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12
Quillen Closed Model Structures for Sheaves
, 1995
"... In this chapter I give a general procedure of transferring closed model structures along adjoint functor pairs. As applications I derive from a global closed model structure on the category of simplicial sheaves closed model structures on the category of sheaves of 2groupoids, the category of bisim ..."
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In this chapter I give a general procedure of transferring closed model structures along adjoint functor pairs. As applications I derive from a global closed model structure on the category of simplicial sheaves closed model structures on the category of sheaves of 2groupoids, the category of bisimplicial sheaves and the category of simplicial sheaves of groupoids. Subsequently, the homotopy theories of these categories are related to the homotopy theory of simplicial sheaves. 1 Introduction There are two ways of trying to generalize the well known closed model structure on the category of simplicial sets to the category of simplicial objects in a Grothendieck topos. One way is to concentrate on the local aspect, and to use the Kanfibrations as a starting point. In [14] Heller showed that for simplicial presheaves there is a local (there called right) closed model structure. In [2] K. Brown showed that for a topological space X the category of "locally fibrant" sheaves of spectra on ...
Types are weak ωgroupoids
, 2008
"... We define a notion of weak ωcategory internal to a model of MartinLöf type theory, and prove that each type bears a canonical weak ωcategory structure obtained from the tower of iterated identity types over that type. We show that the ωcategories arising in this way are in fact ωgroupoids. ..."
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We define a notion of weak ωcategory internal to a model of MartinLöf type theory, and prove that each type bears a canonical weak ωcategory structure obtained from the tower of iterated identity types over that type. We show that the ωcategories arising in this way are in fact ωgroupoids.
PROPRIÉTÉS UNIVERSELLES ET EXTENSIONS DE KAN DÉRIVÉES Contents
"... to the homotopy theory of presheaves of homotopy types on A, is characterized by a natural universal property. In particular, the theory of Kan extensions extends to ..."
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to the homotopy theory of presheaves of homotopy types on A, is characterized by a natural universal property. In particular, the theory of Kan extensions extends to
Homotopy algebras for operads
"... We present a definition of homotopy algebra for an operad, and explore its consequences. The paper should be accessible to topologists, category theorists, and anyone acquainted with operads. After a review of operads and monoidal categories, the definition of homotopy algebra is given. Specifically ..."
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We present a definition of homotopy algebra for an operad, and explore its consequences. The paper should be accessible to topologists, category theorists, and anyone acquainted with operads. After a review of operads and monoidal categories, the definition of homotopy algebra is given. Specifically, suppose that M is a monoidal category in which it makes sense to talk about algebras for some operad P. Then our definition says what a homotopy Palgebra in M is, provided only that some of the morphisms in M have been marked out as ‘homotopy equivalences’. The bulk of the paper consists of examples of homotopy algebras. We show that any loop space is a homotopy monoid, and, in fact, that any nfold loop space is an nfold homotopy monoid in an appropriate sense. We try to compare weakened algebraic structures such as A∞spaces, A∞algebras and nonstrict monoidal categories to our homotopy algebras, with varying degrees of success. We also prove results on ‘change of base’, e.g. that the classifying space of a homotopy monoidal category is a homotopy topological monoid. Finally, we
Internal categorical structure in homotopical algebra
 Proceedings of the IMA workshop ?nCategories: Foundations and Applications?, June 2004, (to appear). CROSSED MODULES AND PEIFFER CONDITION 135 [Ped95] [Por87
, 1995
"... Abstract. This is a survey on the use of some internal higher categorical structures in algebraic topology and homotopy theory. After providing a general view of the area and its applications, we concentrate on the algebraic modelling of connected (n + 1)types through cat ngroups. 1. ..."
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Abstract. This is a survey on the use of some internal higher categorical structures in algebraic topology and homotopy theory. After providing a general view of the area and its applications, we concentrate on the algebraic modelling of connected (n + 1)types through cat ngroups. 1.
Flexible sheaves
, 1996
"... This is an unfinished explanation of the notion of “flexible sheaf”, that is a homotopical notion of sheaf of topological spaces (homotopy types) over a site. See “Homotopy over the complex numbers and generalized de Rham cohomology ” (submitted to proceedings of the Taniguchi conference on vector b ..."
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This is an unfinished explanation of the notion of “flexible sheaf”, that is a homotopical notion of sheaf of topological spaces (homotopy types) over a site. See “Homotopy over the complex numbers and generalized de Rham cohomology ” (submitted to proceedings of the Taniguchi conference on vector bundles, preprint of Toulouse 3, and also available at my homepage 2) for a more detailed introduction, and also for a further development of the ideas presented here. The present paper was finished in December 1993 while I was visiting MIT. Since writing this, I have realized that the theory sketched here is essentially equivalent to JardineIllusie’s theory of presheaves of topological spaces (although they talk about presheaves of simplicial sets which is the same thing). This is the point of view adopted in “Homotopy over the complex numbers and generalized de Rham cohomology”. 1 Added in August 1996: I am finally sending this to “Duke eprints ” because it now seems that there may be several useful features of the treatment given here. First of all, the direct construction of the homotopysheafification (by doing a certain operation n+2 times) seems to be useful, for example I need
Presentations of OmegaCategories By Directed Complexes
, 1997
"... The theory of directed complexes is extended from free !categories to arbitrary ! categories by defining presentations in which the generators are atoms and the relations are equations between molecules. Our main result relates these presentations to the more standard algebraic presentations; we ..."
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The theory of directed complexes is extended from free !categories to arbitrary ! categories by defining presentations in which the generators are atoms and the relations are equations between molecules. Our main result relates these presentations to the more standard algebraic presentations; we also show that every !category has a presentation by directed complexes. The approach is similar to that used by Crans for pasting presentations. 1991 Mathematics Subject Classification: 18D05. 1 Introduction There are at present three combinatorial structures for constructing !categories: pasting schemes, defined in 1988 by Johnson [8], parity complexes, introduced in 1991 by Street [16, 17] and directed complexes, given by Steiner in 1993 [15]. These structures each consist of cells which have collections of lower dimensional cells as domain and codomain; see for example Definition 2.2 below. They also have `local' conditions on the cells, ensuring that a cell together with its boundin...
Localizations of Transfors
, 1998
"... Let C , D and E be ndimensional teisi, i.e., higherdimensional Graycategorical structures. The following questions can be asked. Does a left qtransfor C ! D , i.e., a functor 2 q\Omega C ! D , induce a right qtransfor C ! D , i.e., a functor C\Omega 2 q ! D ? More generally, doe ..."
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Let C , D and E be ndimensional teisi, i.e., higherdimensional Graycategorical structures. The following questions can be asked. Does a left qtransfor C ! D , i.e., a functor 2 q\Omega C ! D , induce a right qtransfor C ! D , i.e., a functor C\Omega 2 q ! D ? More generally, does a functor C\Omega D ! E induce a functor D\Omega C ! E? For c; c 0 elements of C whose (k \Gamma 1)sources and (k \Gamma 1)targets agree, does a qtransfor C ! D induce a qtransfor C (c;c 0 ) ! D (d;d 0 ), for appropriate d;d 0 2 D ? For c; c 0 2 C and d;d 0 2 D whose (k \Gamma 1)sources and (k \Gamma 1)targets agree, does a qtransfor C\Omega D ! E induce a (q+k+1)transfor C (c;c 0 )\Omega D (d;d 0 ) ! E(e;e 0 ), for appropriate e; e 0 2 E? I give answers to these questions in the cases where ndimensional teisi and their tensor product have been defined, i.e., for n 3, and in some cases for n up to 5 which do not need all data and axioms...
Euler Characteristics of Categories and Homotopy Colimits
, 2010
"... In a previous article, we introduced notions of finiteness obstruction, Euler characteristic, and L 2Euler characteristic for wide classes of categories. In this sequel, we prove the compatibility of those notions with homotopy colimits of Iindexed categories where I is any small category admitti ..."
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In a previous article, we introduced notions of finiteness obstruction, Euler characteristic, and L 2Euler characteristic for wide classes of categories. In this sequel, we prove the compatibility of those notions with homotopy colimits of Iindexed categories where I is any small category admitting a finite ICWmodel for its Iclassifying space. Special cases of our Homotopy Colimit Formula include formulas for products, homotopy pushouts, homotopy orbits, and transport groupoids. We also apply our formulas to Haefliger complexes of groups, which extend Bass–Serre graphs of groups to higher dimensions. In particular, we obtain necessary conditions for developability of a finite complex of groups from an action of a finite group on a finite category without loops.
ASPHERICITY STRUCTURES, SMOOTH FUNCTORS, AND
"... This paper was translated from French by Jonathan Chiche. Abstract. The aim of this paper is to generalize Grothendieck’s theory of smooth functors in order to include within this framework the theory of fibered categories. We obtain in particular a new characterization of fibered categories. ..."
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This paper was translated from French by Jonathan Chiche. Abstract. The aim of this paper is to generalize Grothendieck’s theory of smooth functors in order to include within this framework the theory of fibered categories. We obtain in particular a new characterization of fibered categories.