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11
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 ...
Homotopytheoretic aspects of 2–monads
 J. Homotopy Relat. Struct
"... We study 2monads and their algebras using a Catenriched version of Quillen model categories, emphasizing the parallels between the homotopical and 2categorical points of view. Every 2category with finite limits and colimits has a canonical model structure in which the weak equivalences are the e ..."
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We study 2monads and their algebras using a Catenriched version of Quillen model categories, emphasizing the parallels between the homotopical and 2categorical points of view. Every 2category with finite limits and colimits has a canonical model structure in which the weak equivalences are the equivalences; we use these to construct more interesting model structures on 2categories, including a model structure on the 2category of algebras for a 2monad T, and a model structure on a 2category of 2monads on a fixed 2category K. 1
A 2categories companion
"... Abstract. This paper is a rather informal guide to some of the basic theory of 2categories and bicategories, including notions of limit and colimit, 2dimensional universal algebra, formal category theory, and nerves of bicategories. 1. Overview and basic examples This paper is a rather informal gu ..."
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Abstract. This paper is a rather informal guide to some of the basic theory of 2categories and bicategories, including notions of limit and colimit, 2dimensional universal algebra, formal category theory, and nerves of bicategories. 1. Overview and basic examples This paper is a rather informal guide to some of the basic theory of 2categories and bicategories, including notions of limit and colimit, 2dimensional universal algebra, formal category theory, and nerves of bicategories. As is the way of these things, the choice of topics is somewhat personal. No attempt is made at either rigour or completeness. Nor is it completely introductory: you will not find a definition of bicategory; but then nor will you really need one to read it. In keeping with the philosophy of category theory, the morphisms between bicategories play more of a role than the bicategories themselves. 1.1. The key players. There are bicategories, 2categories, and Catcategories. The latter two are exactly the same (except that strictly speaking a Catcategory should have small homcategories, but that need not concern us here). The first two are nominally different — the 2categories are the strict bicategories, and not every bicategory is strict — but every bicategory is biequivalent to a strict one, and biequivalence is the right general notion of equivalence for bicategories and for 2categories. Nonetheless, the theories of bicategories, 2categories, and Catcategories have rather different flavours.
A homotopical algebra of graphs related to zeta series
 Homology, Homotopy and its Applications 10 (2008), 1–13. MR2506131 (2010f:18010). [C95] Crans, Sjoerd E. Quillen
"... Abstract: The purpose of this paper is to develop a homotopical algebra for graphs, relevant to zeta series and spectra of finite graphs. More precisely, we define a Quillen model structure in a category of graphs (directed and possibly infinite, with loops and multiple arcs allowed). The weak equiv ..."
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Abstract: The purpose of this paper is to develop a homotopical algebra for graphs, relevant to zeta series and spectra of finite graphs. More precisely, we define a Quillen model structure in a category of graphs (directed and possibly infinite, with loops and multiple arcs allowed). The weak equivalences for this model structure are the Acyclics (graph morphisms which preserve cycles). The cofibrations and fibrations for the model are determined from the class of Whiskerings (graph morphisms produced by grafting trees). Our model structure seems to fit well with the importance of acyclic directed graphs in many applications. In addition to the weak factorization systems which form this model structure, we also describe two FreydKelly factorization systems based on Folding, Injecting, and Covering graph morphisms. 0. Introduction. In this paper we develop a notion of homotopy within graphs, and demonstrate its relevance to the study of zeta series and spectrum of a finite graph. We will work throughout with a particular category of graphs, described in Section 1 below. Our graphs will be directed and possibly infinite, with loops and multiple arcs allowed.
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...
Homotopy theories of diagrams
, 2011
"... The work which is displayed in this paper arose from a preliminary study of the homotopy theory of dynamical systems. In general, a dynamical system consists of an action X × S → X ..."
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The work which is displayed in this paper arose from a preliminary study of the homotopy theory of dynamical systems. In general, a dynamical system consists of an action X × S → X
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.
Dynamical systems and diagrams
, 2010
"... In general, a dynamical system consists of an action ..."
unknown title
, 2000
"... these four categories have equivalent homotopy categories but, in fact, their proof contains an error and the homotopy categories are not equivalent with the weak equivalences they use in the comma categories. We show here that the correct weak ..."
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these four categories have equivalent homotopy categories but, in fact, their proof contains an error and the homotopy categories are not equivalent with the weak equivalences they use in the comma categories. We show here that the correct weak
FCEyN, Universidad de Buenos Aires
, 810
"... Grothendieck fibrations have played an important role in homotopy theory. Among others, they were used by Thomason to describe homotopy colimits of small categories and by Quillen to derive long exact sequences of higher Ktheory groups. We construct simplicial objects, namely the fibred and the cle ..."
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Grothendieck fibrations have played an important role in homotopy theory. Among others, they were used by Thomason to describe homotopy colimits of small categories and by Quillen to derive long exact sequences of higher Ktheory groups. We construct simplicial objects, namely the fibred and the cleaved nerve, to characterize the homotopy type of a Grothendieck fibration by using the additional structure. From these, we derive long exact sequences of homotopy groups and spectral sequences for homology groups, establishing new results and placing those of Thomason and Quillen into our framework.