## Strict model structures for pro–categories. Categorical decomposition techniques in algebraic topology (2004)

Venue: | Isle of Skye |

Citations: | 19 - 4 self |

### BibTeX

@ARTICLE{Isaksen04strictmodel,

author = {Daniel C. Isaksen},

title = {Strict model structures for pro–categories. Categorical decomposition techniques in algebraic topology},

journal = {Isle of Skye},

year = {2004}

}

### OpenURL

### Abstract

Abstract. We show that if C is a proper model category, then the pro-category pro-C has a strict model structure in which the weak equivalences are the levelwise weak equivalences. This is related to a major result of [10]. The strict model structure is the starting point for many homotopy theories of pro-objects such as those described in [5], [17], and [19].

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Citation Context ...] as closely as possible. Other references include [9] and [15]. 2. Preliminaries on Pro-Categories We begin with a review of the necessary background on pro-categories. This material can be found in =-=[1]-=-, [2], [8], [10], and [18]. 2.1. Pro-Categories. Definition 2.1. For a category C, the category pro-C has objects all cofiltering diagrams in C, and Hompro-C(X, Y ) = lim colim HomC(Xt, Ys). s t Compo... |

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Citation Context ... strict model structure exists when C is proper. Finally, we consider whether the strict model structure is fibrantly generated. We assume familiarity with model categories. The original reference is =-=[23]-=-, but we follow the notation and terminology of [14] as closely as possible. Other references include [9] and [15]. 2. Preliminaries on Pro-Categories We begin with a review of the necessary backgroun... |

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Citation Context ...tly generated. We assume familiarity with model categories. The original reference is [23], but we follow the notation and terminology of [14] as closely as possible. Other references include [9] and =-=[15]-=-. 2. Preliminaries on Pro-Categories We begin with a review of the necessary background on pro-categories. This material can be found in [1], [2], [8], [10], and [18]. 2.1. Pro-Categories. Definition ... |

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Citation Context ... niceness hypothesis of [10, p. 45] is not satisfied by many categories of interest. These include many of the standard models for spectra, such as BousfieldFriedlander spectra [3], symmetric spectra =-=[16]-=-, or S-modules [11]. We shall study strict weak equivalences in pro-C whenever C is a proper model category. Definition 3.1. The strict weak equivalences of pro-C are the essentially levelwise weak eq... |

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Citation Context ...und. We assume that the reader is familiar with model categories, especially in the context of equivariant homotopy theory. The original reference is [18], but we will refer to more modern treatments =-=[11]-=- [12]. This paper is a generalization of [15], and we use specific pro-model category techniques from it. 2. Pro-categories We begin with a review of the necessary background on pro-categories. This m... |

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Citation Context ...objects all cofiltered diagrams in C and has morphisms defined by Hompro-C(X, Y ) = lim colim HomC(Xs, Yt). t s Pro-categories have found many uses over the years in fields such as algebraic geometry =-=[2]-=-, shape theory [20], geometric topology [6], and possibly even applied mathematics [7, Appendix]. When working with pro-categories, one would frequently like to have a homotopy theory of pro-objects. ... |

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Citation Context ... described in [10]. The niceness hypothesis of [10, p. 45] is not satisfied by many categories of interest. These include many of the standard models for spectra, such as BousfieldFriedlander spectra =-=[3]-=-, symmetric spectra [16], or S-modules [11]. We shall study strict weak equivalences in pro-C whenever C is a proper model category. Definition 3.1. The strict weak equivalences of pro-C are the essen... |

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Citation Context ...We assume that the reader is familiar with model categories, especially in the context of equivariant homotopy theory. The original reference is [18], but we will refer to more modern treatments [11] =-=[12]-=-. This paper is a generalization of [15], and we use specific pro-model category techniques from it. 2. Pro-categories We begin with a review of the necessary background on pro-categories. This materi... |

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Citation Context ...s fibrantly generated. We assume familiarity with model categories. The original reference is [23], but we follow the notation and terminology of [14] as closely as possible. Other references include =-=[9]-=- and [15]. 2. Preliminaries on Pro-Categories We begin with a review of the necessary background on pro-categories. This material can be found in [1], [2], [8], [10], and [18]. 2.1. Pro-Categories. De... |

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Citation Context ...that if C is a proper model category, then the pro-category pro-C has a strict model structure in which the weak equivalences are the levelwise weak equivalences. This is related to a major result of =-=[10]-=-. The strict model structure is the starting point for many homotopy theories of pro-objects such as those described in [5], [17], and [19]. 1. Introduction If C is a category, then the category pro-C... |

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Citation Context ...and possibly even applied mathematics [7, Appendix]. When working with pro-categories, one would frequently like to have a homotopy theory of pro-objects. The first attempts at this appear in [2] and =-=[24]-=- in which pro-objects in homotopy categories are considered. The difficulty with this approach is that the diagrams commute only up to homotopy, and this makes it virtually impossible to make sense of... |

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Citation Context ...isfied by our assumption on Wn. If h∗ is a periodic cohomology theory, then the model structure of this example is just the strict model structure associated to the h∗-local model structure on spaces =-=[3]-=-. 8. The underlying model structure for pro-finite groups Recall our goal of finding a model structure for pro-G-spaces in which {E(G/U)} is a cofibrant replacement for ∗, where G is a pro-finite grou... |

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Citation Context ...s morphisms defined by Hompro-C(X, Y ) = lim colim HomC(Xs, Yt). t s Pro-categories have found many uses over the years in fields such as algebraic geometry [2], shape theory [20], geometric topology =-=[6]-=-, and possibly even applied mathematics [7, Appendix]. When working with pro-categories, one would frequently like to have a homotopy theory of pro-objects. The first attempts at this appear in [2] an... |

20 | A model structure on the category of pro-simplicial sets
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Citation Context ... the levelwise weak equivalences. This is related to a major result of [10]. The strict model structure is the starting point for many homotopy theories of pro-objects such as those described in [5], =-=[17]-=-, and [19]. 1. Introduction If C is a category, then the category pro-C has as objects all cofiltered diagrams in C and has morphisms defined by Hompro-C(X, Y ) = lim colim HomC(Xs, Yt). t s Pro-categ... |

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Citation Context ...ered diagrams in C and has morphisms defined by Hompro-C(X, Y ) = lim colim HomC(Xs, Yt). t s Pro-categories have found many uses over the years in fields such as algebraic geometry [2], shape theory =-=[20]-=-, geometric topology [6], and possibly even applied mathematics [7, Appendix]. When working with pro-categories, one would frequently like to have a homotopy theory of pro-objects. The first attempts ... |

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Citation Context ...heories when G is a pro-finite group. 1.2. Background. We assume that the reader is familiar with model categories, especially in the context of equivariant homotopy theory. The original reference is =-=[18]-=-, but we will refer to more modern treatments [11] [12]. This paper is a generalization of [15], and we use specific pro-model category techniques from it. 2. Pro-categories We begin with a review of ... |

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Citation Context ...ories. The non-equivariant analogue of this phenomenon is explained in detail in [16]. In the context of equivariant model categories that behave well with respect to continuous cohomology, the paper =-=[10]-=- should also be mentioned. 1.1. Organization. We begin with a review of pro-categories, including a technical discussion of essentially levelwise properties. Afterwards, we define filtered model struc... |

8 | Completions of pro-spaces
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Citation Context ...wise weak equivalences. This is related to a major result of [10]. The strict model structure is the starting point for many homotopy theories of pro-objects such as those described in [5], [17], and =-=[19]-=-. 1. Introduction If C is a category, then the category pro-C has as objects all cofiltered diagrams in C and has morphisms defined by Hompro-C(X, Y ) = lim colim HomC(Xs, Yt). t s Pro-categories have... |

8 | Calculating limits and colimits in pro-categories, Fund - Isaksen |

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Citation Context ...egory is small if it has only a set of objects and a set of morphisms. A diagram is said to be cofiltering if its indexing category is so. Beware that some material on pro-categories, such as [2] and =-=[21]-=-, consider cofiltering categories that are not small. All of our pro-objects will be indexed by small categories. Objects of pro-C are functors from cofiltering categories to C. We use both set theore... |

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Citation Context ...actually commuting cofiltered diagrams (of spaces or simplicial sets or spectra or whatever) and then to define a notion of weak equivalence between such pro-objects. This approach was first taken by =-=[12]-=- in a restricted context. It was also applied much more generally in [10]. The idea is to start with a model structure (i.e., a homotopy theory) on a category C and then to construct a strict model st... |

4 |
Generalized Cohomology of Pro-Spectra, preprint
- Isaksen
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Citation Context ...hypotheses of Proposition 4.19 are satisfied, and the associated filtered model structure is proper. The resulting model structure on the category of pro-spectra is the same as the model structure of =-=[16]-=-. The weak equivalences can be described in terms of pro-homotopy groups, but the reformulation is not quite as obvious as one might expect. See [16] for details. Example 7.5. Let C be the category of... |

3 |
Pro-objects (after Verdier), Dualité de Poincaré (Seminaire Heidelberg-Strasbourg 1966/67
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Citation Context ...ly as possible. Other references include [9] and [15]. 2. Preliminaries on Pro-Categories We begin with a review of the necessary background on pro-categories. This material can be found in [1], [2], =-=[8]-=-, [10], and [18]. 2.1. Pro-Categories. Definition 2.1. For a category C, the category pro-C has objects all cofiltering diagrams in C, and Hompro-C(X, Y ) = lim colim HomC(Xt, Ys). s t Composition is ... |

2 |
Calculating limits and colimits in pro-categories, preprint
- Isaksen
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Citation Context ... Other references include [9] and [15]. 2. Preliminaries on Pro-Categories We begin with a review of the necessary background on pro-categories. This material can be found in [1], [2], [8], [10], and =-=[18]-=-. 2.1. Pro-Categories. Definition 2.1. For a category C, the category pro-C has objects all cofiltering diagrams in C, and Hompro-C(X, Y ) = lim colim HomC(Xt, Ys). s t Composition is defined in the n... |

2 |
J.P May (with an appendix by M
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Citation Context ...of model structures on pro-categories that can be established with Theorem 5.15. Example 7.1. We give a concrete example that is an application of Proposition 4.18. Let C be the category of S-modules =-=[7]-=-, and let A = N. For each n, let En be a generalized homology theory such that if a map of spectra is an En-homology equivalence, then it is also an Em-homology equivalence for all m ≤ n. Let Wn be th... |

2 | t-model structures
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Citation Context ...terest. A t-model structure is a stable model structure C with a t-structure on its (triangulated) homotopy category, together with a lift of the t-structure to C. This notion is studied in detail in =-=[8]-=-, where it is shown that a particularly well-behaved filtered model structure on C (and thus a model structure on pro-C) can be associated to any t-model structure. This paper grew out of an attempt t... |

1 |
Localization of Model Categories, preprint dated 4/12/2000
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Citation Context ...ally, we consider whether the strict model structure is fibrantly generated. We assume familiarity with model categories. The original reference is [23], but we follow the notation and terminology of =-=[14]-=- as closely as possible. Other references include [9] and [15]. 2. Preliminaries on Pro-Categories We begin with a review of the necessary background on pro-categories. This material can be found in [... |

1 |
On the two definitions of Ho(proC
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Citation Context ...owed by a levelwise acyclic fibration. Therefore, every levelwise weak equivalence is a weak equivalence in the sense of [10, §3.3]. The preceding proposition is closely related to the main result of =-=[22]-=-. However, we make a useful observation missed there. Namely, it is not necessary to saturate the essentially levelwise weak equivalences; they are already saturated when C is proper. �� W � Z 4. Stri... |

1 |
Equivariant stable homotopy theory for profinite groups, in preparation
- Fausk, Isaksen
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Citation Context ...nite group. Section 8 contains a detailed description of a model structure on G-spaces as an illustration of our general theory. Analogous results for pro-G-spectra are presented in detail in [8] and =-=[9]-=-. We now summarize our interest in pro-G-spaces when G is a pro-finite group. Let G be a finite group. There is an obvious generalization to pro-G-spaces of the model structure for pro-non-equivariant... |