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Stable model categories are categories of modules
 TOPOLOGY
, 2003
"... A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for ..."
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Cited by 78 (16 self)
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A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent ‘the same homotopy theory’. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a ‘ring spectrum with several objects’, i.e., as spectrum valued diagram categories. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can be realized by smashing with a pair of bimodules. Finally, we characterize stable model categories which represent the derived category of a ring. This is a slight generalization of Rickard’s work on derived equivalent rings. We also include a proof of the model category equivalence of modules over the EilenbergMac Lane spectrum HR and (unbounded) chain complexes of Rmodules for a ring R.
Perturbation theory in differential homological algebra
 Illinois J. Math
, 1989
"... Perturbation theory is a particularly useful way to obtain relatively small differential complexes representing a given chain homotopy type. An important part of the theory is “the basic perturbation lemma ” [RB], [G1], [LS] which is stated in terms of modules M and N of the same homotopy type. It ..."
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Cited by 66 (10 self)
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Perturbation theory is a particularly useful way to obtain relatively small differential complexes representing a given chain homotopy type. An important part of the theory is “the basic perturbation lemma ” [RB], [G1], [LS] which is stated in terms of modules M and N of the same homotopy type. It has been known for some time that it would be useful to have a perturbation
Lie theory for nilpotent L∞algebras
 Ann. Math
"... Let R be a commutative algebra over a field K of characteristic 0. The spectrum Spec(R) of R is the set Hom(R, K) of all homomorphisms from R to K. Let Ω • be the simplicial differential graded (dg) commutative algebra whose nsimplices Ωn are the dg algebra of differential forms on the geometric n ..."
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Cited by 22 (0 self)
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Let R be a commutative algebra over a field K of characteristic 0. The spectrum Spec(R) of R is the set Hom(R, K) of all homomorphisms from R to K. Let Ω • be the simplicial differential graded (dg) commutative algebra whose nsimplices Ωn are the dg algebra of differential forms on the geometric nsimplex ∆ n. In [20], Sullivan reformulated Quillen’s
Resolutions via homological perturbation
 J. Symbolic Comp
, 1991
"... The purpose of this paper is to review an algorithm for computing “small ” resolutions in homological algebra, to provide examples of its use as promised in [L1], [LS], and to illustrate the use of computer algebra in an area not usually associated with that subject. Comparison of the complexes prod ..."
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Cited by 14 (5 self)
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The purpose of this paper is to review an algorithm for computing “small ” resolutions in homological algebra, to provide examples of its use as promised in [L1], [LS], and to illustrate the use of computer algebra in an area not usually associated with that subject. Comparison of the complexes produced by the method discussed here with those produced by other methods shows
M.: Cohomology with coefficients in symmetric catgroups. An extension of Eilenberg–MacLane’s classification theorem
 Math. Proc. Cambridge Philos. Soc. 114
, 1993
"... Abstract. In this paper we introduce and study a cohomology theory {H n (−, A)} for simplicial sets with coefficients in symmetric categorical groups A. We associate to a symmetric categorical group A a sequence of simplicial sets {K(A,n)}n≥0, which allows us to give a representation theorem for our ..."
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Cited by 12 (0 self)
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Abstract. In this paper we introduce and study a cohomology theory {H n (−, A)} for simplicial sets with coefficients in symmetric categorical groups A. We associate to a symmetric categorical group A a sequence of simplicial sets {K(A,n)}n≥0, which allows us to give a representation theorem for our cohomology. Moreover, we prove that for any n ≥ 3, the functor K(−,n) is right adjoint to the functor ℘n, where℘n(X•) is defined as the fundamental groupoid of the nloop complex � n (X•). Using this adjunction, we give another proof of how symmetric categorical groups model all homotopy types of spaces Y with πi(Y) = 0foralli = n, n + 1andn ≥ 3; and also we obtain a classification theorem for those spaces: [−,Y] ∼ = H n (−,℘n(Y)).
Computing homotopy types using crossed ncubes of groups
 in Adams Memorial Symposium on Algebraic Topology
, 1992
"... Dedicated to the memory of Frank Adams ..."
The twisted EilenbergZilber theorem
 In ‘Simposio di Topologia (Messina, 1964)’, Edizioni Oderisi, Gubbio
, 1965
"... The purpose of this paper is to give a simpler proof of a theorem of E.H. Brown [Bro59], that if F → E → B is a fibre space, then there is a differential on the graded group X = C(B) ⊗Λ C(F) such that X with this differential is chain equivalent to to C(E) (where C(E) denotes the normalised singular ..."
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Cited by 7 (0 self)
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The purpose of this paper is to give a simpler proof of a theorem of E.H. Brown [Bro59], that if F → E → B is a fibre space, then there is a differential on the graded group X = C(B) ⊗Λ C(F) such that X with this differential is chain equivalent to to C(E) (where C(E) denotes the normalised singular chains of E over a ring Λ). We work in the context of (semisimplicial) twisted cartesian products (thus we assume as do the proofs of the theorem given in [Gug60, Shi62, Szc61] the results of [BGM59] on the relation between fibre spaces and twisted cartesian products). In fact we prove a general result on filtered chain complexes; this result applies to give proofs not only of Brown’s theorem but also of a theorem of G. Hirsch, [Hir53]. Our proof is suggested by the formulae (1) of [Shi62, Ch. II, §1]. Let (X,d), (Y,d) be chain complexes over a ring Λ. Let (Y,d) ∇ f − → (X,d) − → (Y,d) be chain maps and let Φ: X → X be a chain homotopy such that Let X,Y have filtrations (1.1) f ∇ = 1; (1.2) ∇f = 1 + dΦ + Φd; (1.3) fΦ = 0; (1.4) Φ ∇ = 0; (1.5) Φ 2 = 0; (1.6) ΦdΦ = −Φ. and let ∇,f,Φ all preserve these filtrations. 0 = F −1 X ⊆ F 0 X ⊆ · · · ⊆ F p X ⊆ F p+1 X ⊆ · · · (1) 0 = F −1 Y ⊆ F 0 Y ⊆ · · · ⊆ F p Y ⊆ F p+1 Y ⊆ · · · (2) Example 1 Let B,F be (semisimplicial) complexes, let (X,d) = C(B × F), the normalised chains of B × F, let (Y,d) = C(B) ⊗Λ C(F), and let ∇,f,Φ be the natural maps of the EilenbergZilber theorem as constructed explicitly in [EML53]. The relations (1.1)(1.4) are proved in [EML53] while (1.5), (1.6) are easily proved (cf. [Shi62, p.114]). The filtrations on X,Y are induced by the filtration of B by its skeletons. The fact that ∇,f,Φ preserve filtrations is a consequence of naturality of these maps (cf. [Moo56, Ch. 5, p.13]). We now wish to compare C(B ×F) with C(B ×τ F) where B ×τ F coincides with B ×F as a complex except that ∂0 in B ×τ F is given by ∂0(b,x) = (∂0b,τ(b,x)), b ∈ Bp,x ∈ Fp. Then the filtered groups of C(B × F) and C(B ×τ F) coincide but the latter has a differential d τ. If τ satisfies the normalisation condition τ(s0b ′,x) = ∂0x,
Classification of Stable Model Categories
"... A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for ..."
Abstract

Cited by 6 (5 self)
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A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent `the same homotopy theory'. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a `ring spectrum with several objects', i.e., as spectrum valued diagram categories. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can be realized by smashing with a pair of bimodules. Finally, we characterize stable model categories which represent the derived category of a ring. This is a slight generalization of Rickard's work on derived equivalent rings. We also include a proof of the model category equivalence of modules over the EilenbergMac Lane spectrum HR and (unbounded) chain complexes of Rmodules for a ring R. 1.