## Dold-Kan Type Theorem for Γ-Groups (1998)

Citations: | 1 - 1 self |

### BibTeX

@TECHREPORT{Pirashvili98dold-kantype,

author = {Teimuraz Pirashvili and A. M. Razmadze and Math Inst Alexidze},

title = {Dold-Kan Type Theorem for Γ-Groups},

institution = {},

year = {1998}

}

### OpenURL

### Abstract

Introduction \Gamma-spaces were introduced by Segal [S], who proved that they are combinatorial models for connective spectra (see also [A], [BF]). Based on Kan-Thurston theorem we show that any \Gamma-space is stably weak equivalent to a discrete \Gamma-group. By a well-known theorem of Dold-Kan the Moore normalization establishes the equivalence between the category of simplicial abelian groups and the category of chain complexes (see [DP]). mimicking the construction of normalization of simplicial groups, we give a similar construction for \Gamma-groups. This construction is based on the notion of cross-effects of functors [BP], which is a generalizatin of the classical definition of Eilenberg and Mac Lane [EM] to the non-abelian setup. Finally a Dold-Kan type theorem for the category of \Gamma-groups is proved. In abelian case our theorem claims that the category of abelian \Gamma-groups is equivalent to the category of functors Ab\Omega , where\Om

### Citations

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Citation Context ...plicial objects, it follows that for any pointed simplicial set K, the map j(K) yields also an isomorphism on homology. Therefore j(S n ) is a weak equivalence for n ? 1. It follows from Lemma 5.2 of =-=[W]-=-, that for any pointwise fibration of \Gamma-spaces B 0 ! B ! B 00 and for any pointed connected simplicial set K the induced sequence B 0 (K) ! B(K) ! B 00 (K) is still a fibration up to homotopy. We... |

45 |
Categories and cohomology theories
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Citation Context ... DOLD-KAN TYPE THEOREM FOR \Gamma -GROUPS By TEIMURAZ PIRASHVILI A.M. Razmadze Math. Inst. Alexidze str. 1, Tbilisi, 380093. Republic of Georgia 0. Introduction \Gamma-spaces were introduced by Segal =-=[S]-=-, who proved that they are combinatorial models for connective spectra (see also [A], [BF]). Based on Kan-Thurston theorem we show that any \Gamma-space is stably weak equivalent to a discrete \Gamma-... |

42 |
Homologie nicht additiver Funktoren
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Citation Context ...screte \Gamma-group. By a well-known theorem of Dold-Kan the Moore normalization establishes the equivalence between the category of simplicial abelian groups and the category of chain complexes (see =-=[DP]-=-). mimicking the construction of normalization of simplicial groups, we give a similar construction for \Gamma-groups. This construction is based on the notion of cross-effects of functors [BP], which... |

40 | A combinatorial definition of homotopy groups - Kan - 1958 |

32 |
Higher Dimensional Crossed Modules and the Homotopy
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Citation Context ...ps T j��j modulo the relation [x; y] = (f ��; )sfx; yg; x 2 T j��j ; y 2 T jj ; where �� andsrun over all nonempty subsets of ! m ?. Now the theorem is a consequence of Theorem 1.1 and=-= Theorem 1.2 of [ES]-=-. In order to apply the result of [ES], one needs only to remark that for a \Gamma-group T and ns0, the group G = T ([n]) has n-commuting projections, which are induced by the maps s i : [n] ! [n]; i ... |

18 |
Pirashvili: Quadratic endofunctors of the category of groups, Adv
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Citation Context ...s (see [DP]). mimicking the construction of normalization of simplicial groups, we give a similar construction for \Gamma-groups. This construction is based on the notion of cross-effects of functors =-=[BP]-=-, which is a generalizatin of the classical definition of Eilenberg and Mac Lane [EM] to the non-abelian setup. Finally a Dold-Kan type theorem for the category of \Gamma-groups is proved. In abelian ... |

18 |
Homotopy theory of \Gamma-spaces, spectra, and bisimplicial sets
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(Show Context)
Citation Context .... Alexidze str. 1, Tbilisi, 380093. Republic of Georgia 0. Introduction \Gamma-spaces were introduced by Segal [S], who proved that they are combinatorial models for connective spectra (see also [A], =-=[BF]-=-). Based on Kan-Thurston theorem we show that any \Gamma-space is stably weak equivalent to a discrete \Gamma-group. By a well-known theorem of Dold-Kan the Moore normalization establishes the equival... |

15 |
Commutator calculus and groups of homotopy classes
- Baues
- 1981
(Show Context)
Citation Context ...�j ). Here f �� is the unique order preserving injection from [j �� j] to [n], whose image is f0g [ ��. 4.4. Remark. I learned recently that based on ideas of Baues (see relation (1.13) =-=on page 40 of [B]-=-) Dreckmann in his thesis [D] proved a theorem from which one can deduce our Theorem 4.3. 4.5. Remarks on polynomial \Gamma -groups. We will say that the degree of a functor T :\Omega ! Gr is less or ... |

12 |
Γ-(Co)homology of commutative algebras and some related representations of the symmetric groups,” Doctoral dissertation
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Citation Context ... (S 1 j \Delta \Delta \Delta j S n+1 ) = 0, for any S 1 ; \Delta \Delta \Delta ; S n+1 . 2.6. Generators and relations for\Omega . We recall the presentation of\Omega (see for example Section II.1 of =-=[Wh], co-=-mpare also with E.6.1.7 of [L2]). In what follows, \Sigma n denotes the symmetric group on n letters. We set �� k = (k; k + 1) 2 \Sigma n ; where k = 1; \Delta \Delta \Delta ; n \Gamma 1. We need ... |

7 |
Chain functors and homology theories
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Citation Context ... Inst. Alexidze str. 1, Tbilisi, 380093. Republic of Georgia 0. Introduction \Gamma-spaces were introduced by Segal [S], who proved that they are combinatorial models for connective spectra (see also =-=[A]-=-, [BF]). Based on Kan-Thurston theorem we show that any \Gamma-space is stably weak equivalent to a discrete \Gamma-group. By a well-known theorem of Dold-Kan the Moore normalization establishes the e... |

6 |
Distributivgesetze in der Homotopietheorie
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(Show Context)
Citation Context ... order preserving injection from [j �� j] to [n], whose image is f0g [ ��. 4.4. Remark. I learned recently that based on ideas of Baues (see relation (1.13) on page 40 of [B]) Dreckmann in his=-= thesis [D]-=- proved a theorem from which one can deduce our Theorem 4.3. 4.5. Remarks on polynomial \Gamma -groups. We will say that the degree of a functor T :\Omega ! Gr is less or equal to n if T k = 0 for k ?... |

5 |
On the Construction FK
- Milnor
- 1956
(Show Context)
Citation Context ...A precomposition with F gives a simplicial \Gamma-group F(A) and a morphism of \Gamma-spaces j : A ! F(A): 1.3. Lemma. The morphism j is a stable weak equivalence. Proof. Thanks to the Milnor theorem =-=[M]-=- we know that F(A)(S n ) has the same homotopy type as \Omega\Gamma A(S n ). Since A sends points to points, one sees that the simplicial object A(S n ) is trivial in dimensions ! n. Thus A(S n ) is (... |

4 |
Every connected space has the homology of a K(��; 1), Topology 15
- Kan, Thurston
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Citation Context ...f. According to the well known theorem of Kan and Thurston, for any connected space X there exists a group G and a map BG ! X which induces isomorphism on homology. The proof of this theorem given in =-=[KT] shows t-=-hat one can choose G functorially in X. Therefore there exist a \Gamma - group A 0 and a map of \Gamma-spaces j : �� WA 0 ! �� WA which pointwise yields an isomorphism on homology. By usual ar... |

1 |
On the groups H(��; n) II.Ann
- Eilenberg, Lane
(Show Context)
Citation Context ...ve a similar construction for \Gamma-groups. This construction is based on the notion of cross-effects of functors [BP], which is a generalizatin of the classical definition of Eilenberg and Mac Lane =-=[EM]-=- to the non-abelian setup. Finally a Dold-Kan type theorem for the category of \Gamma-groups is proved. In abelian case our theorem claims that the category of abelian \Gamma-groups is equivalent to t... |

1 |
with projections and applications to homotopy theory
- Groups
- 1980
(Show Context)
Citation Context ... we will do is to enrich the category Gr\Omega in order to get still an equivalence of categories. It turns out that these additional data are very close to the notion of groups with projections (see =-=[E]-=-, [St]). In this section we use additive notation for the group structure. We also fix some notation. For maps f 2 Hom\Omega (! n ?; ! n 0 ?); g 2 Hom\Omega (! m ?; ! m 0 ?) and h 2 Hom\Omega (! m ?; ... |

1 |
A note on groups with projections
- Steiner
- 1982
(Show Context)
Citation Context ...ill do is to enrich the category Gr\Omega in order to get still an equivalence of categories. It turns out that these additional data are very close to the notion of groups with projections (see [E], =-=[St]-=-). In this section we use additive notation for the group structure. We also fix some notation. For maps f 2 Hom\Omega (! n ?; ! n 0 ?); g 2 Hom\Omega (! m ?; ! m 0 ?) and h 2 Hom\Omega (! m ?; ! n 0 ... |