## ON HOMOTOPY VARIETIES (2005)

### BibTeX

@MISC{Rosicky05onhomotopy,

author = {J. Rosicky},

title = {ON HOMOTOPY VARIETIES},

year = {2005}

}

### OpenURL

### Abstract

Given an algebraic theory T, a homotopy T-algebra is a simplicial set where all equations from T hold up to homotopy. All homotopy T-algebras form a homotopy variety. We will give a characterization of homotopy varieties analogous to the characterization of varieties. We will also study homotopy models of limit theories which leads to homotopy locally presentable categories. These were recently considered by Simpson, Lurie, Toën and Vezzosi.

### Citations

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Théorie des topos et cohomologie étale des schémas
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Citation Context ... colimits. Thus it is a free completion of C under homotopy λ-filtered colimits. Hence HInd λ(C) is analogous to Grothendieck’s free completion Ind λ(C) of a category C under λ-filtered colimits (see =-=[5]-=-). 4. Homotopy varieties Definition 4.1. A small category D will be called homotopy sifted provided that homotopy colimits over D homotopy commute in SSet with finite products. Explicitly, D is homoto... |

377 | Basic concepts of enriched category theory, volume 64
- Kelly
- 1982
(Show Context)
Citation Context ...ategories or Segal categories. In particular, homotopy (co)limits are a special case of weighted (co)limits and the crucial fact is that SPre(C) is a free completion of C under weighted colimits (see =-=[24]-=- and [28]). We expect that our results can be later placed to the context of quasi-categories which is under creation by A. Joyal (see [22] and [23]). 2. Simplicial categories A simplicial category K ... |

326 |
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Citation Context ...iltered category is homotopy sifted. But it also follows from the fact that every filtered category D is aspherical because it is a filtered colimit of categories d ↓ D having the initial object (see =-=[36]-=-). (c) Every category D with finite coproducts is homotopy sifted (see [34], 7.4). It immediately follows from the fact that d1 ∐d2 is the initial object in (d1, d2) ↓ D. (d) Every homotopy sifted cat... |

318 |
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Citation Context ...ak equivalence for each finite product X1 × · · ·×Xn in T . These homotopy algebras have been considered in recent papers [6], [7] and [9] but the subject is much older (see, e.g., [16], [8], [35] or =-=[38]-=-). A category is called a variety if it is equivalent to the category Alg(T ) of all T -algebras in Set for some algebraic theory T . There is a characterization of varieties proved by F. W. Lawvere i... |

196 |
Locally Presentable and Accessible Categories
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- 1994
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Citation Context ...M 0021622409. 12 J. ROSICK ´ Y (Barr) exact. There is a characterization of varieties proved by F. W. Lawvere in the single-sorted case which can be immediately extended to varieties in general (cf. =-=[2]-=-, 3.25). Recent papers [3] and [1] reformulated this characterization by using the concept of a sifted colimit. Sifted colimits generalize filtered colimits – while a category D is filtered if colimit... |

166 |
Functorial Semantics of Algebraic Theories
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Citation Context ...es which leads to homotopy locally presentable categories. These were recently considered by Simpson, Lurie, Toën and Vezzosi. 1. Introduction Algebraic theories were introduced by F. W. Lawvere (see =-=[26]-=- and also [27]) in order to provide a convenient approach to study algebras in general categories. An algebraic theory is a small category T with finite products. Having a category K with finite produ... |

138 |
Handbook of Categorical Algebra
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(Show Context)
Citation Context ...implicial maps. Morphisms between simplicial categories are simplicial functors, i.e., F : K → L is given by simplicial maps hom(K, L) → hom(FK, FL) compatible with composition and unit (see [20], or =-=[9]-=- for basic facts about enriched categories in general). We recall that an appropriate4 J. ROSICK ´ Y concept of (co)limits are weighted (co)limits. Given simplicial functors D : D → K and G : D → SSe... |

50 | Combinatorial model categories have presentations
- Dugger
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Citation Context ...egal categories and called them, a little bit unfortunately, ∞-pretoposes. He characterized them as categories of fibrant and cofibrant objects of cofibrantly generated model categories. By D. Dugger =-=[18]-=-, homotopy locally presentable categories correspond inON HOMOTOPY VARIETIES 3 this way to combinatorial model categories (i.e., cofibrantly generated and locally presentable). B. Toën and G. Vezzosi... |

48 | A model category structure on the category of simplicial categories
- Bergner
(Show Context)
Citation Context ...bjects K1 and K2 of K and (2) each object L of Ho(L) is isomorphic in Ho(L) to Ho(F)(K) for some object K of K. Let SCat denote the category of small simplicial categories and simplicial functors. By =-=[10]-=-, there is a model category structure on SCat whose weak equivalences are the just defined equivalences. Fibrations are simplicial functors F : C → D satisfying two conditions (F1) and (F2) where the ... |

43 |
The theory of quasi-categories I
- Joyal
- 2008
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Citation Context ...a free completion of C under weighted colimits (see [24] and [28]). We expect that our results can be later placed to the context of quasi-categories which is under creation by A. Joyal (see [22] and =-=[23]-=-). 2. Simplicial categories A simplicial category K is a category enriched over the category SSet of simplicial set. This means that hom-sets hom(K, L) are simplicial sets and the composition of morph... |

37 | Universal homotopy theories
- Dugger
(Show Context)
Citation Context ...orphisms (see [18]). Moreover, SPre(C) is a free completion of C under homotopy colimits, which is analogous to Pre(C) being a free completion of C under colimits. This is a reformulation of [29] and =-=[17]-=- - the main result of [17] says, in model category terms, that SPre(C) is a free homotopy locally presentable category over C. We will show that homotopy locally presentable categories are precisely c... |

33 |
Model Categories and their
- HIRSCHHORN
- 2003
(Show Context)
Citation Context ...iven by simplicial maps. Morphisms between simplicial categories are simplicial functors, i.e., F : K → L is given by simplicial maps hom(K, L) → hom(FK, FL) compatible with composition and unit (see =-=[20]-=-, or [9] for basic facts about enriched categories in general). We recall that an appropriate4 J. ROSICK ´ Y concept of (co)limits are weighted (co)limits. Given simplicial functors D : D → K and G :... |

33 |
Quasi-categories and Kan complexes
- Joyal
(Show Context)
Citation Context ...re(C) is a free completion of C under weighted colimits (see [24] and [28]). We expect that our results can be later placed to the context of quasi-categories which is under creation by A. Joyal (see =-=[22]-=- and [23]). 2. Simplicial categories A simplicial category K is a category enriched over the category SSet of simplicial set. This means that hom-sets hom(K, L) are simplicial sets and the composition... |

31 |
Structures defined by finite limits in the enriched context 1,” Cahiers de Topologie et Géométrie Différentielle
- Kelly
- 1980
(Show Context)
Citation Context ...locally presentable (cf. [2]) and its model structure is cofibrantly generated. There is a well developed theory of simplicial locally presentable categories and simplicial accessible categories (cf. =-=[25]-=-, [10] and [11]). Simplicial locally presentable categories then correspond to weighted limit theories while simplicial accessible categories to theories specified by both weighted limits and weighted... |

31 |
Segal topoi and stacks over Segal categories. Available for download: math.AG/0212330
- Toën, Vezzosi
(Show Context)
Citation Context ...ible and locally presentable categories (see [2]), homotopy algebra belongs to the recently created theory of homotopy accessible and homotopy locally presentable categories (see [33], [37], [40] and =-=[41]-=-). J. Lurie [33] introduced homotopy accessible and homotopy locally presentable categories under the name of accessible ∞-categories and presentable ∞-categories. He works with CWcomplexes instead of... |

29 | Homotopical algebraic Geometry I: Topos Theory
- Toën, Vezzosi
- 2005
(Show Context)
Citation Context ...d in the homotopy Giraud theorem characterizing homotopy Grothendieck toposes as homotopy locally presentable categories satisfying some properties of homotopy limits and homotopy colimits (see [29], =-=[34]-=- and [35]). Our main result is a characterization of homotopy varieties analogous to the just mentioned characterization of varieties. It uses the concept of a homotopy sifted colimit – a category D i... |

27 |
Farjoun, Cellular Spaces, Null Spaces and Homotopy Lacalization
- Dror
- 1996
(Show Context)
Citation Context ...et Mλ consisting of all morphisms mD where D : D → C is λ-small and holim D exists in C. Hence HCont λ(C) is closed in SPre(C op ) under homotopy limits and homotopy λ-filtered colimits. Moreover, by =-=[16]-=-, the inclusion G : HCont λ(C) → SPre(C op ) has a homotopy left adjoint F : SPre(C op ) → HCont λ(C). Since G preserves homotopy λ-filtered colimits, F preserves homotopy λ-presentable objects. Indee... |

23 |
Rigidification of algebras over multi-sorted theories. Algebraic and Geometric Topoogy 7
- Bergner
- 2007
(Show Context)
Citation Context ... is the projection to the homotopy category. This means that A preserves finite products up to a weak equivalence. These homotopy universal algebras have been considered in recent papers [6], [7] and =-=[8]-=-. In particular, there is proved a rigidification theorem saying that every homotopy T -algebra is weakly equivalent to a simplicial T -algebra ([6] deals with single-sorted algebraic theories and [8]... |

22 | Homotopy Coherent Category Theory
- Cordier, Porter
(Show Context)
Citation Context ...gebraic theories. A delicate point is the definition of the concept of being homotopy equivalent. Homotopy universal algebra is a part of a broader subject of a homotopy coherent category theory (see =-=[15]-=-). More precisely, while usual universal algebra form a part of the theory of accessible and locally presentable categories (see [2]), homotopy universal algebra belongs to the recently created theory... |

17 | Homotopical and higher categorical structures in algebraic geometry, Hablitation Thesis available at math.AG/0312262 - Toën |

13 | Algebraic theories in homotopy theory
- Badzioch
(Show Context)
Citation Context ...et → Ho(SSet) is the projection to the homotopy category. This means that A preserves finite products up to a weak equivalence. These homotopy universal algebras have been considered in recent papers =-=[6]-=-, [7] and [8]. In particular, there is proved a rigidification theorem saying that every homotopy T -algebra is weakly equivalent to a simplicial T -algebra ([6] deals with single-sorted algebraic the... |

13 |
La théorie de l’homotopie de Grothendieck
- Maltsiniotis
- 2005
(Show Context)
Citation Context ...olimit – a category D is homotopy sifted if homotopy colimits over D commute with finite products in Ho(SSet). Homotopy sifted categories coincide with totally coaspherical categories in the sense of =-=[30]-=-. Our result is closely related to the rigidification theorem of [6] and [8] – in fact, we generalize their results from algebraic theories to simplicial algebraic theories. A delicate point is the de... |

12 | On the duality between varieties and algebraic theories. Algebra universalis
- Adámek, Lawvere, et al.
(Show Context)
Citation Context ...arr) exact. There is a characterization of varieties proved by F. W. Lawvere in the single-sorted case which can be immediately extended to varieties in general (cf. [2], 3.25). Recent papers [3] and =-=[1]-=- reformulated this characterization by using the concept of a sifted colimit. Sifted colimits generalize filtered colimits – while a category D is filtered if colimits over D commute with finite limit... |

12 | On ∞-topoi
- Lurie
- 2003
(Show Context)
Citation Context ...as used in the homotopy Giraud theorem characterizing homotopy Grothendieck toposes as homotopy locally presentable categories satisfying some properties of homotopy limits and homotopy colimits (see =-=[29]-=-, [34] and [35]). Our main result is a characterization of homotopy varieties analogous to the just mentioned characterization of varieties. It uses the concept of a homotopy sifted colimit – a catego... |

11 | Enriched accessible categories
- Borceux, Quintero
- 1996
(Show Context)
Citation Context ...y presentable (cf. [2]) and its model structure is cofibrantly generated. There is a well developed theory of simplicial locally presentable categories and simplicial accessible categories (cf. [25], =-=[10]-=- and [11]). Simplicial locally presentable categories then correspond to weighted limit theories while simplicial accessible categories to theories specified by both weighted limits and weighted colim... |

11 |
Homotopical algebraic geometry
- Toën, Vezzosi
(Show Context)
Citation Context ...of accessible and locally presentable categories (see [2]), homotopy algebra belongs to the recently created theory of homotopy accessible and homotopy locally presentable categories (see [33], [37], =-=[40]-=- and [41]). J. Lurie [33] introduced homotopy accessible and homotopy locally presentable categories under the name of accessible ∞-categories and presentable ∞-categories. He works with CWcomplexes i... |

10 | Limits of small functors
- Day, Lack
(Show Context)
Citation Context ... or Segal categories. In particular, homotopy (co)limits are a special case of weighted (co)limits and the crucial fact is that SPre(C) is a free completion of C under weighted colimits (see [24] and =-=[28]-=-). We expect that our results can be later placed to the context of quasi-categories which is under creation by A. Joyal (see [22] and [23]). 2. Simplicial categories A simplicial category K is a cate... |

9 |
A Giraud-type characterization of the simplicial categories associated to closed model categories as ∞-pretopoi
- Simpson
(Show Context)
Citation Context ...ccessible and locally presentable categories (see [2]), homotopy universal algebra belongs to the recently created theory of homotopy accessible and homotopy locally presentable categories (see [29], =-=[32]-=-, [34] and [35]). J. Lurie [29] introduced homotopy accessible and homotopy locally presentable categories under the name of accessible ∞-categories and presentable ∞-categories. He works with CW-comp... |

9 | Homotopy theory of small diagrams over large categories. http://www.nd.edu/∼wgd
- Chorny, Dwyer
- 2005
(Show Context)
Citation Context ...e cofibrantly generated by images in FC, C ∈ C, of (generating) cofibrations in SSet and the same for trivial cofibrations. This procedure is described in [25], 11.6.1, for an ordinary category C and =-=[18]-=- extends it to the simplicial category of small simplicial functors C op → SSet for an arbitrary simplicial category C. The consequence is that all hom-functors hom(−, C) are cofibrant. Remark 3.1. (a... |

8 | On sifted colimits and generalized varieties
- Adámek, Rosick´y
- 2001
(Show Context)
Citation Context ...K ´ Y (Barr) exact. There is a characterization of varieties proved by F. W. Lawvere in the single-sorted case which can be immediately extended to varieties in general (cf. [2], 3.25). Recent papers =-=[3]-=- and [1] reformulated this characterization by using the concept of a sifted colimit. Sifted colimits generalize filtered colimits – while a category D is filtered if colimits over D commute with fini... |

8 |
A general formulation of homotopy limits
- Bourn, Cordier
- 1983
(Show Context)
Citation Context ...inition of homotopy limits and homotopy colimits adopted in [20], 18.1.8 and 18.1.1 make them a special case of weighted limits and weighted colimits (see [20], 18.3.1); this observation goes back to =-=[12]-=-. The corresponding weights form a homotopy invariant approximations of constant diagrams at a point. The same definitions work in any simplicial category; in what follows, B(X) denotes the nerve of t... |

7 | The orthogonal subcategory problem in homotopy theory, in: An Alpine Anthology of Homotopy Theory
- Casacuberta, Chorny
- 2006
(Show Context)
Citation Context ...o M. Since map(B, A) is weakly equivalent to hom(B, A) whenever B is cofibrant and A is fibrant and all morphisms from M have cofibrant domains and codomains, we have HAlg(T ) = Pre(T op ) ∩ M ⊥ . By =-=[17]-=-, 1.1, there is a functor L : SSet T → M ⊥ preserving weak equivalences and equipped with a simplicial natural transformation η : Id → L which is idempotent up to homotopy and, moreover, it is a point... |

6 | Fibrations and homotopy colimits of simplicial sheaves, preprint available at the author home page: http://www.math.uiuc.edu/ rezk/papers.html - Rezk |

6 |
On H-spaces and infinite loop spaces
- Beck
- 1969
(Show Context)
Citation Context ...A(Xn) is a weak equivalence for each finite product X1 × · · ·×Xn in T . These homotopy algebras have been considered in recent papers [6], [7] and [9] but the subject is much older (see, e.g., [16], =-=[8]-=-, [35] or [38]). A category is called a variety if it is equivalent to the category Alg(T ) of all T -algebras in Set for some algebraic theory T . There is a characterization of varieties proved by F... |

2 | From Γ-spaces to algebraic theories
- Badzioch
(Show Context)
Citation Context ...Ho(SSet) is the projection to the homotopy category. This means that A preserves finite products up to a weak equivalence. These homotopy universal algebras have been considered in recent papers [6], =-=[7]-=- and [8]. In particular, there is proved a rigidification theorem saying that every homotopy T -algebra is weakly equivalent to a simplicial T -algebra ([6] deals with single-sorted algebraic theories... |

2 | A theory of enriched sketches
- Borceux, Quinteiro, et al.
- 1998
(Show Context)
Citation Context ...able (cf. [2]) and its model structure is cofibrantly generated. There is a well developed theory of simplicial locally presentable categories and simplicial accessible categories (cf. [25], [10] and =-=[11]-=-). Simplicial locally presentable categories then correspond to weighted limit theories while simplicial accessible categories to theories specified by both weighted limits and weighted colimits. The ... |

2 |
Homotopy colimits
- Bousfield, Kan
- 1972
(Show Context)
Citation Context ... preserves homotopy λ-filtered colimits. Under a homotopy λ-filtered colimit we mean a homotopy colimit over a diagram D : D → M where D is a λ-filtered category (cf.ON HOMOTOPY VARIETIES 7 [2]). By =-=[13]-=-, XII., 3.5(ii), homotopy λ-filtered colimits are weakly equivalent to λ-filtered colimits in SSet (and thus in SPre(C) as well). A homotopy colimit of a diagram D → K is called λ-small if the categor... |

2 |
Small diagrams over large categories
- Chorny
- 2004
(Show Context)
Citation Context ...28], SPre(C) has all weighted limits provided that C has this property. There is always the BousfieldKan model category structure on SPre(C), i.e., weak equivalences and fibrations are pointwise (see =-=[14]-=-). We have the Yoneda embedding YC : C → SPre(C) given by Y (C) = hom(−, C). It is a free completion of C under weighted colimits and preserves all existing weighted limits (cf. [24] and [28]). Dually... |

2 |
Geometry of Iterated Loop
- May
- 1972
(Show Context)
Citation Context ... is a weak equivalence for each finite product X1 × · · ·×Xn in T . These homotopy algebras have been considered in recent papers [6], [7] and [9] but the subject is much older (see, e.g., [16], [8], =-=[35]-=- or [38]). A category is called a variety if it is equivalent to the category Alg(T ) of all T -algebras in Set for some algebraic theory T . There is a characterization of varieties proved by F. W. L... |