Equivariant homotopy equivalence of homotopy colimits of G-functors

Homotopic Descent over Monoidal Model Categories

This thesis provides a framework to study the homotopy theory of stratified spaces, in a way that is compatible with previous approaches. In particular our approach will be closely related to the work of Frank Quinn on homotopically stratified sets. We introduce a stratified analogue of the geometric realisation-singular simplicial set adjunction, allowing us to relate simplicial sets to stratified spaces. This allows us to cofibrantly transfer the Joyal model structure from simplicial sets to the category of fibrant stratified spaces. We have chosen to use the Joyal model structure on simplicial sets over the Quillen model structure. This choice allows a partial (ordered) composition of simplices, which under the stratified adjunction corresponds to concatenation of stratified paths. One of the biggest advantages of working in a simplicially enriched model structure is the ability to exploit the combinatorial nature of simplicial sets, which helps us to prove results about stratified spaces. By studying the cofibrations and fibrations that we transfer to the category of stratified spaces, we see that the cofibrant stratified spaces satisfy one condition that Quinn imposed for homotopically stratified sets, and that the fibrant stratified spaces satisfy the other condition. Consequently, the cofibrant-fibrant stratified spaces in our model structure are closely related to homotopically stratified sets. To use our framework to study homotopy theory, we need a notion of basepoint for a stratified space. We define the basing of a stratified space to be a factoring of the counit on the underlying poset through a choice of continuous map to the underlying topological space. The requirement of a stratified space to be based provides a restriction on the stratified spaces, and as such there are examples of cofibrant-fibrant stratified spaces which cannot be based. To justify this approach we are able construct an adjunction between stratified suspension and loop space functors. In addition, we are able to construct an indexed family of categories for a based fibrant stratified space, which we call the homotopy categories of a stratified space. Importantly, in the case of a trivially stratified connected space, the homotopy categories coincide with the homotopy groups of the underlying topological space. The homotopy categories of a based fibrant stratified space behave analogously to homotopy groups. For example, we are able to extract a long exact sequence of homotopy categories from a stratified fibration. Furthermore, we are able to provide partial results towards construction of a Postnikov Tower of a based fibrant stratified space. Further research is required to complete this construction, which would hopefully lead to a stratified analogue of Eilenberg-Mac Lane spaces.

We extend the notion of simplicial set with effective homology presented in [22] to diagrams of simplicial sets. Further, for a given finite diagram of simplicial sets $X \colon \mathcal{I}\rightarrow \mbox{sSet}$ such that each simplicial set $X(i)$ has effective homology, we present an algorithm computing the homotopy colimit $\mbox{hocolim}\,X$ as a simplicial set with effective homology. We also give an algorithm computing the cofibrant replacement $X^{\mbox{cof}}$ of $X$ as a diagram with effective homology. This is applied to computing of equivariant cohomology operations.

This book provides an introduction to modern homotopy theory through the lens of higher categories after Joyal and Lurie, giving access to methods used at the forefront of research in algebraic topology and algebraic geometry in the twenty-first century. The text starts from scratch - revisiting results from classical homotopy theory such as Serre's long exact sequence, Quillen's theorems A and B, Grothendieck's smooth/proper base change formulas, and the construction of the Kan–Quillen model structure on simplicial sets - and develops an alternative to a significant part of Lurie's definitive reference Higher Topos Theory, with new constructions and proofs, in particular, the Yoneda Lemma and Kan extensions. The strong emphasis on homotopical algebra provides clear insights into classical constructions such as calculus of fractions, homotopy limits and derived functors. For graduate students and researchers from neighbouring fields, this book is a user-friendly guide to advanced tools that the theory provides for application.

This thesis consists of two independent parts. In the first part we ask how traces in monoidal categories behave under homotopical operations. In order to investigate this question we define traces in closedmonoidal derivators and establish some of their properties. In the stable setting we derive an explicit formula for the trace of the homotopy colimit over finite categories in which every endomorphism is invertible. In the second part, we study motives of algebraic varieties over a subfield of the complex numbers, as defined by Nori on the one hand and by Voevodsky, Levine, and Hanamura on the other. Ayoub attached to the latter theory a motivic Galois group using the Betti realization, based on a weak Tannakian formalism. Our main theorem states that Nori’s and Ayoub’s motivic Galois groups are isomorphic. In the process of proving this result we construct well-behaved functors relating the two theories which are of independent interest.

The classifying space BG of a topological group G can be filtered by a sequence of subspaces B(q,G) using the descending central series of free groups. If G is finite, describing them as homotopy colimits is convenient when applying homotopy theoretic methods. In this thesis we introduce natural subspaces B(q,G)_p of B(q,G) defined for a fixed prime p. We show that B(q,G) is stably homotopy equivalent to a wedge sum of B(q,G)_p as p runs over the primes dividing the order of G. Colimits of abelian groups play an important role in understanding the homotopy type of these spaces. Extraspecial p-groups are key examples, for which these colimits turn out to be finite. We prove that for extraspecial p-groups of rank at least 4 the space B(2,G) does not have the homotopy type of a K(π,1) space. Furthermore, we give a group theoretic condition, applicable to symmetric groups and general linear groups, which implies the space B(2,G) not having the homotopy type of a K(π,1) space. For a finite group G, we compute the complex K-theory of B(2,G) modulo torsion.

The aim of this note is (i) to give (in §2) a precise statement and proof of the (to some extent well-known) fact that the most elementary homotopy theory of “simplicial sets on which a fixed simplicial group H acts” is equivalent to the homotopy theory of “simplicial sets over the classifying complex W ¯ H \bar WH ", and (ii) to use this (in §1) to prove a classification theorem for simplicial sets with an H-action, which provides classifying complexes for their equivariant maps which are self homotopy equivalences.

Thomason’s Homotopy Colimit Theorem has been extended to bicategories and this extension can be adapted, through the delooping principle, to a corresponding theorem for diagrams of monoidal categories. In this version, we show that the homotopy type of the diagram can also be represented by a genuine simplicial set nerve associated with it. This suggests the study of a homotopy colimit theorem, for diagrams B of braided monoidal categories, by means of a simplicial set nerve of the diagram. We prove that it is weak homotopy equivalent to the homotopy colimit of the diagram, of simplicial sets, obtained from composing B with the geometric nerve functor of braided monoidal categories.

In this paper we prove that various quasi-categories whose objects are $$\infty $$-categories in a very general sense are complete: admitting limits indexed by all simplicial sets. This result and others of a similar flavor follow from a general theorem in which we characterize the data that is required to define a limit cone in a quasi-category constructed as a homotopy coherent nerve. Since all quasi-categories arise this way up to equivalence, this analysis covers the general case. Namely, we show that quasi-categorical limit cones may be modeled at the point-set level by pseudo homotopy limit cones, whose shape is governed by the weight for pseudo limits over a homotopy coherent diagram but with the defining universal property up to equivalence, rather than isomorphism, of mapping spaces. Our applications follow from the fact that the $$(\infty ,1)$$-categorical core of an $$\infty $$-cosmos admits weighted homotopy limits for all flexible weights, which includes in particular the weight for pseudo cones.

We develop here a version of abstract homotopical algebra based onhomotopy kernels andcokernels, which are particular homotopy limits and colimits. These notions are introduced in anh-category, a sort of two-dimensional context more general than a 2-category, abstracting thenearly 2-categorical properties of topological spaces, continuous maps and homotopies. A setting which applies also, at different extents, to cubical or simplicial sets, chain complexes, chain algebras, ... and in which homotopical algebra can be established as a two-dimensional enrichment of homological algebra. Actually, a hierarchy of notions ofh-,h1-, ...h4-categories is introduced, through progressive enrichment of thevertical structure of homotopies, so that the strongest notion,h4-category, is a sort of relaxed 2-category. After investigating homotopy pullbacks and homotopical diagrammatical lemmas in these settings, we introduceright semihomotopical categories, ash-categories provided with terminal object and homotopy cokernels (mapping cones), andright homotopical categories, provided also with anh4-structure and verifying second-order regularity properties forh-cokernels. In these frames we study the Puppe sequence of a map, its comparison with the sequence of iterated homotopy cokernels and theh-cogroup structure of the suspension endofunctor. Left (semi-) homotopical categories, based on homotopy kernels, give the fibration sequence of a map and theh-group of loops. Finally, the self-dual notion of homotopical categories is considered, together with their stability properties.

Let [math] be a finite group. We define a suitable model-categorical framework for [math] –equivariant homotopy theory, which we call [math] –model categories. We show that the diagrams in a [math] –model category which are equipped with a certain equivariant structure admit a model structure. This model category of equivariant diagrams supports a well-behaved theory of equivariant homotopy limits and colimits. We then apply this theory to study equivariant excision of homotopy functors.

Homotopy theory and simplicial groupoids

We show that the classifying space functor $$B:\mathcal {M}on \rightarrow {\mathcal {T}\! op}^*$$ from the category of topological monoids to the category of based spaces is left adjoint to the Moore loop space functor $$\Omega ':{\mathcal {T}\! op}^*\rightarrow \mathcal {M}on$$ after we have localized $$\mathcal {M}on$$ with respect to all homomorphisms whose underlying maps are homotopy equivalences and $${\mathcal {T}\! op}^*$$ with respect to all based maps which are (not necessarily based) homotopy equivalences. It is well-known that this localization of $${\mathcal {T}\! op}^*$$ exists, and we show that the localization of $$\mathcal {M}on$$ is the category of monoids and homotopy classes of homotopy homomorphisms. To make this statement precise we have to modify the classical definition of a homotopy homomorphism, and we discuss the necessary changes. The adjunction is induced by an adjunction up to homotopy $$B:\mathcal {H}\mathcal {M}on^{w}\leftrightarrows {\mathcal {T}\! op}^w:\Omega '$$ between the category of well-pointed monoids and homotopy homomorphisms and the category of well-pointed spaces. This adjunction is shown to lift to diagrams. As a consequence, the well-known derived adjunction between the homotopy colimit and the constant diagram functor can also be seen to be induced by an adjunction up to homotopy before taking homotopy classes. As applications we among other things deduce a more algebraic version of the group completion theorem and show that the classifying space functor preserves homotopy colimits up to natural homotopy equivalences.

We study diagrams associated with a finite simplicial complex Kin various algebraic and topological categories. We relate their colimits to familiar structures in algebra, combinatorics, geometry and topology. These include: right-angled Artin and Coxeter groups (and their complex analogues, which we call circulation groups);Stanley-Reisner algebras and coalgebras; Davis and Januszkiewicz’s spaces DJ(K) associated with toric manifolds and their generalisations; and coordinate subspace arrangements. When K is a flag complex, we extend well-known results on Artin and Coxeter groups by confirming that the relevant circulation group is homotopy equivalent to the space of loops Ω DJ(K). We define homotopy colimits for diagrams of topological monoids and topological groups, and show they commute with the formation of classifying spaces in a suitably generalised sense. We deduce that the homotopy colimit of the appropriate diagram of topological groups is a model for Ω DJ(K) for an arbitrary complex K,and that the natural projection onto the original colimit is a homotopy equivalence when K is flag. In this case, the two models are compatible.

It is well known that localization of the category Top at homotopy equivalences yields the homotopy category H(Top) (see Theorem 4.35). The analogous procedure for inverse systems, indexed by Λ, is the localization of the category TopΛ at level homotopy equivalences. The resulting category will be denoted by Ho(TopΛ). One also considers the corresponding localization of pro -Top and denotes it by Ho(pro-Top). The main results of this section show that the localized categories, obtained in this manner, are isomorphic to the corresponding coherent homotopy categories CH(TopΛ) and CH(pro-Top), respectively. Therefore, the coherent homotopy categories can be viewed as concrete realizations of the rather abstract localized categories. The first subsection is devoted to an isomorphism theorem in coherent homotopy, which enables one to obtain functors between the categories in question. The second subsection defines cotelescopes (homotopy limits), a tool needed to prove that these functors are isomorphisms of categories.KeywordsNatural TransformationHomotopy ClassLevel MappingWeak EquivalenceInverse SystemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.