## Homological perturbations, equivariant cohomology, and Koszul duality

Citations: | 10 - 6 self |

### BibTeX

@MISC{Huebschmann_homologicalperturbations,,

author = {Johannes Huebschmann},

title = {Homological perturbations, equivariant cohomology, and Koszul duality},

year = {}

}

### OpenURL

### Abstract

Dedicated to the memory of V.K.A.M. Gugenheim Abstract. Our main objective is to demonstrate how homological perturbation theory (HPT) results over the last 40 years immediately or with little extra work give some of the Koszul duality results that have appeared in the last decade. Higher homotopies typically arise when a huge object, e. g. a chain complex defining various invariants of a certain geometric situation, is cut to a small model, and the higher homotopies can then be dealt with concisely in the language of sh-structures (strong homotopy structures). This amounts to precise ways of handling the requisite additional structure encapsulating the various coherence conditions. Given e. g. two augmented differential graded algebras A1 and A2, an sh-map from A1 to A2 is a twisting cochain from the reduced bar construction BA1 of A1 to A2 and, in this manner, the class of morphisms of augmented differential graded algebras is extended to that of sh-morphisms. In the present paper, we explore small models for equivariant (co)homology via differential homological algebra techniques including homological perturbation theory which, in turn, is a standard tool to handle

### Citations

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Citation Context ...or 1 ≤ j ≤ n so that, when x1, . . ., xn are interpreted as cohomology classes of G in the obvious manner, the real cohomology H ∗ G is the exterior algebra on x1, . . ., xn, ξ1, . . ., ξn; see e. g. =-=[1]-=- or [35] where such generators are constructed for a group of the kind Map 0 (Σ, K) where Σ is a closed surface of arbitrary genus rather than just the 2-sphere S 2 (for genus higher than zero additio... |

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Citation Context ...t formality. See [23] Theorem 1.1 for a discussion and, in particular, Example 5.2 in that paper. Below we shall come back to the difference between the two properties. According to an observation in =-=[47]-=- (proof of Proposition 6.8), a smooth compact symplectic manifold, endowed with a hamiltonian action of a compact Lie group, is equivariantly formal over the reals. This fact is also an immediate cons... |

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Citation Context ...we exploit the techniques developed or reproduced in earlier sections to deduce the main Koszul duality results in a conceptual manner. Our constructions of small models extend some of the results in =-=[21,22,24]-=- related with Koszul duality by placing the latter in the sh-context in the sense isolated in the seminal paper [55] of Stasheff and Halperin; the theory of sh-modules was then exploited in [29,51,52]... |

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Citation Context ...ompact Lie group, and let G = Map 0 (S 2 , K), otherwise known as Ω 2 K, the group of based smooth maps from S 2 to K, with pointwise multiplication. Endowed with the induced differentiable structure =-=[10]-=-, [11], G is a Lie group. Let x1, . . ., xn be odd degree real cohomology classes of K so that the real cohomology H ∗ K is the exterior algebra on these classes. Then there are real cohomology classe... |

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Citation Context ... the −→ Ep+1,0 p (p ≥ 2) in the cohomology spectral sequence (E∗, d∗) of the fibration determines an additive relation differential dp: E 0,p p τ: E 0,p 2 ⇀ Ep+1,0 2 referred to as transgression, cf. =-=[48]-=- (XI.4 p. 332); this is not the original definition of transgression but it is an equivalent notion. The elements of E 0,p p ⊆ E 0,p 2 ∼ = H p (F) are then referred to as transgressive. When f is the ... |

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Citation Context ...ltration preserving, the contraction is said to be filtered. The requirements (2.1.3) are referred to as annihilation properties or side conditions. The notion of contraction was introduced in §12 of =-=[16]-=-; it is among the basic tools in homological perturbation theory, cf. [44] and the literature there. Remark 2.1.4. It is well known that the side conditions (2.1.3) can always be achieved. This fact r... |

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Citation Context ...ogical group or a Lie group). Recall that a G-space is called equivariantly formal over the ground ring R when the spectral sequence from ordinary cohomology to Gequivariant cohomology collapses, cf. =-=[2]-=- and [20]. Thus when the G-space X is equivariantly formal (over R), the graded object E∞(X) associated with the Gequivariant cohomology H∗ G (X) relative to the Serre filtration (the filtration comin... |

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Citation Context ...ion of twisting cochain in differential homological algebra,HPT, EQUIVARIANT COHOMOLOGY, KOSZUL DUALITY 3 introduced in [6], is intimately related to that of connection in differential geometry, cf. =-=[8]-=-, [9], [12], [36], as well as to the Maurer-Cartan or master equation, cf. [45]. We now give a brief overview of the paper. Section 1 is preliminary in character; is contains a eview of various differ... |

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Citation Context ... of X as a free left S1-space. We remind the reader that S2n−1 is the Stiefel manifold G(n, n − 1) of unitary complex 1-frames in Cn ; cell decompositions of general Stiefel manifolds can be found in =-=[56]-=- (Chap. 4). Somewhat more formally: Pick a 1-cocycle representing the generator of H1 (S1) and view this 1-cocycle as a morphism ϑ: C∗(S1 ) → Λ[v] of differential graded algebras, necessarily inducing... |

65 | Perturbation theory in differential homological algebra II
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Citation Context ... and h are filtration preserving, the contraction is said to be filtered. This notion of contraction was introduced in §12 of [12]; it is among the basic tools in homological perturbation theory, cf. =-=[21,32]-=- and the literature there. The argument in [23] (4.1) establishes the following: Proposition 2.2∗. Let C and C ′ be coaugmented differential graded coalgebras, let (C π −−→ ←−− ′ C , h) be a contracti... |

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Citation Context ...f twisting cochain in differential homological algebra,HPT, EQUIVARIANT COHOMOLOGY, KOSZUL DUALITY 3 introduced in [6], is intimately related to that of connection in differential geometry, cf. [8], =-=[9]-=-, [12], [36], as well as to the Maurer-Cartan or master equation, cf. [45]. We now give a brief overview of the paper. Section 1 is preliminary in character; is contains a eview of various differentia... |

50 |
Small models for chain algebras
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Citation Context ... be done by other methods. Equivariant cohomology may be viewed as being part of group cohomology and, in this spirit, the present paper pushes further some of the ideas developed in [30]–[35] and in =-=[44]-=-. In a follow-up paper [41], we have worked out a related approach to equivariant de Rham theory in the framework of suitable relative derived functors, and in [42] we have extended this approach to e... |

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Twisted tensor products
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Citation Context ...t here, in particular, to construct twisting cochains and contractions. The notion of twisting cochain in differential homological algebra,HPT, EQUIVARIANT COHOMOLOGY, KOSZUL DUALITY 3 introduced in =-=[6]-=-, is intimately related to that of connection in differential geometry, cf. [8], [9], [12], [36], as well as to the Maurer-Cartan or master equation, cf. [45]. We now give a brief overview of the pape... |

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Citation Context ... given a differential graded left A-module N, the A-action on N induces the chain map (1.1.5) C ⊗τ A ⊗ N −→ C ⊗τ N which induces an isomorphism (1.1.6) (C ⊗τ A) ⊗A N −→ C ⊗τ N of chain complexes, cf. =-=[25]-=- (2.6∗ Proposition). In fact, cf. [25] (2.4∗ Proposition), the twisted differential on C ⊗τ N is the unique differential on C ⊗ N which makes (1.1.5) into a chain map. We recall that the twisting coch... |

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Citation Context ...imilar illustration is given in Example 7.1 below. This example serves, in4 JOHANNES HUEBSCHMANN particular, as an illustration for the notion of twisting cochain. According to a result of Frankel’s =-=[20]-=- and Kirwan’s [47], a smooth compact symplectic manifold, endowed with a hamiltonian action of a compact Lie group, is equivariantly formal over the reals. Both examples show that the compactness hypo... |

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Citation Context ...e write N∂ and MD for the new chain complexes, n≥0 (2.3.5) (MD g∂ −−−→ ←−−− ∇∂ N∂, h∂) constitute a new filtered contraction that is natural in terms of the given data. Proof. Details may be found in =-=[7]-=- and in Lemmata 3.1 and 3.2 of [25]. □ Remark 2.4. Given an arbitrary augmented differential graded algebra A, the construction in [50] (2.14 Proposition, 2.15 Corollary) yields a contraction (ΩBA ∇ −... |

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Citation Context ...as a cochain complex, is then a chain complex which is concentrated in non-positive degrees. Differential homological algebra terminology and notation will essentially be the same as that in [29] and =-=[46]-=-. We will write the reduced bar and cobar functors as B and Ω, respectively, rather than as B and Ω; see e. g. [50] for explicit descriptions. In Section 6 we will reproduce an explicit description of... |

30 |
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Citation Context ...e reduced bar construction. Definitions of the differential Cotor-functor may be found in [19] (p. 206) and in [46] (Chap. 1), and definitions of the differential Tor and Ext functors may be found in =-=[28]-=- (p. 3 and p. 11); see also [18] (p. 7). The Eilenberg-Koszul sign convention is in force thoughout. The degree of a homogeneous element x of a graded object is written as |x|. Given the chain complex... |

28 |
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Citation Context ...and Moore involves appropriate resolutions in the differential graded category. Let N be a (C∗G)-module; then (N, ζG) is an sh-module over H∗G and, for the sake of consistency with the definitions in =-=(5)-=- above, we will write this sh-module as (N, τ Ω ) so that N is considered as a differential graded (ΩH∗(BG))-module via ζG: H∗(BG) → C∗G, cf. (4.4.1 ∗ ). The small model (6.5) t ∞ (N, τ Ω ) = (H∗(BG))... |

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Citation Context ...ion of the term ‘extension’, this yields functorial extensions of the originalt Tor-, Cotor-, and Ext-functors, and the functoriality of those extensions is then referred to as extended functoriality =-=[29]-=-. We maintain the hypothesis that G is of strictly exterior type in such a way that the duals of the exterior homology generators are universally transgressive. In the category DASHh of differential n... |

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Citation Context ...e to the group and not for the simplicial structure. Well known categorical machinery formalizes this situation but we shall not need this kind of formalization. For intelligibility we recall that in =-=[3]-=- the functor which we write as E is denoted by P. 1.4. The geometric resolution and nerve construction Let G be a topological group. The universal simplicial G-bundle EG → NG is a special case of the ... |

21 | Formal solution of the master equation via HPT and deformation theory
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(Show Context)
Citation Context ...COHOMOLOGY, KOSZUL DUALITY 3 introduced in [6], is intimately related to that of connection in differential geometry, cf. [8], [9], [12], [36], as well as to the Maurer-Cartan or master equation, cf. =-=[45]-=-. We now give a brief overview of the paper. Section 1 is preliminary in character; is contains a eview of various differential homological algebra techniques and includes in particular a discussion o... |

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Citation Context ...d observation that an exact sequence of chain complexes which splits as an exact sequence of graded modules and which has a contractible quotient necessarily splits in the category of chain complexes =-=[13]-=- (2.18). Here is an essential piece of machinery. Definition 2.1. A contraction (2.1.1) (M π −−→ ←−− ∇ N, h) of chain complexes consists of chain complexes N and M, chain maps π: N → M and ∇: M → N, a... |

19 | On a perturbation theory for the homology of the loop space - Gugenheim - 1982 |

19 |
Cohomology of metacyclic groups
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Citation Context ...op small models of the kind given below via HPT techniques during the 80’s; the HPT-techniques include refinements of reasoning usually carried out in the literature via spectral sequences. In [17] – =-=[20]-=-, I have constructed and exploited suitable small models encapsulating the appropriate sh-structures in the (co)homology of a discrete group, and by means of these small models, I did explicit numeric... |

18 |
The Eilenberg–Moore spectral sequence and strongly homotopy multiplicative maps
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Citation Context ...al algebra terminology and notation will essentially be the same as that in [29] and [46]. We will write the reduced bar and cobar functors as B and Ω, respectively, rather than as B and Ω; see e. g. =-=[50]-=- for explicit descriptions. In Section 6 we will reproduce an explicit description of the reduced bar construction. Definitions of the differential Cotor-functor may be found in [19] (p. 206) and in [... |

18 |
Perturbation theory and free resolutions for nilpotent groups of class 2
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Citation Context ...o develop small models of the kind given below via HPT techniques during the 80’s; the HPT-techniques include refinements of reasoning usually carried out in the literature via spectral sequences. In =-=[17]-=- – [20], I have constructed and exploited suitable small models encapsulating the appropriate sh-structures in the (co)homology of a discrete group, and by means of these small models, I did explicit ... |

18 | Cohomology of nilpotent groups of class 2 - HUEBSCHMANN |

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Citation Context ... N(G, X) is a simplicial space whose geometric realization |N(G, X)| is the Borel construction EG ×G X; with reference to the obvious filtration, EG × X is, in particular, a free acyclic G-resolution =-=[53]-=- of X in the category of left G-spaces. In particular, when X is a point, N(G, X) comes down to the ordinary nerve NG of G, the (lean) realization of which is the ordinary classifying space BG of G. (... |

15 | The mod p cohomology rings of metacyclic groups - Huebschmann - 1989 |

11 |
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Citation Context ...ptual manner. Our constructions of small models extend some of the results in [21,22,24] related with Koszul duality by placing the latter in the sh-context in the sense isolated in the seminal paper =-=[55]-=- of Stasheff and Halperin; the theory of sh-modules was then exploited in [29,51,52] and pushed further in our paper [30]. The ‘up to homotopy’ interpretation of Koszul duality has been known for a wh... |

9 |
Homologie nicht-additiver Funktoren; Anwendugen
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Citation Context ... resolution of M; see (5.3) below for details. Koszul duality reflects the old observation that, for any k ≥ 0, the non-abelian left derived functor of the k-th exterior power functor in the sense of =-=[14]-=- is the k-th symmetric copower on the suspension (the invariants in the k-th tensor power on the suspension with respect to the symmetric group on k letters). In Section 6 we offer some unification: F... |

9 |
Koszul duality and equivariant cohomology
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(Show Context)
Citation Context ...we exploit the techniques developed or reproduced in earlier sections to deduce the main Koszul duality results in a conceptual manner. Our constructions of small models extend some of the results in =-=[21,22,24]-=- related with Koszul duality by placing the latter in the sh-context in the sense isolated in the seminal paper [55] of Stasheff and Halperin; the theory of sh-modules was then exploited in [29,51,52]... |

9 | Exact cohomology sequences with integral coefficients for torus actions
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Citation Context ... extension problem is trivial, for example when the ground ring is a field. The property that the equivariant cohomology is such an induced module is strictly stronger than equivariant formality. See =-=[23]-=- Theorem 1.1 for a discussion and, in particular, Example 5.2 in that paper. Below we shall come back to the difference between the two properties. According to an observation in [47] (proof of Propos... |

9 |
Extended moduli spaces, the Kan construction, and lattice gauge theory. Topology 38
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Citation Context ...day cannot be done by other methods. Equivariant cohomology may be viewed as being part of group cohomology, and in this spirit, the present paper pushes further some of the ideas developed in [16] – =-=[21]-=- and in [23]. In a follow-up paper [22], a related approach to duality in equivariant de Rham theory is worked out in the framework of suitable relative derived functors. 1. Preliminaries Let R be a c... |

8 |
Perturbation theory and small models for the chains of certain induced fibre spaces
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Citation Context ...lacing the latter in the sh-context in the sense isolated in the seminal paper [55] of Stasheff and Halperin; the theory of sh-modules was then exploited in [29,51,52] and pushed further in our paper =-=[30]-=-. The ‘up to homotopy’ interpretation of Koszul duality has been known for a while in the context of operads as well and is also behind e. g. [21]. Yet we believe that our approach in terms of HPT cla... |

6 |
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Citation Context ... . . . ⊆ L be a filtered chain complex. Recall that ({Fp}, L) is said to be complete when the canonical map L → limp L/Fp into the projective limit limp L/Fp is an isomorphism of chain complexes, cf. =-=[17]-=- (Section 4) where the terminology P-complete is used and [46] (I.3). For completeness we recall that ({Fp}, L) is said to be cocomplete when the canonical map limp Fp → L from the injective limit lim... |

5 |
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Citation Context ...y here to make this adjointness precise; see e. g. [29] for details. We will say that the twisting cochain τ is acyclic when C ⊗τ A is an acyclic complex (and hence an acyclic construction, cf. e. g. =-=[49]-=- for this notion). Subject to mild appropriate additional hypotheses of the kind that C and/or A be projective as R-modules—which will always hold in the paper—the adjoints τ: C → BA and τ: ΩC → A of ... |

5 |
Relative homological algebra, homological perturbations, equivariant de Rham theory, and Koszul duality, math.AG/0401161
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Citation Context ...uivariant cohomology may be viewed as being part of group cohomology, and in this spirit, the present paper pushes further some of the ideas developed in [16] – [21] and in [23]. In a follow-up paper =-=[22]-=-, a related approach to duality in equivariant de Rham theory is worked out in the framework of suitable relative derived functors. 1. Preliminaries Let R be a commutative ring with 1, taken hencefort... |

3 |
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Citation Context ...[16,17,18] related with Koszul duality by placing the latter in the sh-context in the sense isolated in the seminal paper [42] of Stasheff and Halperin; the theory of sh-modules was then exploited in =-=[23,38,39]-=- and pushed further in our paper [24]. The ‘up to homotopy’ interpretation of Koszul duality has been known for a while in the context of operads as well and is also behind e. g. [16]. Yet we believe ... |

3 |
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Citation Context ...ct Lie group, which may even be infinite dimensional. In the infinite dimensional case, the group is still required to be of finite homological type; all that is needed is, then, a differentiable [3],=-=[4]-=- or diffeological structure [28]. Thus, our theory applies, for example, to groups of gauge transformations. Again, the first step (within our approach) is to describe equivariant de Rham theory as a ... |

2 |
duality and equivariant cohomology
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Citation Context ...we exploit the techniques developed or reproduced in earlier sections to deduce the main Koszul duality results in a conceptual manner. Our constructions of small models extend some of the results in =-=[21,22,24]-=- related with Koszul duality by placing the latter in the sh-context in the sense isolated in the seminal paper [55] of Stasheff and Halperin; the theory of sh-modules was then exploited in [29,51,52]... |

2 |
moduli spaces, the Kan construction, and lattice gauge theory, Topology 38
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(Show Context)
Citation Context ...today cannot be done by other methods. Equivariant cohomology may be viewed as being part of group cohomology and, in this spirit, the present paper pushes further some of the ideas developed in [30]–=-=[35]-=- and in [44]. In a follow-up paper [41], we have worked out a related approach to equivariant de Rham theory in the framework of suitable relative derived functors, and in [42] we have extended this a... |

2 | free multi models for chain algebras, in - Minimal |

2 |
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Citation Context ... infinite dimensional. In the infinite dimensional case, the group is still required to be of finite homological type; all that is needed is, then, a differentiable [3],[4] or diffeological structure =-=[28]-=-. Thus, our theory applies, for example, to groups of gauge transformations. Again, the first step (within our approach) is to describe equivariant de Rham theory as a suitable differential graded der... |

2 |
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Citation Context ...hains on G acts from the left on the normalized singular chains C∗(X) and from the right on the normalized singular cochains C∗ (X) on X. Definitions of the differential Cotor-functor may be found in =-=[13]-=- (p. 206) and in [34] (Chap. 1), and definitions of the differential Tor and Ext functors may be found in [22] (p. 3 and p. 11); see also [14] (p. 7). The construction of the small models for ordinary... |

2 |
Formal solution of the master equation via
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(Show Context)
Citation Context ...ing cochain in differential homological algebra is intimately related to that of connection in differential geometry, cf. [6], [7], [10], [30], as well as to the Maurer-Cartan or master equation, cf. =-=[33]-=-. Section 1 is preliminary in character. In Section 2, we explain briefly the requisite HPT-techniques. The small models for (singular) equivariant G-(co)homology for a topological group G having as h... |

1 |
indices and Chern classes, Adv
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Citation Context ... Lie group, and let G = Map 0 (S 2 , K), otherwise known as Ω 2 K, the group of based smooth maps from S 2 to K, with pointwise multiplication. Endowed with the induced differentiable structure [10], =-=[11]-=-, G is a Lie group. Let x1, . . ., xn be odd degree real cohomology classes of K so that the real cohomology H ∗ K is the exterior algebra on these classes. Then there are real cohomology classes ξ1, ... |

1 |
A note on principal constructions
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(Show Context)
Citation Context ... a connected coaugmented differential graded coalgebra is given in [46] (II.4.5 Theorem p. 148). The special cases for a connected algebra A and a simply connected coalgebra C can be found already in =-=[15]-=-.3. Duality HPT, EQUIVARIANT COHOMOLOGY, KOSZUL DUALITY 19 Let C be a coaugmented differential graded coalgebra, A an augmented differential graded algebra, and τ: C → A an acyclic twisting cochain. ... |