## PARAMETRIZED SPACES ARE LOCALLY CONSTANT HOMOTOPY SHEAVES (706)

Citations: | 1 - 1 self |

### BibTeX

@MISC{Shulman706parametrizedspaces,

author = {Michael Shulman},

title = {PARAMETRIZED SPACES ARE LOCALLY CONSTANT HOMOTOPY SHEAVES},

year = {706}

}

### OpenURL

### Abstract

Abstract. We prove that the homotopy theory of parametrized spaces embeds fully and faithfully in the homotopy theory of simplicial presheaves, and that its essential image consists of the locally homotopically constant objects. This gives a homotopy-theoretic version of the classical identification of covering spaces with locally constant sheaves. We also prove a new version of the classical result that spaces parametrized over X are equivalent to spaces with an action of ΩX. This gives a homotopy-theoretic version of the correspondence between covering spaces and π1-sets. We then use these two equivalences to study base change functors for parametrized spaces. Contents

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Citation Context ...ixed point spaces for all H ∈ H; we may call this the qH-model structure. This is most frequently used in equivariant homotopy theory when H is the set of all closed subgroups of G; see, for example, =-=[May96]-=-. However, we will be interested instead in the case when H consists only of the trivial subgroup {e}. We call this the qe-model structure and refer to its weak equivalences as the e-equivalences. Thi... |

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Citation Context ...ation of the Kan loop group, which is a grouplike topological monoid; then A can be reconstructed, up to q-equivalence, as the classifying space of ΩA. Moreover, if A is m-cofibrant, then so is ΩA by =-=[Mil59]-=-. Since the homotopy theory of parametrized spaces is invariant under q-equivalences of the base space, it is harmless to assume that A is m-cofibrant. Thus, for the rest of this section we make the f... |

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Citation Context ...o be replaced by a topologically enriched category, in which case the use of spaces rather than simplicial sets would become important. Remark 4.3. The above construction is the same idea followed in =-=[Lur07]-=-, although there the localization is done using quasi-categories rather than model categories. However, the elements of the model-categorical approach can be found in [Lur07,8 MICHAEL SHULMAN §7.1]. ... |

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Citation Context ...structure for topological spaces discovered by Cole [Col06], obtained by mixing the ‘standard’ model structure constructed by Quillen [Qui67] with the ‘classical’ model structure constructed by Strøm =-=[Str72]-=-. In Cole’s model structure the weak equivalences are the weak homotopy equivalences, while the fibrations are the Hurewicz fibrations, and the cofibrant objects are the spaces of the homotopy type of... |

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Citation Context ...07,8 MICHAEL SHULMAN §7.1]. This model structure for homotopy sheaves is not equivalent to that of [Jar87], which is constructed by localizing with respect to the larger class of hypercoverings (see =-=[DHI04]-=-). Several arguments for using coverings rather than hypercoverings can be found in [Lur07]; the results of this paper can be taken as another. As we mentioned in §1, however, there is also a model st... |

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Citation Context ... too restrictive; for example, it disallows diagonal maps ∆: B → B ×B. Another solution is to use a topological quasitopos, such as pseudotopological spaces (see [Wyl91]) or subsequential spaces (see =-=[Joh79]-=-). We adopt instead the solution used in [MS06]: we require base spaces to be compactly generated, but allow total spaces to be arbitrary k-spaces, not necessarily weak Hausdorff. The references given... |

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Citation Context ...ative theory in the literature: one involving ‘sheaves of spaces on B’, or homotopy sheaves, such as that of [Jar87, Lur07], and one involving ‘spaces over B’, or parametrized spaces, such as that of =-=[MS06]-=-. To date there has been no formal comparison of the two. In this paper, we state and prove such a comparison; our slogan is that parametrized spaces are equivalent to locally constant homotopy sheave... |

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Citation Context ... ij-cofibrant object QX. Since this comes with an ij-equivalence QX ∼ −→ X, we have a canonical map (7.3) f ∗ QX → f ∗ X which represents a map (7.4) Lijf ∗ (ι⋆X) → ι⋆(Rqf ∗ X). In the terminology of =-=[Shua]-=-, this is the derived natural transformation of the pointset level equality f ∗ ◦ Id = Id ◦f ∗ . We claim that (7.4) is an isomorphism, or equivalently that (7.3) is an ijequivalence. The proof is ver... |

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Citation Context ...spaces. Finally, in §9 we show that this equivalence preserves all the base change functors. An important technical tool in our work is a new model structure for topological spaces discovered by Cole =-=[Col06]-=-, obtained by mixing the ‘standard’ model structure constructed by Quillen [Qui67] with the ‘classical’ model structure constructed by Strøm [Str72]. In Cole’s model structure the weak equivalences ar... |

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Citation Context ... covering spaces as the local homeomorphisms which are ‘locally constant’. Analogously, we will prove the equivalence between (a) and (b) by using a different model structure on spaces over B, due to =-=[IJ02]-=-, whose homotopy theory is equivalent to that of homotopy sheaves and in which all objects are fibrant. We will show that a model structure for spaces parametrized over B embeds into this model struct... |

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Citation Context ...s with a G-action to the homotopy theory of spaces parametrized over BG. The results in this section are basically folklore. A bijection between equivalence classes can be found in the survey article =-=[Sta71]-=-, and a full equivalence of homotopy theories using simplicial fibrations can be found in [DDK80, DK85]; our use of the m-model structure on K /BG will allow us to prove the strong result while using ... |

2 |
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Citation Context ...T HOMOTOPY SHEAVES 5 Theorem 2.1 ([IJ02, 6.6]). If X is a CW complex with the above properties, then any open subspace of X has the homotopy type of a CW complex. Proof. Given a class C of spaces, in =-=[Hym68]-=- a space is defined to be an ANR(C) (absolute neighborhood retract) if it is a neighborhood retract of every space in C that contains it as a closed subset. If C is the class of metric spaces, an ANR(... |

2 |
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Citation Context ...[IJ02]. However, in some cases this is too restrictive; for example, it disallows diagonal maps ∆: B → B ×B. Another solution is to use a topological quasitopos, such as pseudotopological spaces (see =-=[Wyl91]-=-) or subsequential spaces (see [Joh79]). We adopt instead the solution used in [MS06]: we require base spaces to be compactly generated, but allow total spaces to be arbitrary k-spaces, not necessaril... |