paper cover classical topics of deformation theory of associative algebras and necessary background material. We then analyze algebraic structures of the Hochschild cohomology and describe the relation between deformations and solutions of the correspo ..."
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Abstract. First three sections of this overview paper cover classical topics of deformation theory of associative algebras and necessary background material. We then analyze algebraic structures of the Hochschild cohomology and describe the relation between deformations and solutions of the corresponding MaurerCartan equation. In Section 6 we generalize the MaurerCartan equation to strongly homotopy Lie algebras and prove the homotopy invariance of the moduli space of solutions of this equation. In the last section we indicate the main ideas of Kontsevich’s proof of the existence of deformation quantization of Poisson manifolds.
Realizing coalgebras over the Steenrod algebra
 Topology
"... (co)algebra K ∗ over the mod p Steenrod algebra as the (co)homology of a topological space, and for distinguishing between the phomotopy types of different realizations. The theories are expressed in terms of the Quillen cohomology of K∗. 1. ..."
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Cited by 3 (2 self)
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(co)algebra K ∗ over the mod p Steenrod algebra as the (co)homology of a topological space, and for distinguishing between the phomotopy types of different realizations. The theories are expressed in terms of the Quillen cohomology of K∗. 1.
Derived Smooth Manifolds
"... ... and it is closed under taking arbitrary intersections in a manifold. A derived manifold is a space together with a sheaf of local C∞rings that is obtained by patching together homotopy zerosets of smooth functions on Euclidean spaces. We show that derived manifolds come equipped with a stable ..."
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... and it is closed under taking arbitrary intersections in a manifold. A derived manifold is a space together with a sheaf of local C∞rings that is obtained by patching together homotopy zerosets of smooth functions on Euclidean spaces. We show that derived manifolds come equipped with a stable normal bundle and can be imbedded into Euclidean space. We define a cohomology theory called derived cobordism, and use a PontrjaginThom argument to show that the derived cobordism theory is isomorphic to the classical cobordism theory. This allows us to define fundamental classes in cobordism for all derived manifolds. In particular, the intersection A ∩ B of submanifolds A, B ⊂ X exists on the categorical level in our theory, and a cup product formula [A] ⌣ [B] = [A ∩ B] holds, even if the submanifolds are not transverse. One can thus consider the theory of derived manifolds as a categorification of intersection theory.
Operads and ΓHomology of Commutative Rings
, 1998
"... this paper, we construct and investigate the natural homology theory for coherently homotopy commutative dgalgebras, usually known as E1 algebras. We call the theory \Gammahomology for historical reasons (see, for instance, [3]). Since discrete commutative rings are E1 rings, we obtain by special ..."
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this paper, we construct and investigate the natural homology theory for coherently homotopy commutative dgalgebras, usually known as E1 algebras. We call the theory \Gammahomology for historical reasons (see, for instance, [3]). Since discrete commutative rings are E1 rings, we obtain by specialization a new homology theory for commutative rings. This special case is far from trivial. It has the following application in stable homotopy theory, which was our original motivation and which will be treated in a sequel to this paper. The obstructions to an E1 multiplicative structure on a spectrum lie (under mild hypotheses) in the \Gammacohomology of the corresponding dual Steenrod algebra, just as the obstructions to an A1 structure lie in the Hochschild cohomology of that algebra [15]. The \Gammahomology of a discrete commutative algebra B can be understood as a refinement of Harrison homology, which was originally defined as the homology of the quotient of the Hochschild complex by the subcomplex generated by nontrivial shuffle products. It is better defined as the homology of a related complex which one obtains by tensoring each term