## Homotopy limits and colimits and enriched homotopy theory (2006)

Citations: | 16 - 2 self |

### BibTeX

@TECHREPORT{Shulman06homotopylimits,

author = {Michael Shulman},

title = {Homotopy limits and colimits and enriched homotopy theory},

institution = {},

year = {2006}

}

### OpenURL

### Abstract

Abstract. Homotopy limits and colimits are homotopical replacements for the usual limits and colimits of category theory, which can be approached either using classical explicit constructions or the modern abstract machinery of derived functors. Our first goal is to explain both and show their equivalence. Our second goal is to generalize this result to enriched categories and homotopy weighted limits, showing that the classical explicit constructions still give the right answer in the abstract sense, thus partially bridging the gap between classical homotopy theory and modern abstract homotopy theory. To do this we introduce a notion of “enriched homotopical categories”, which are more general than enriched model categories, but are still a good place to do enriched homotopy theory. This demonstrates that the presence of enrichment often simplifies rather than complicates matters, and goes some way toward achieving a better understanding of “the role of homotopy in homotopy theory.” Contents

### Citations

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Citation Context ...iarity with that work is required if the reader is willing to take a few of its results on faith. In various places, we use the terminology and techniques of enriched category theory, as described in =-=[Kel82]-=- and [Dub70]. Their use is concentrated in the second half of the paper, however, and we attempt to explain these concepts as they arise. Enriched category theory, being the sort of category theory wh... |

375 | Homotopy limits, completions and localizations - Bous, Kan - 1972 |

286 | The Geometry of Iterated Loop Spaces
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Citation Context ...mma 8.1. □ A more general version of this result will be proven in §23. Results of this sort are as old as the bar construction; it has always been thought of as a sort of cofibrant replacement. May (=-=[May72]-=- and [May75]) shows that under suitable conditions, simplicial bar constructions (on topological spaces) are “proper simplicial spaces,” which is equivalent to being “Reedy h-cofibrant.” We will have ... |

243 |
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(Show Context)
Citation Context ...ram categories are more often present in the context of colimits. The reader is assumed to have some knowledge of category theory, including the formalisms of ends and Kan extensions, as described in =-=[ML98]-=-. Some familiarity with model categories is also expected, especially in the second part of the paper. This includes an acquaintance with the various model structures that exist on diagram categories.... |

223 | Model categories
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(Show Context)
Citation Context ...t of it in full generality. For example, essentially the same result for the case V = M can be found in [SS03a, 6.1]. In the case when D has one object and V = M, it reduces to a result like those of =-=[Hov98]-=- and [SS00] for modules over a monoid in a monoidal model category. It can also be viewed as a special case of the model structures considered in [BM07] for algebras over colored operads. Theorem 24.4... |

131 | Model categories of diagram spectra
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(Show Context)
Citation Context ...ng K giving K ∗ : M E → M D , and this functor has left and right adjoints LanK and RanK. The homotopical behavior of these functors is important in many different contexts, such as the comparison in =-=[MMSS01]-=- of various types of diagram spectra, including symmetric spectra and orthogonal spectra. The “prolongation” functors in that paper are left Kan extensions for which the enrichment, as considered here... |

96 |
Model categories and their localizations, volume 99
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(Show Context)
Citation Context ...l spaces, or over simplicial sets. In this case there are well-known explicit constructions of homotopy limits and colimits dating back to the classical work [BK72]; for a more modern exposition, see =-=[Hir03]-=-. These homotopy limits are objects satisfying a “homotopical” version of the usual universal property: instead of representing commuting cones over a diagram, they represent “homotopy coherent” cones... |

73 |
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Citation Context ...] for enriched model categories to the more general homotopical categories of [DHKS04]. The main points are an enriched analogue of the theory of “two-variable adjunctions” given6 MICHAEL SHULMAN in =-=[Hov99]-=-, and a general theory of enrichments of homotopy categories derived from enriched homotopical categories. This abstract framework may be seen as a proposed axiomatization of enriched homotopy theory,... |

68 |
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(Show Context)
Citation Context ...expected, especially in the second part of the paper. This includes an acquaintance with the various model structures that exist on diagram categories. For good introductions to model categories, see =-=[DS95]-=-, [Hov99], and [Hir03]. We assume that all model categories are complete and cocomplete and have functorial factorizations. We also use a number of ideas and results from [DHKS04], but no prior famili... |

67 | Sheafifiable homotopy model categories - Beke |

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48 |
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Citation Context ... more general version of this result will be proven in §23. Results of this sort are as old as the bar construction; it has always been thought of as a sort of cofibrant replacement. May ([May72] and =-=[May75]-=-) shows that under suitable conditions, simplicial bar constructions (on topological spaces) are “proper simplicial spaces,” which is equivalent to being “Reedy h-cofibrant.” We will have more to say ... |

36 |
Kan extensions in enriched category theory
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(Show Context)
Citation Context ...that work is required if the reader is willing to take a few of its results on faith. In various places, we use the terminology and techniques of enriched category theory, as described in [Kel82] and =-=[Dub70]-=-. Their use is concentrated in the second half of the paper, however, and we attempt to explain these concepts as they arise. Enriched category theory, being the sort of category theory which nearly a... |

34 |
Doctrinal adjunction
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(Show Context)
Citation Context ... is the given one; (ii) A colax V -module structure on F0; and (iii) A lax V -module structure on G0. This result can be viewed as a special case of the theory of “doctrinal adjunctions” presented in =-=[Kel74]-=-. In view of (i), we call such data an enrichment of the adjunction F0 ⊣ G0. An adjunction F0 ⊣ G0 together with (ii) or (iii) is also known as a V -module adjunction. We summarize the above propositi... |

27 | Simplicial objects in algebraic topology, Chicago Lectures in Mathematics - May - 1992 |

24 | Homotopy Coherent Category Theory - Cordier, Porter - 1997 |

21 | A model structure for the category of pro-simplicial sets - Isaksen |

19 | and Ieke Moerdijk, Axiomatic homotopy theory for operads,Comment - Berger |

18 | Resolution of coloured operads and rectification of homotopy algebras - Berger, Moerdijk |

18 | Spectral enrichments of model categories - Dugger |

17 |
Parametrized Homotopy Theory, volume 132 of Mathematical Surveys and Monographs
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(Show Context)
Citation Context ...s a much more subtle question, and even under the best of circumstances (such as left and right Quillen functors) composition need not be preserved. See, for example, [MS06, Counterexample 0.0.1]. In =-=[MS06]-=- this question is dealt with using the tools of “middle derived functors”, which we introduce in the next section. 4. Middle derived functors We have defined derived functors by a universal property a... |

16 | closed model structures for sheaves
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(Show Context)
Citation Context ...ence of projective model structures on enriched diagram categories. This is a straightforward application of the principle of transfer of model structures along adjunctions, which was first stated in =-=[Cra95]-=-. Various special cases of this result are scattered throughout the literature, but we have been unable to find a statement of it in full generality. For example, essentially the same result for the c... |

16 |
Function complexes in homotopical algebra
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(Show Context)
Citation Context ...h are locally Kan. In our terminology, a DKequivalence is a sS-functor K which is homotopically fully faithful and such that (hK)0 is an equivalence. The notion of DK-equivalence was first defined in =-=[DK80]-=-, where it was called a “weak equivalence of simplicial categories;” see also [Ber07].HOMOTOPY LIMITS AND COLIMITS AND ENRICHED HOMOTOPY THEORY 65 We can now prove the following generalization of Pro... |

14 | The geometry of iterated loop spaces, volume 271 - May |

9 |
A general formulation of homotopy limits
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(Show Context)
Citation Context ...ll homotopical categories, and of subcategories of “good objects,” which they call deformation retracts. They do not consider notions of homotopy at all. On the other side of the coin, the authors of =-=[CP97]-=- study abstract notions of homotopy, modeled by simplicially enriched categories, but without reference to weak equivalences. We believe that a synthesis of such approaches is needed, especially to de... |

4 |
Homotopical Algebra, volume 43 of Lecture Notes in Mathematics
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(Show Context)
Citation Context ...no further discussion of this approach would be necessary, at least for model categories. There have been steps in the direction of a notion of model category with these properties; see, for example, =-=[Wei01]-=-. But the more common notion of Quillen model category does not have them. However, there are important special cases in which the diagram category does have a model structure and limit or colimit fun... |

3 |
Homotopy meaningful constructions: Homotopy colimits
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(Show Context)
Citation Context ...preserves all weak equivalences between diagrams of the form LanT F ′, where F ′ is a Reedy cofibrant ∆D-diagram. These proofs can be found in [DHKS04, §23]. □ A very similar procedure is followed in =-=[CS98]-=-, using essentially the Reedy model structure on M ∆D , but restricted to the subcategory of “bounded” diagrams, which are easier to analyze explicitly. These technical results give homotopy limit and... |

2 | M A Mandell, Modules in monoidal model categories
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(Show Context)
Citation Context ...Quillen pair, and so on. Moreover, it is proven that the homotopy category of a V -model category is enriched over Ho V , as is the derived adjunction of a V -Quillen pair, and so on. Remark 14.1. In =-=[LM04]-=- Lewis and Mandell have described a more general notion of an enriched model category which does not require that it be tensored and cotensored. However, since model categories are usually assumed to ... |

1 |
With an appendix by M
- Rings
- 1997
(Show Context)
Citation Context ...he cofibrations of the model structure are not necessarily the best cofibrations to use for this purpose. In this section, we use the term “cofibration” without prejudice as to meaning, adopting from =-=[EKMM97]-=- and [MS06] the terminology q-cofibration for the model structure cofibrations. Another common type of cofibration is an h-cofibration (for “Hurewicz”), generally defined by a homotopy lifting propert... |

1 | Errata for “model categories”. Available online at http://claude.math.wesleyan.edu/ mhovey/papers - Hovey |