## Universal homotopy theories

Venue: | Adv. Math |

Citations: | 37 - 3 self |

### BibTeX

@ARTICLE{Dugger_universalhomotopy,

author = {Daniel Dugger},

title = {Universal homotopy theories},

journal = {Adv. Math},

year = {},

pages = {144--176}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. Begin with a small category C. The goal of this short note is to point out that there is such a thing as a ‘universal model category built from C’. We describe applications of this to the study of homotopy colimits, the Dwyer-Kan theory of framings, to sheaf theory, and to the homotopy theory of schemes. Contents

### Citations

358 | Homotopical Algebra - Quillen - 1967 |

348 |
Homotopy Limits, Completions and Localizations
- Bous, Kan
- 1972
(Show Context)
Citation Context ...Define ˜ QF to be the simplicial presheaf whose nth level is ( ˜ ∐ QF)n = (rXn) rXn→···→rX0→F and whose face and degeneracy maps are the obvious candidates (di means omit Xi, etc.) In the language of =-=[BK]-=-, this is the simplicial replacement of the canonical diagram (C ↓ F) → Pre(C). (Notice that ˜ QF is in some sense the formal homotopy colimit of this diagram.) Also note that there is a natural map ˜... |

221 | Model categories - Hovey - 1999 |

215 | Locally presentable and accessible categories - Adámek, Rosický - 1994 |

160 | Etale homotopy - Artin, Mazur - 1969 |

158 |
Homotopy theory of Γ-spaces, spectra, and bisimplicial sets, in, “Geometric applications of homotopy theory
- Bousfield, Friedlander
- 1977
(Show Context)
Citation Context ... ‘obvious’ maps one can write down: e.g., projections p1, p2: S × S → S, inclusions into a factor i1, i2: S → S × S, diagonal maps S → S × S, etc. Now C is almost the same as the category called Γ in =-=[BF]-=-—the only difference is that Γ contains an extra object corresponding to the trivial spectrum ∗. In any case the inclusion C ֒→ Spectra extends to a Quillen pair Re : U∗C ⇄ Spectra : Sing, and the cat... |

119 |
Simplicial presheaves
- Jardine
- 1987
(Show Context)
Citation Context ...1], using very different methods.) Finally, in Sections 7 and 8 we deal with some elementary applications. The first of these is an interpretation of Jardine’s model category of simplicial presheaves =-=[J2]-=-: we point out that giving a Grothendieck topology on a category C amounts to specifying certain ‘‘homotopy-colimit’’ type relations, and studying the sheaf theory of C is precisely studying the model... |

68 | Accessible Categories: the Foundations of Categorical Model Theory - Makkai, Parè - 1989 |

65 | Sheafifiable homotopy model categories - Beke |

59 |
Function complexes in homotopical algebra, Topology 19
- Dwyer, Kan
- 1980
(Show Context)
Citation Context ...X ∐ X X1 −→ X, where the first map is a cofibration and the second a weak equivalence. These maps can be assembled into the beginning of a cosimplicial object: X The Dwyer-Kan theory of resolutions =-=[DK]-=- is a massive generalization of this, which gives a way of talking about objects which ‘look and feel’ like ‘X × ∆n ’ for any n. This is actually what is called a cosimplicial resolution. There are al... |

52 | Combinatorial model categories have presentations
- Dugger
(Show Context)
Citation Context ...announce in (6.3) the result that every combinatorial model category is equivalent to a localization of some UC; the proof is too involved to include here, but is instead given in the companion paper =-=[D3]-=-. One immediate consequence is that every combinatorial model category is equivalent to a model category which is simplicial and left proper�� � �� �� UNIVERSAL HOMOTOPY THEORIES 3 (the ‘simplicial’ ... |

44 | Algebraic K-theory as generalized sheaf cohomology - Brown, Gersten - 1973 |

39 | Algebraic K-theory and étale cohomology, Ann - Thomason - 1985 |

38 | Simplicial objects in a Grothendieck topos. In Applications of algebraic K-theory to algebraic geometry and number theory, Part I - Jardine - 1983 |

37 |
Bousfield, The localization of spaces with respect to homology, Topology 14
- K
- 1975
(Show Context)
Citation Context ...under control. We regard the process of localization as a way of ‘imposing relations’ in a model category, hence the notation M/S—other authors have used S −1 M or LSM for the same concept. Bousfield =-=[Bo]-=- was the first to give a systematic approach to what S-localizations might look like, and we now recall this. Definition 5.3. (a) An S-local object of M is a fibrant object X such that for every map A... |

33 |
Model categories and more general abstract homotopy theory. In preparation, available at http://www-math.mit.edu/ psh
- Dwyer, Hirschhorn, et al.
- 1997
(Show Context)
Citation Context ...tes a homotopy function complex from X to Y . Finally, we must say something about our conventions regarding homotopy colimits. To define these in general model categories, the approach taken in both =-=[DHK]-=- and [H] is to chose a framing on the model category and then to define homotopy colimits via an explicit formula. The subtlety is that this yields a construction which is only homotopy invariant for ... |

30 |
Localization of Model Categories
- Hirschhorn
- 2000
(Show Context)
Citation Context ... refer back to when necessary. 1.3. Notation and terminology. Our conventions regarding model categories, framings, and other elements of abstract homotopy theory generally follow those of Hirschhorn =-=[H]-=-. Hovey’s book [Ho] is also a good reference. In particular, model categories are assumed to contain small limits and colimits, and to have functorial factorizations. Following [Ho] we will define a m... |

29 |
A1-homotopy theory of schemes
- Morel, Voevodsky
- 1999
(Show Context)
Citation Context ...2 DANIEL DUGGER some known phenomenon: in fact our original motivation was to ‘explain’ (if that can be considered the right word) Morel and Voevodsky’s construction of a homotopy theory for schemes =-=[MV]-=-. On the other hand, however, this can be a technique for understanding a model category one already has, by asking what kinds of objects and relations are needed to reconstruct that homotopy theory f... |

22 |
Letter to A. Grothendieck
- Joyal
- 1984
(Show Context)
Citation Context ... property, however. It is sometimes considered more ‘natural’ to work with simplicial sheaves than with simplicial presheaves, although they give rise to the same homotopy theory— this was what Joyal =-=[Jo]-=- originally did, and simplicial sheaves were also used in [MV]. But from the viewpoint of universal model categories simplicial presheaves18 DANIEL DUGGER are very natural. By working with sheaves on... |

21 |
Homotopy theories
- Heller
- 1988
(Show Context)
Citation Context ... localize: in this paper we have consistently used the Bousfield-Kan model structure, in which the fibrations and weak equivalences are detected objectwise, but there is also a Heller model structure =-=[He]-=- in which the cofibrations and weak equivalences are detected objectwise. The Heller structure doesn’t seem to enjoy any kind of universal property, however. It is sometimes considered more ‘natural’ ... |

19 |
Stable homotopical algebra and Γ-spaces
- Schwede
- 1999
(Show Context)
Citation Context ...ur Quillen pair descends to give U∗C/W ⇄ Spectra. The model category U∗C/W turns out to be precisely one of the well-known model structures for the category of Γ-spaces: it is the one used by Schwede =-=[Sch]-=-, and the identification of the appropriate maps to localize is implicit in that paper. The fibrant objects can be seen to be the ‘very special’ Γ-spaces (see [Sch, bottom paragraph on Page 349] for a... |

14 | Handbook of categorical algebra II, Categories and structures - Borceux - 1994 |

7 |
Hypercovers and simplicial presheaves, preprint
- Dugger, Hollander, et al.
(Show Context)
Citation Context ...0. In a general hypercover one takes the iterated fibered products at each level but then is allowed to refine that object further, by taking a cover of it. We refer the reader to [AM, Section 8] and =-=[DHI]-=- for further discussion of hypercovers. For the category of topological spaces one has the following very useful property: if U Q X is any hypercover of the space X—where in this * context we now cons... |

5 |
Grothendieck topologies, Seminar notes
- Artin
- 1962
(Show Context)
Citation Context ...te is a small category C equipped with finite limits, together with a topology: a collection of families {Uα → X} called covering families, which are required to satisfy various reasonable properties =-=[Ar]-=-. (There is also a more general approach involving covering sieves, which we have foregone only for ease of presentation). The prototype for all Grothendieck sites is the category of topological space... |

4 |
Replacing model categories by simplicial ones
- Dugger
(Show Context)
Citation Context ...category is equivalent to a model category which is simplicial and left proper�� � �� �� UNIVERSAL HOMOTOPY THEORIES 3 (the ‘simplicial’ part was proven under slightly more restrictive hypotheses in =-=[D1]-=-, using very different methods.) Finally, in sections 7 and 8 we deal with some elementary applications. The first of these is an interpretation of Jardine’s model category of simplicial presheaves [J... |

3 | Homotopy meaningful constructions: Homotopy colimits - Chacholski, Scherer - 1998 |

3 |
Combinatorial model categories
- Smith
(Show Context)
Citation Context ...e localizations are always known to exist (for any set of maps S). These are the left proper, cellular model categories of Hirschhorn [H], and the left proper, combinatorial model categories of Smith =-=[Sm]-=-. We will not recall the definitions of these classes here, but suffice it to say that the model categories UC belong to both of them, and so we are free to localize. In general, the model categories ... |

2 |
Homotopy theory of C-spaces, spectra, and bisimplicial sets, in ‘‘Geometric Applications of Homotopy Theory
- Bousfield, Friedlander
- 1977
(Show Context)
Citation Context ...ed by the ‘‘obvious’’ maps one can write down: e.g., projections p1,p2 :S×SQS; inclusions into a factor i1,i2 :SQ S×S; diagonal maps S Q S×S; etc. Now C is almost the same as the category called C in =-=[BF]-=-—the only difference is that C contains an extra object corresponding to the trivial spectrum f. In any case the inclusion C + Spectra extends to a Quillen pair Re : U C \ Spectra : Sing, and the cate... |

1 |
Simplicial presheaves, revisited
- Dugger
(Show Context)
Citation Context ...check that the maps we are localizing are weak equivalences in Jardine’s sense, but this is easy). The essence of the following proposition could almost be considered folklore—a proof can be found in =-=[D2]-=-: Proposition 7.3. The above map UC/T → sPre(C)Jardine is a Quillen equivalence. Remark 7.4. The model categories UC/T and sPre(C)Jardine are of course not that different: they share the same underlyi... |

1 |
Combinatorial model categories. Toexist
- Smith
(Show Context)
Citation Context ...e localizations are always known to exist (for any set of maps S). These are the left proper, cellular model categories of Hirschhorn [H], and the left proper, combinatorial model categories of Smith =-=[Sm]-=-. We will not recall the definitions of these classes here, but suffice it to say that the model categories UC belong to both of them, and so we are free to localize. In general, the model categories ... |

1 |
Homotopy Theories,’’ Memoirs
- Heller
- 1988
(Show Context)
Citation Context ... localize: in this paper we have consistently used the Bousfield–Kan model structure, in which the fibrations and weak equivalences are detected objectwise, but there is also a Heller model structure =-=[He]-=- in which the cofibrations and weak equivalences are detected objectwise. The Heller structure doesn’t seem to enjoy any kind of universal property, however. It is sometimes considered more ‘‘natural’... |

1 |
unpublished letter to A
- Joyal
(Show Context)
Citation Context ...property, however. It is sometimes considered more ‘‘natural’’ to work with simplicial sheaves than with simplicial presheaves, although they give rise to the same homotopy theory—this was what Joyal =-=[Jo]-=- originally did, and simplicial sheaves were also used in [MV]. But from the viewpoint of universal model categories simplicial presheaves are very natural. By working with sheaves one allows oneself ... |