A model category structure for equivariant algebraic models

In the equivariant category of spaces with an action of a finite group, algebraic `minimal models' exist which describe the rational homotopy for -spaces which are 1-connected and of finite type. These models are diagrams of commutative differential graded algebras. In this paper we prove that a model category structure exists on this diagram category in such a way that the equivariant minimal models are cofibrant objects. We show that with this model structure, there is a Quillen equivalence between the equivariant category of rational -spaces satisfying the above conditions and the algebraic category of the models.

     

单纯广群的准层和同伦纤维

显示了在设置C上的单纯广群的准层的范畴是个封闭模型范畴.证明了在一个单纯广群的准层G上的单纯函子X是局部弱等价于同伦纤维.     

Stable Homotopy Monomorphisms

It is well known that the concept of monomorphism in a category can be defined using an appropriate pullback diagram. In the homotopy category of TOP pullbacks do not generally exist. This motivated Michael Mather to introduce another notion of homotopy pullback which does exist. The aim of this paper is to investigate the modified notion of homotopy monomorphism obtained by applying the pullback characterization using Mather's homotopy pullback. The main result of Section 1 shows that these modified homotopy monomorphisms are exactly those homotopy monomorphisms (in the usual sense) which are homotopy pullback stable, hence the terminology “stable” homotopy monomorphism. We also link these stable homotopy monomorphisms to monomorphisms and products in the track homotopy category over a fixed space. In Section 2 we answer the question: when is a (weak) fibration also a stable homotopy monomorphism? In the final section it is shown that the class of (weak) fibrations with this additional property coincides with the class of “double” (weak) fibrations. The double (weak) covering homotopy property being introduced here is a stronger version of the (W) CHP in which the final maps of the homotopies involved play the same role as the initial maps.     

Flow does not Model Flows up to Weak Dihomotopy

We prove that the category of flows cannot be the underlying category of a model category whose corresponding homotopy types are the flows up to weak dihomotopy. Some hints are given to overcome this problem. In particular, a new approach of dihomotopy involving simplicial presheaves over an appropriate small category is proposed. This small category is obtained by taking a full subcategory of a locally presentable version of the category of flows. Mathematics Subject Classifications (2000) 55P99, 68Q85, 18A32, 55U35.     

Closed Simplicial Model Structures for Exterior and Proper Homotopy Theory

The notion of exterior space consists of a topological space together with a certain nonempty family of open subsets that is thought of as a system of open neighbourhoods at infinity while an exterior map is a continuous map which is continuous at infinity. The category of spaces and proper maps is a subcategory of the category of exterior spaces.In this paper we show that the category of exterior spaces has a family of closed simplicial model structures, in the sense of Quillen, depending on a pair {T,T} of suitable exterior spaces. For this goal, for a given exterior space T, we construct the exterior T-homotopy groups of an exterior space under T. Using different spaces T we have as particular cases the main proper homotopy groups: the Brown–Grossman, erin–Steenrod, p-cylindrical, Baues–Quintero and Farrell–Taylor–Wagoner groups, as well as the standard (Hurewicz) homotopy groups.The existence of this model structure in the category of exterior spaces has interesting applications. For instance, using different pairs {T,T}, it is possible to study the standard homotopy type, the homotopy type at infinity and the global proper homotopy type.     

奇点模型范畴的Quillen等价

令R是左Gorenstein环.我们构造了奇点反导出模型范畴和奇点余导出模型范畴(见文[Models for singularity categories,Adv Math.,2014,254:187-232])之间的Quillen等价.作为应用,给出了投射,内射模的正合复形的同伦范畴之间的一个具体的等价■.     

A 2-groupoid Characterisation of the Cubical Homotopy Pushout

We give a 2-track-theoretical characterisation of the homotopy pushout of a 3-corner by recognising the mapping 2-simplex as an initial object in a coherent homotopy category of Hausdorff spaces under a 3-corner with morphisms expressed in terms of the 1-morphisms and 2-morphisms of a homotopy 2-groupoid.     

Coherent Homotopical Algebras: “Special Gamma-Categories”

We introduce a theory of coherence for symmetric monoidal categories inthe spirit of Segal and show that it is equivalent, in an appropriate sense,to MacLanes original notion. More precisely, we prove thatspecial categories, the analogue ofspecial spaces, and coherently symmetric monoidalcategories are one and the same. This is analogous to the situation intopology where special spaces are precisely homotopicalcommutative monoids. In light of the obervation that the category of smallcategories Cat bears a functorial Quillen model structure with respect tothe class of categorical equivalences: in fact, is a homotopy theory in thesense of Heller, we may reinterpret the theorem as stating that coherentlysymmetric monoidal categories are precisely the homotopical commutativemonoids within this new homotopy theory.     

Invariants of Directed Spaces

Directed spaces are the objects of study within directed algebraic topology. They are characterised by spaces of directed paths associated to a source and a target, both elements of an underlying topological space. The algebraic topology of these path spaces and their connections are studied from a categorical perspective. In particular, we study the preorder category associated to a directed space and various “quotient” categories arising from algebraic topological functors. Furthermore, we propose and study a new notion of directed homotopy equivalence between directed spaces.      

The Left and Right Triangulated Structures of Stable Categories

Beligiannis and Marmaridis in 1994, constructed the one-sided triangulated structures on the stable categories of additive categories induced from some homologically finite subcategories. We extend their results to slightly more general settings. As an application of our results, we give some new examples of one-sided triangulated categories arising from abelian model categories. An interesting outcome is that we can describe the pretriangulated structures of the homotopy categories of abelian model categories via those of stable categories.     

Stable t-structures and Homotopy Category of Strongly Copure Projective Modules

In this paper, we study the homotopy category of unbounded complexes of strongly copure projective modules with bounded relative homologies K~(∞,bscp)(SCP).We show that the existence of a right recollement of K~(∞,bscp)(SCP) with respect to K~(-,bscp)(SCP), K_(scpac)(SCP) and K~(∞,bscp)(SCP) has the homotopy category of strongly copure projective acyclic complexes as a triangulated subcategory in some case.     

Homotopy Categories for Simply Connected Torsion Spaces

For each n > 1 and each multiplicative closed set of integers S, we study closed model category structures on the pointed category of topological spaces, where the classes of weak equivalences are classes of maps inducing isomorphism on homotopy groups with coefficients in determined torsion abelian groups, in degrees higher than or equal to n. We take coefficients either on all the cyclic groups with s ∈ S, or in the abelian group where is the group of fractions of the form with s ∈ S. In the first case, for n > 1 the localized category is equivalent to the ordinary homotopy category of (n − 1)-connected CW-complexes whose homotopy groups are S-torsion. In the second case, for n > 1 we obtain that the localized category is equivalent to the ordinary homotopy category of (n − 1)-connected CW-complexes whose homotopy groups are S-torsion and the nth homotopy group is divisible. These equivalences of categories are given by colocalizations , obtained by cofibrant approximations on the model structures. These colocalization maps have nice universal properties. For instance, the map is final (in the homotopy category) among all the maps of the form Y → X with Y an (n − 1)-connected CW-complex whose homotopy groups are S-torsion and its nth homotopy group is divisible. The spaces , are constructed using the cones of Moore spaces of the form M(T, k), where T is a coefficient group of the corresponding structure of models, and homotopy colimits indexed by a suitable ordinal. If S is generated by a set P of primes and S ^p is generated by a prime p ∈ P one has that for n > 1 the category is equivalent to the product category . If the multiplicative system S is generated by a finite set of primes, then localized category is equivalent to the homotopy category of n-connected Ext-S-complete CW-complexes and a similar result is obtained for .     

Λ-Cofibration Categories and the Homotopy Categories of Global Actions and Simplicial Complexes

We investigate the homotopy category of a -cofibration category and compare the homotopy categories of Global Actions, Simplicial Complexes and Topological Spaces.     

The coherent homotopy category over a fixed space is a category of fractions

In this note we prove that the coherent homotopy category over a fixed space B with morphisms represented by certain homotopy commutative squares (see [8]) is isomorphic to the category obtained by formally inverting those maps in the category Top_B of topological spaces over B which are ordinary homotopy equivalences.     

Weak complicial sets I. Basic homotopy theory

This paper develops the foundations of a simplicial theory of weak ω-categories, which builds upon the insights originally expounded by Ross Street in his 1987 paper on oriented simplices. The resulting theory of weak complicial sets provides a common generalisation of the theories of (strict) ω-categories, Kan complexes and Joyal's quasi-categories. We generalise a number of results due to the current author with regard to complicial sets and strict ω-categories to provide an armoury of well behaved technical devices, such as joins and Gray tensor products, which will be used to study the weak ω-category theory of these structures in a series of companion papers. In particular, we establish their basic homotopy theory by constructing a Quillen model structure on the category of stratified simplicial sets whose fibrant objects are the weak complicial sets. As a simple corollary of this work we provide an independent construction of Joyal's model structure on simplicial sets for which the fibrant objects are the quasi-categories.     

Local Projective Model Structures on Simplicial Presheaves

We introduce different model structures on the categories of simplicial presheaves and simplicial sheaves on some essentially small Gro-then-dieck site T and give some applications of these simplified model categories. In particular, we prove that the stable homotopy categories SH((Sm/k)Nis,A1) and SH((Sch/k)cdh,A1) are equivalent. This result was first proven by Voevodsky and our proof uses many of his techniques, but it does not use his theory of -closed classes.     

涉及迹同伦范畴的几对伴随函子

Ｋ．Ａ．Ｈａｒｄｉｅ与Ｋ．Ｈ．Ｋａｍｐｓ研究过固定空间Ｂ上的迹同伦范畴（［１］）．他们引进了两对伴随函子ＰＢ┤ＮＢ与ｍ┤ｍ，此处ｍ：ＡＢ是固定映射，ＰＢ：ＨＢＨＢ与ｍ：ＨＡＨＢ是函子．我们在［２］中引进了分裂的范畴纤维化Ｌ：ＨｂＨＢ，并且证明了Ｌ┤Ｊ，Ｊ┤Ｌ．本文首先将ＰＢ┤ＮＢ推广到ＰＢｂ┤ＮＢｂ＃，其中ｂ：ＢＢ是任一固定映射，并且我们还得到涉及迹同伦范畴Ｈｂ与Ｈｂ的两对伴随函子，此处Ｈｂ是Ｈｂ的对偶．特别，Ｎｂ┤Ｐｂ不同于ＰＢ┤ＮＢ．     

Three models for the homotopy theory of homotopy theories

Given any model category, or more generally any category with weak equivalences, its simplicial localization is a simplicial category which can rightfully be called the “homotopy theory” of the model category. There is a model category structure on the category of simplicial categories, so taking its simplicial localization yields a “homotopy theory of homotopy theories”. In this paper we show that there are two different categories of diagrams of simplicial sets, each equipped with an appropriate definition of weak equivalence, such that the resulting homotopy theories are each equivalent to the homotopy theory arising from the model category structure on simplicial categories. Thus, any of these three categories with the respective weak equivalences could be considered a model for the homotopy theory of homotopy theories. One of them in particular, Rezk’s complete Segal space model category structure on the category of simplicial spaces, is much more convenient from the perspective of making calculations and therefore obtaining information about a given homotopy theory.     

A closed model category for ()-connected spaces

For each integer , we give a distinct closed model category structure to the category of pointed spaces such that the corresponding localized category is equivalent to the standard homotopy category of -connected CW-complexes. The structure of closed model category given by Quillen to is based on maps which induce isomorphisms on all homotopy group functors and for any choice of base point. For each , the closed model category structure given here takes as weak equivalences those maps that for the given base point induce isomorphisms on for .