## A CARTAN-EILENBERG APPROACH TO HOMOTOPICAL ALGEBRA (707)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Santos707acartan-eilenberg,

author = {F. Guillén Santos and V. Navarro and P. Pascual and Agustí Roig},

title = {A CARTAN-EILENBERG APPROACH TO HOMOTOPICAL ALGEBRA},

year = {707}

}

### OpenURL

### Abstract

Abstract. In this paper we propose an approach to homotopical algebra where the basic ingredient is a category with two classes of distinguished morphisms: strong and weak equivalences. These data determine the cofibrant objects by an extension property analogous to the classical lifting property of projective modules. We define a Cartan-Eilenberg category as a category with strong and weak equivalences such that there is an equivalence between its localization with respect to weak equivalences and the localised category of cofibrant objets with respect to strong equivalences. This equivalence allows us to extend the classical theory of derived additive functors to this non additive setting. The main examples include Quillen model categories and functor categories with a triple, in the last case we find examples in which the class of strong equivalences is not determined by a homotopy relation. Among other applications, we prove the existence of filtered minimal models for cdg algebras over a zero-characteristic field and we formulate an acyclic models theorem for non additive functors.

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Citation Context ...he localisation provided by some quotient categories. 1.3.1. Let C be a category and ∼ a congruence on C, that is, an equivalence relation between morphisms of C which is compatible with composition (=-=[ML]-=-, page 51). We denote by C/∼ the� � � �� �� �� �� A CARTAN-EILENBERG APPROACH 7 quotient category, and by π : C −→ C/∼ the universal canonical functor. We denote by S the class of morphisms f : X −→ ... |

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Citation Context ...tegories. 2.4. Relation with reflective subcategories. In some cases, localisation of categories may be realised through reflective subcategories or, equivalently, by idempotent functors, see [A] and =-=[Bo]-=-. The following result relates Cartan-Eilenberg categories with these concepts. Observe that in order to be consistent with our laterality assumptions we consider the dual notions: coreflective subcat... |

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Citation Context ...et of axioms of model categories is, in some sense, somewhat strong because there are interesting categories in which to do homotopy theory that do not satisfy all of them. Several authors (see [Br], =-=[Ba]-=- and others) have developed simpler alternatives, all of them focused on laterality, asking only for a left- (or right-) handed version of Quillen’s set of axioms. All these alternatives are very clos... |

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Citation Context ..., we define a structure of Cartan-Eilenberg category on the functor category Cat(X,C+(A)) (see theorem 6.1.3). We apply this result to obtain theorems of the acyclic models kind, extending results in =-=[B]-=- and [GNPR2]. We stress that in these examples the class of strong equivalences S does not come from a homotopy relation. We also prove a cubical version of acyclic models used in [GN] without proof. ... |

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Citation Context ...y; introduce Cartan-Eilenberg categories, and give some criteria to prove that a given category is Cartan-Eilenberg. We also relate these notions with Adams’ study of localisation in homotopy theory, =-=[A]-=-. 2.1. Models in a category with strong and weak equivalences. In this section we introduce models of objects and diagrams in categories with two distinguished classes of morphisms. Let C be a categor... |

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Citation Context ...h cofibrant objects. This problem is equivalent to the orthogonal category problem for (C[S −1 ], δ(W)). This problem has been studied by Casacuberta and Chorny in the context of homotopy theory (see =-=[CCh]-=-). If the subcategory of cofibrant objects is a left model subcategory of C, the category C will be called a left CartanEilenberg category. It is a non additive generalisation for the category of comp... |

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Citation Context ... The set of axioms of model categories is, in some sense, somewhat strong because there are interesting categories in which to do homotopy theory that do not satisfy all of them. Several authors (see =-=[Br]-=-, [Ba] and others) have developed simpler alternatives, all of them focused on laterality, asking only for a left- (or right-) handed version of Quillen’s set of axioms. All these alternatives are ver... |

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Citation Context ...ine a structure of Cartan-Eilenberg category on the functor category Cat(X,C+(A)) (see theorem 6.1.3). We apply this result to obtain theorems of the acyclic models kind, extending results in [B] and =-=[GNPR2]-=-. We stress that in these examples the class of strong equivalences S does not come from a homotopy relation. We also prove a cubical version of acyclic models used in [GN] without proof. Acknowledgem... |

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Citation Context ...emark two specific achievements obtained in this paper. The first one corresponds to the homotopy theory of filtered cdg algebras, first developed by S. Halperin and D. Tanré by perturbation methods (=-=[HT]-=-). Let FAlg 1(k) be the category of 1-connected filtered cdg algebras over a field k of characteristic zero, (see section 5.4 for the specific assumptions we impose to the filtration). We extend the c... |

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Citation Context ...d triangles. Proof. This lemma and its proof are the filtered version of lemma 8.6.4 of [GNPR1]. Remark 5.1.8. The induced equivalence of categories K+FP(A) −→ DF+(A) has been obtained by Illusie (see=-=[I]-=- Cor. (V.1.4.7)) for complexes with a finite filtration. 5.2. Filtered complexes of vector spaces. As a special case of the results above, take k a field and A the category of k-vector spaces. We writ... |

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Citation Context ... C[W −1 ] is called a left derived functor of F with respect to W, and denoted by θF : LWF −→ F. If W has a right calculus of fractions, this definition agrees with the definition given by Deligne in =-=[D3]-=-. Recall that the category Cat(C[W −1 ], D) is identified, by means of the functor γ ∗ : Cat(C[W −1 ], D) −→ Cat(C, D), with the full subcategory CatW(C, D) of Cat(C, D) whose objects are the functors... |

3 | Moduli spaces and formal operads - Santos, Navarro, et al. |

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Citation Context ...σ), ∀ α ∈ □, with the obvious differential and convenient coaugmentation and comultiplication. 6.3.2. Let FNC+(Z) be the category of filtered complexes of abelian groups with filtrations supported in =-=[0, N]-=-. Given a double complex X∗∗ in FNC+(Z), we consider the total complex filtered by Wp(Tot X∗∗)n = ⊕ WpXrs. r+s=n We let Σ denote the class of filtered homotopy equivalences in FNC+(Z). In this way we ... |

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Citation Context ...gories with a congruence. There are some situations where it is possible to give an easiest presentation of morphisms of the category C[W −1 ], for example, when there is a calculus of fractions (see =-=[GZ]-=-). In this section we present an even simpler situation which will occur later, the localisation provided by some quotient categories. 1.3.1. Let C be a category and ∼ a congruence on C, that is, an e... |

1 |
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Citation Context ...F ′ : C[W −1 ] −→ C ′ [W ′ −1 ]. Because of its potential applications, there has been interest in giving sufficient conditions that assure that F ′ is an equivalence of categories. Kahn and Sujatha (=-=[KS]-=-) have given a solution in the style of Quillen’s theorem A. In this paper we propose a different approach. 1.2. Hammocks. In this section we describe the localisation of categories by using hammocks ... |