## Model structures for homotopy of internal categories

Venue: | Theory Appl. Categ |

Citations: | 5 - 0 self |

### BibTeX

@ARTICLE{Everaert_modelstructures,

author = {T. Everaert and R. W. Kieboom and T. Van and Der Linden},

title = {Model structures for homotopy of internal categories},

journal = {Theory Appl. Categ},

year = {},

volume = {15}

}

### OpenURL

### Abstract

CatC of internal categories and functors in a given finitely complete categoryC. Several non-equivalent notions of internal equivalence exist; to capture these notions, the model structures are defined relative to a given Grothendieck topology on C. Under mild conditions on C, the regular epimorphism topology determines a modelstructure where we is the class of weak equivalences of internal categories (in the sense of Bunge and Par'e). For a Grothendieck topos C we get a structure that, thoughdifferent from Joyal and Tierney's, has an equivalent homotopy category. In case C is semi-abelian, these weak equivalences turn out to be homology isomorphisms, and themodel structure on CatC induces a notion of homotopy of internal crossed modules. Incase C is the category

### Citations

448 |
Théorie des topos et cohomologie étale des schémas
- Artin, Grothendieck, et al.
- 1977
(Show Context)
Citation Context ...on concerning internal categories (and, of course, topos theory) may be found in Johnstone [27]. Other works on topos theory we used are Mac Lane and Moerdijk [36], Johnstone's Elephant [29] and SGA4 =-=[1]-=-. The standard work on "all things semi-abelian" is Borceux and Bourn's book [7]. Acknowledgements. Twice the scope of this paper has been considerably widened: first by Stephen Lack, who pointed out ... |

196 | Locally Presentable and Accessible Categories - Adámek, Rosicky - 1994 |

188 |
Sheaves in Geometry and Logic. A First Introduction to Topos Theory
- Lane, Moerdijk
- 1992
(Show Context)
Citation Context ...gory theoretic results. Lots of information concerning internal categories (and, of course, topos theory) may be found in Johnstone [27]. Other works on topos theory we used are Mac Lane and Moerdijk =-=[36]-=-, Johnstone's Elephant [29] and SGA4 [1]. The standard work on "all things semi-abelian" is Borceux and Bourn's book [7]. Acknowledgements. Twice the scope of this paper has been considerably widened:... |

149 |
Sketches of an elephant: a topos theory compendium
- Johnstone
- 2002
(Show Context)
Citation Context ...s of information concerning internal categories (and, of course, topos theory) may be found in Johnstone [27]. Other works on topos theory we used are Mac Lane and Moerdijk [36], Johnstone's Elephant =-=[29]-=- and SGA4 [1]. The standard work on "all things semi-abelian" is Borceux and Bourn's book [7]. Acknowledgements. Twice the scope of this paper has been considerably widened: first by Stephen Lack, who... |

73 |
Two dimensional monad theory
- Blackwell, Kelly, et al.
- 1989
(Show Context)
Citation Context ... this is still the case for strong equivalences of internal categories. In fact, in any 2-category, an equivalence between two objects is always an adjoint equivalence; see Blackwell, Kelly and Power =-=[5]-=-. More precisely, the following holds. 4.8. Proposition. [Blackwell, Kelly and Power, [5]] Let C be a 2-category and f : C ,2 D a 1-cell of C. Then f is an adjoint equivalence if and only if for every... |

45 |
Abstract Homotopy Theory and Generalized Sheaf Cohomology
- Brown
- 1973
(Show Context)
Citation Context ...right inverse to a strong trivial fibration) followed by a strong fibration.sMODEL STRUCTURES FOR HOMOTOPY OF INTERNAL CATEGORIES 83 Proof. This is an application of K. S. Brown's Factorization Lemma =-=[13]-=-. To use it, we must show that (CatC, F , W), where F is the class of strong fibrations and W is the class of strong equivalences, forms a category of fibrant objects. Condition (A) is just Propositio... |

35 |
protomodular, homological and semi-abelian categories
- Borceux, Bourn, et al.
- 2004
(Show Context)
Citation Context ... Johnstone [27]. Other works on topos theory we used are Mac Lane and Moerdijk [36], Johnstone's Elephant [29] and SGA4 [1]. The standard work on "all things semi-abelian" is Borceux and Bourn's book =-=[7]-=-. Acknowledgements. Twice the scope of this paper has been considerably widened: first by Stephen Lack, who pointed out the difference between notions of internal equivalence, and incited us to consid... |

28 |
Fibred and cofibred categories
- Gray
- 1965
(Show Context)
Citation Context ...tities (s^s'1f )*(1f s't^) = 1f and (1g s's^)*(t^s'1g ) = 1g hold. Then f is left adjoint to g , g right adjoint to f , s^ the counit and t^ the unit of the adjunction. Using J. W. Gray's terminology =-=[24]-=-, we shall call lali a left adjoint left inverse functor, and, dually, rari a right adjoint right inverse functor. In case f is left adjoint left inverse to g , we denote the situation f = lali g or g... |

19 |
Strong stacks and classifying spaces, Category theory
- Joyal, Tierney
- 1990
(Show Context)
Citation Context ...in e such that p(s^) = r'; this notion was introduced for groupoids by R. Brown in [14]. We are unaware of who first proved this fact; certainly, it is a special case of Joyal and Tierney's structure =-=[30]-=-, but it was probably known before. A very explicit proof may be found in an unpublished paper by Rezk [39]. Other approaches to model category structures on Cat exist: Golasi'nski uses the homotopy t... |

17 |
Homotopical algebra”, Lecture Notes in Mathematics No. 43, Springer-Verlag 1967. 203 1] ”Revêtements étales et groupes fondamental
- Quillen
(Show Context)
Citation Context ... copy for private use granted. 66sMODEL STRUCTURES FOR HOMOTOPY OF INTERNAL CATEGORIES 67 1. Introduction It is very well-known that the following choices of morphisms define a Quillen model category =-=[38]-=- structure--known as the "folk" structure--on the category Cat of small categories and functors between them: we is the class of equivalences of categories, cof the class of functors, injective on obj... |

16 | Weak factorization systems and topological functors
- Adámek, Herrlich, et al.
(Show Context)
Citation Context ...ory [16], [7, Example A.5.17], [10]. 2.3. Weak factorization systems and model categories. In this paper we use the definition of model category as presented by Ad'amek, Herrlich, Rosick'y and Tholen =-=[2]-=-. For us, next to its elegance, the advantage over Quillen's original definition [38] is its explicit use of weak factorization systems. We briefly recall some important definitions. 2.4. Definition. ... |

16 |
C.: A categorical approach to commutator theory
- Pedicchio
- 1995
(Show Context)
Citation Context ...nce GrpdC may be viewed as a subcategory of RGC. As soon as C is, moreover, finitely cocomplete and regular, this subcategory is reflective (see Borceux and Bourn [7, Theorem 2.8.13]). In her article =-=[37]-=-, M. C. Pedicchio shows that, if C is an exact Mal'tsev category with coequalizers, then the category GrpdC is {regular epi}-reflective in RGC. This implies that GrpdC is closed in RGC under subobject... |

14 |
Handbook of Categorical Algebra. Encyclopedia of Math. and its applications
- Borceux
- 1994
(Show Context)
Citation Context ...fibrations have the homotopy lifting property (Proposition 7.3) and cofibrations the homotopy extension property (Proposition 7.6) with respect to the cocylinder defined in Section 3. We used Borceux =-=[6]-=- and Mac Lane [35] for general category theoretic results. Lots of information concerning internal categories (and, of course, topos theory) may be found in Johnstone [27]. Other works on topos theory... |

14 |
A Quillen model structure for bicategories
- Lack
(Show Context)
Citation Context ...el structure on Cat itself [41]. Both are very different from the folk structure. Related work includes folk-style model category structures on categories of 2-categories and bicategories (Lack [34], =-=[33]-=-) and a Thomason-style model category structure for 2-categories (Worytkiewicz, Hess, Parent and Tonks [42]). If E is a Grothendieck topos there are two model structures on the category CatE of intern... |

12 |
Exact categories, in: Exact categories and categories of sheaves, LNM 236
- Barr
- 1971
(Show Context)
Citation Context ...n fibration. Proof. First note that, because a semi-abelian category is always Mal'tsev, every category in C is an internal groupoid. We may now use M. Barr's Embedding Theorem for regular categories =-=[4]-=- in the form of Metatheorem A.5.7 in [7]. Indeed, the properties "some internal functor is a fibration" and "some simplicial morphism is a Kan fibration" may be added to the list of properties [7, 0.1... |

12 | Protomodularity, descent, and semidirect products
- Bourn, Janelidze
(Show Context)
Citation Context ...tion. In his paper [25], Janelidze introduces a notion of crossed module in an arbitrary semiabelian category C. Its definition is based on Bourn and Janelidze's notion of internal semidirect product =-=[12]-=- and Borceux, Janelidze and Kelly's notion of internal object action [8]. Internal crossed modules also generalize the case where C = Gp in the sense that an equivalence XModC ' CatC still exists. Usi... |

10 |
The “closed subgroup theorem” for localic herds and pregroupoids
- Johnstone
- 1989
(Show Context)
Citation Context ...in Section 3. We used Borceux [6] and Mac Lane [35] for general category theoretic results. Lots of information concerning internal categories (and, of course, topos theory) may be found in Johnstone =-=[27]-=-. Other works on topos theory we used are Mac Lane and Moerdijk [36], Johnstone's Elephant [29] and SGA4 [1]. The standard work on "all things semi-abelian" is Borceux and Bourn's book [7]. Acknowledg... |

9 | Baer invariants in semi-abelian categories I - Everaert, Linden |

8 |
Internal object actions
- Borceux, Janelidze, et al.
(Show Context)
Citation Context ...in an arbitrary semiabelian category C. Its definition is based on Bourn and Janelidze's notion of internal semidirect product [12] and Borceux, Janelidze and Kelly's notion of internal object action =-=[8]-=-. Internal crossed modules also generalize the case where C = Gp in the sense that an equivalence XModC ' CatC still exists. Using this equivalence, we may transport thes88 T. EVERAERT, R.W. KIEBOOM A... |

8 |
Internal graphs and internal groupoids in Mal’cev categories. Category Theory
- Carboni, Pedicchio, et al.
- 1991
(Show Context)
Citation Context ...es, groupoids) in C. Let GrpdC J ,2 CatC I ,2 RGC denote the forgetful functors. It is well-known that J embeds GrpdC into CatC as a coreflective subcategory. Carboni, Pedicchio and Pirovano prove in =-=[17]-=- that, if C is Mal'tsev, then I is full, and J is an isomorphism. Moreover, an internal reflexive graphs70 T. EVERAERT, R.W. KIEBOOM AND T. VAN DER LINDEN carries at most one structure of internal gro... |

6 |
The denormalized 3 × 3 lemma
- Bourn
(Show Context)
Citation Context ...the arrow H1f is an isomorphism if and only if the square i is a pullback (Lemmas 4.2.4 and 4.2.5 in [7]); square ii is a pullback if and only if iii is a joint pullback, which (by Proposition 1.1 in =-=[9]-=-) is the case exactly when H0f is mono. This already shows one implication of 1. To prove the other, note that if f is fully faithful, then H1f is an isomorphism; hence i is a pullback, ii is a pullba... |

4 | Kan-Bedingungen und abstrakte Homotopietheorie - Kamps - 1972 |

3 | On the direct image of intersections in exact homological categories - Bourn |

3 |
Special reflexive graphs in modular varieties, Algebra univers
- Gran, Rosicky
(Show Context)
Citation Context ... this result that CatC is closed in RGC under quotients. It follows that CatC is Birkhoff [26] in RGC. This, in turn, implies that if C is semi-abelian, so is CatC [18, Remark 5.4]. Gran and Rosick'y =-=[23]-=- extend these results to the context of modular varieties. For any variety V, the category RGV is equivalent to a variety. They show that, if, moreover, V is modular, V is Mal'tsev if and only if Grpd... |

3 |
A model category for categories, unpublished
- Rezk
- 1996
(Show Context)
Citation Context ...who first proved this fact; certainly, it is a special case of Joyal and Tierney's structure [30], but it was probably known before. A very explicit proof may be found in an unpublished paper by Rezk =-=[39]-=-. Other approaches to model category structures on Cat exist: Golasi'nski uses the homotopy theory of cubical sets to define a model structure on the category of pro-objects in Cat [21]; Thomason uses... |

2 |
Fibrations of groupoids, J. Algebra 15
- Brown
- 1970
(Show Context)
Citation Context ... B such that for any object e of E and any isomorphism r' : b ,2 p(e) in B there exists an isomorphism s^ with codomain e such that p(s^) = r'; this notion was introduced for groupoids by R. Brown in =-=[14]-=-. We are unaware of who first proved this fact; certainly, it is a special case of Joyal and Tierney's structure [30], but it was probably known before. A very explicit proof may be found in an unpubl... |

2 |
Homotopy theory for (braided
- Garz'on, Miranda
- 1997
(Show Context)
Citation Context ...gory of groups and homomorphisms, we obtain the model structures on the category CatGp of categorical groups and the category XMod of crossed modules of groups, as described by Garz'on and Miranda in =-=[20]-=-. The second case models the situation in Cat, equipped with the folk model structure, in the sense that here, weak equivalences are homotopy equivalences, fibrations have the homotopy lifting propert... |

2 |
Cat as a closed model category, Cah
- Thomason
- 1980
(Show Context)
Citation Context ...s the homotopy theory of cubical sets to define a model structure on the category of pro-objects in Cat [21]; Thomason uses an adjunction to simplicial sets to acquire a model structure on Cat itself =-=[41]-=-. Both are very different from the folk structure. Related work includes folk-style model category structures on categories of 2-categories and bicategories (Lack [34], [33]) and a Thomason-style mode... |

2 |
A model structure `a la Thomason on 2-Cat, math.AT/0411154, 2005. Vakgroep Wiskunde Faculteit Wetenschappen Vrije Universiteit Brussel Pleinlaan 2 1050 Brussel Belgium Email: teveraer@vub.ac.be rkieboom@vub.ac.be tvdlinde@vub.ac.be This article may be acc
- Worytkiewicz, Hess, et al.
(Show Context)
Citation Context ...k-style model category structures on categories of 2-categories and bicategories (Lack [34], [33]) and a Thomason-style model category structure for 2-categories (Worytkiewicz, Hess, Parent and Tonks =-=[42]-=-). If E is a Grothendieck topos there are two model structures on the category CatE of internal categories in E . On can define the cofibrations and weak equivalences "as in Cat", and then define the ... |

1 |
Homotopies of small categories, Fund
- Golasi'nski
- 1981
(Show Context)
Citation Context ... paper by Rezk [39]. Other approaches to model category structures on Cat exist: Golasi'nski uses the homotopy theory of cubical sets to define a model structure on the category of pro-objects in Cat =-=[21]-=-; Thomason uses an adjunction to simplicial sets to acquire a model structure on Cat itself [41]. Both are very different from the folk structure. Related work includes folk-style model category struc... |

1 |
Internal crossed modules, Georgian
- Janelidze
- 1994
(Show Context)
Citation Context ...ogy isomorphisms, and the fibrations, Kan fibrations. Moreover, the category of internal categories in a semi-abelian category C is equivalent to Janelidze's category of internal crossed modules in C =-=[25]-=-. Reformulating the model structure in terms of internal crossed modules (as is done in Theorem 6.7) simplifies its description. If C is the category of groups and homomorphisms, we obtain the model s... |