Marco Grandis
In homotopy theory, exact sequences and spectral sequences consist of groups and pointed sets, linked by actions. We prove that the theory of such exact and spectral sequences can be established in a categorical setting which is based on the existence of kernels and cokernels with respect to an assigned ideal of null morphisms, a generalisation of abelian categories and Puppe-exact categories.
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This paper contains some contributions to the study of the relationship between 2-categories and the homotopy types of their classifying spaces. Mainly, generalizations are given of both Quillen's Theorem B and Thomason's Homotopy Colimit Theorem to 2-functors.
This paper has been withdrawn by the author, due a critical mistake on page 3.
Working in homotopy type theory, we introduce the notion of $n$-exactness for a short sequence $F\to E\to B$ of pointed types, and show that any fiber sequence $F\hookrightarrow E \twoheadrightarrow B$ of arbitrary types induces a short sequence $\|F\|_{n-1} \to \|E\|_{n-1} \to \|B\|_{n-1}$ that is $n$-exact at $\|E\|_{n-1}$. We explain how the indexing makes sense when interpreted in terms of $n$-groups, and we compare our definition to the existing definitions of an exact sequence of $n$-groups for $n=1,2$. As the main application, we obtain the long $n$-exact sequence of homotopy $n$-groups…
Small Bénabou's bicategories and, in particular, Mac Lane's monoidal categories, have well-understood classifying spaces, which give geometric meaning to their cells. This paper contains some contributions to the study of the relationship between bicategories and the homotopy types of their classifying spaces. Mainly, generalizations are given of Quillen's Theorems A and B to lax functors between bicategories.
The classifying space of a crossed complex generalises the construction of Eilenberg-Mac Lane spaces. We show how the theory of fibrations of crossed complexes allows the analysis of homotopy classes of maps from a free crossed complex to such a classifying space. This gives results on the homotopy classification of maps from a CW-complex to the classifying space of a crossed module and also, more generally, of a crossed complex whose homotopy groups vanish in dimensions between 1 and n. The results are analogous to those for the obstruction to an abstract kernel in group extension theory.
The paper summarizes the construction of pairings on some standard spectral sequences in algebraic topology.
This mostly expository paper records some basic facts about towers of homotopy fiber sequences. We give a proof that a pairing of towers induces a pairing of associated spectral sequences, for towers of spaces and towers of spectra.
This paper defines an invariant associated to Whitehead's certain exact sequence of a simply connected CW-complex which is much more elementary - and less powerful - than the boundary invariant of Baues. Nevertheless, in good cases, it classifies the homotopy types of CW-complexes.
The focus of this paper is the comparison of two unstable homotopy spectral sequences-- the unstable mod p Adams spectral sequence that computes the unstable homotopy of a p-complete space, and the Goerss--Hopkins spectral sequence, which computes the unstable homotopy of the space of E-infinity maps between Hk-algebras, where k is the algebraic closure of the field with p elements and p is an odd prime. Using an adjunction between p-complete nilpotent spaces and a subset of Hk-algebras, this paper shows that the unit of this adjunction provides an isomorphism between these spectral sequences.
Given a CW-complex A we define an `A-shaped' homology theory which behaves nicely towards A-homotopy groups allowing the generalization of many classical results. We also develop a relative version of the Federer spectral sequence for computing A-homotopy groups. As an application we derive a generalization of the Hopf-Whitney theorem.
These are notes for a five lecture series intended to uncover large-scale phenomena in the homotopy groups of spheres using the Adams-Novikov Spectral Sequence. The lectures were given in Strasbourg, May 7-11, 2007.
We propose a simplified definition of Quillen's fibration sequences in a pointed model category that fully captures the theory, although it is completely independent of the concept of action. This advantage arises from the understanding that the homotopy theory of the model category's arrow category contains all homotopical information about its long fibration sequences.
Homotopy Groups and Puppe Sequence of Digraphs
本文引入了能直观描述网格结构的有向图同伦群作为GLMY同伦群的变体，并证明了任意基于有向图的映射均存在同伦群长正合序列，即有向图版本的Puppe序列。
Anick proved that every q-mild Hopf algebra up to homotopy is isomorphic to a primitively-generated chain Hopf algebra. We provide a new proof, that involves extensive use of the Bockstein spectral sequence.
Homotopy theories via the magnitude-path spectral sequence
引入基于量级-路径谱序列的广义度量空间同伦理论族，定义r-拟同构并证明每页满足Eilenberg-Steenrod公理的度量类比，建立Brown范畴结构以计算同伦余极限。
We develop a general theory of cosimplicial resolutions, homotopy spectral sequences, and completions for objects in model categories, extending work of Bousfield-Kan and Bendersky-Thompson for ordinary spaces. This is based on a generalized cosimplicial version of the Dwyer-Kan-Stover theory of resolution model categories, and we are able to construct our homotopy spectral sequences and completions using very flexible weak resolutions in the spirit of relative homological algebra. We deduce that our completion functors have triple structures and preserve certain fiber squares up to homotopy. …
We give a survey of a generalization of Quillen-Sullivan rational homotopy theory which gives spectral algebra models of unstable v_n-periodic homotopy types. In addition to describing and contextualizing our original approach, we sketch two other recent approaches which are of a more conceptual nature, due to Arone-Ching and Heuts. In the process, we also survey many relevant concepts which arise in the study of spectral algebra over operads, including topological André-Quillen cohomology, Koszul duality, and Goodwillie calculus.
Assume that all spaces and maps are localised at a fixed prime $p$. We study the possibility of generating a universal space $U(X)$ from a space $X$ which is universal in the category of homotopy associative, homotopy commutative H-spaces in the sense that any map f:X->Y to a homotopy associative, homotopy commutative H-space extends to a uniquely determined H-map F: U(X)->Y. Developing a method for recognising certain universal spaces, we show the existence of the universal space F_2(n) of a certain three-cell complex L. Using this specific example, we derive some consequences for the calcula…
We survey computations of stable motivic homotopy groups over various fields. The main tools are the motivic Adams spectral sequence, the motivic Adams-Novikov spectral sequence, and the effective slice spectral sequence. We state some projects for future study.
This paper is an English translation of chapter nine of the book "Adams spectral sequence and stable homotopy groups of spheres" by Jinkun Lin (in Chinese)(Sciences Press, Beijing 2007). In this paper, a sequence of new indecomposable families in the stable homotopy groups of spheres such as h_0h_n,h_nb_n,h_0h_nh_m,h_0(h_mb_{n-1}-b_{m-1}h_n) families was detected. The proof of all the detections was stated in this paper in based on several published papers by the author.
Secondary homotopy groups supplement the structure of classical homotopy groups. They yield a track functor on the track category of pointed spaces compatible with fiber sequences, suspensions and loop spaces. They also yield algebraic models of homotopy types with homotopy groups concentrated in two consecutive dimensions.
We provide a calculational method for rational stable equivariant homotopy theory for a torus G based on the homology of the Borel construction on fixed points. More precisely we define an abelian torsion model, A_t(G) of finite injective dimension, a homology theory \piAt_* taking values in A_t(G) based on the homology of the Borel construction, and a finite Adams spectral sequence Ext_{A_t(G)}^{*,*}(\piAt_*(X), \piAt_*(Y)) ==> [X,Y]^G_* for rational G-spectra X and Y.
We construct a Quillen model structure on the category of spectral categories, where the weak equivalences are the symmetric spectra analogue of the notion of equivalence of categories.
Correspondences and stable homotopy theory
本文提出由一般范畴中的对称环对象生成对应与谱范畴的通用方法，并据此分别恢复稳定同伦理论 SH 与稳定动机同伦理论 SH(k)。
We bound the volume of the homotopy groups of the 2-local Goodwillie approximations of a sphere in terms of the amount of $2$-torsion in the stable stems, providing a Goodwillie-theoretic refinement of a result of Burklund and Senger (arXiv:2203.00670). At the $2^k$-excisive approximation, this bound is obtained by `multiplying the stable answer by a polynomial of degree $k$'. The main tool is Behrens' Goodwillie-EHP Long Exact Sequence.
We analyze in homological terms the homotopy fixed point spectrum of a T-equivariant commutative S-algebra R. There is a homological homotopy fixed point spectral sequence with E^2_{s,t} = H^{-s}_{gp}(T; H_t(R; F_p)), converging conditionally to the continuous homology H^c_{s+t}(R^{hT}; F_p) of the homotopy fixed point spectrum. We show that there are Dyer-Lashof operations beta^epsilon Q^i acting on this algebra spectral sequence, and that its differentials are completely determined by those originating on the vertical axis. More surprisingly, we show that for each class x in the $^{2r}-term …
For a simply connected CW-complex $X$, let $\mathcal{E}(X)$ denote the group of homotopy classes of self-homotopy equivalence of $X$ and let $\mathcal{E}_{\sharp}(X)$ be its subgroup of homotopy classes which induce the identity on homotopy groups. As we know, the quotient group $\frac{\mathcal{E}(X)}{\mathcal{E}_{\sharp}(X)}$ can be identified with a subgroup of $Aut(π_{*}(X))$. The aim of this work is to determine this subgroup for rational spaces. We construct the Whitehead exact sequence associated with the minimal Sullivan model of $X$ which allows us to define the subgroup $\mathrm{Coh.A…
This belongs to a series of papers devoted to the study of the cohomology of classifying spaces of Lie groupoids. Our aim here is to introduce and study the notion of representation up to homotopy of Lie groupoids, the resulting derived category, and to show that the adjoint representation is well defined as a representation up to homotopy. As an application, we extend Bott's spectral sequence converging to the cohomology of classifying spaces from Lie groups to Lie groupoids.
A topological groupoid G is K-pointed, if it is equipped with a homomorphism from a topological group K to G. We describe the homotopy groups of such K-pointed topological groupoids and relate these groups to the ordinary homotopy groups in terms of a long exact sequence. As an application, we give an obstruction to presentability of proper regular Lie groupoids.
Let $R$ be a commutative ring with unit. We consider the homotopy theory of the category of spectral sequences of $R$-modules with the class of weak equivalences given by those morphisms inducing a quasi-isomorphism at a certain fixed page. We show that this admits a structure close to that of a category of fibrant objects in the sense of Brown and in particular the structure of a partial Brown category with fibrant objects. We use this to compare with related structures on the categories of multicomplexes and filtered complexes.
In proper homotopy theory, the original concept of point used in the classical homotopy theory of topological spaces is generalized in order to obtain homotopy groups that study the infinite of the spaces. This idea: "Using any arbitrary object as base point" and even "any morphism as zero morphism" can be developed in most of the algebraic homotopy theories. In particular, categories with a natural cone have a generalized homotopy theory obtained through the relative homotopy relation. Generalized homotopy groups and exact sequences of them are built so that respective classical pointed ones …
Centers of categories capture the natural operations on their objects. Homotopy coherent centers are introduced here as an extension of this notion to categories with an associated homotopy theory. These centers can also be interpreted as Hochschild cohomology type invariants in contexts that are not necessarily linear or stable, and we argue that they are more appropriate to higher categorical contexts than the centers of their homotopy or derived categories. Among many other things, we present an obstruction theory for realizing elements in the centers of homotopy categories, and a Bousfield…
The purpose of this paper is to give some solutions for the classification problem in fibration theory by using the homotopy sequences of fibrations (sequences of $n$-th homotopy groups $ π_{n}(S,s_{o}) $ of total spaces of fibrations). In particular, to show the role of homotopy sequence of $n$-th homotopy to get the required fiber map in Fadell-Dold theorem such that the restriction of this fiber map on some fiber spaces is a homotopy equivalence.
A C-motivic modular forms spectrum mmf has recently been constructed. This article presents detailed computational information on the Adams spectral sequence for mmf. This information is essential for computing with the C-motivic and classical Adams spectral sequences that compute the C-motivic and classical stable homotopy groups of spheres.
We study two different flavours of A^1-homotopy theory in the setting of spectral algebraic geometry, and compare them to classical A^1-homotopy theory. As an application we show that the spectral analogue of Weibel's homotopy invariant K-theory collapses to the classical theory. Along the way we give a new construction of nonconnective algebraic K-theory of stable infinity-categories via a generalization of the Bass-Thomason-Trobaugh construction.
Using techniques in motivic homotopy theory, especially the theorem of Gheorghe, the second and the third author on the isomorphism between motivic Adams spectral sequence for $Cτ$ and the algebraic Novikov spectral sequence for $BP_*$, we compute the classical and motivic stable homotopy groups of spheres from dimension 0 to 90, except for some carefully enumerated uncertainties.
We define inductively a sequence of purely algebraic invariants - namely, classes in the Quillen cohomology of the Pi-algebra π_* X - for distinguishing between different homotopy types of spaces. Another sequence of such cohomology classes allows one to decide whether a given abstract Pi-algebra can be realized as the homotopy Pi-algebra of a space in the first place. The paper is written for a relatively general "resolution model category", so it also applies, for example, to rational homotopy types.
In this work, we build a spectral sequence in motivic homotopy that is analogous to both the Serre spectral sequence in algebraic topology and the Leray spectral sequence in algebraic geometry. Here, we focus on laying the foundations necessary to build the spectral sequence and give a convenient description of its $E_2$-page. Our description of the $E_2$-page is in terms of homology of the local system of fibers, which is given using a theory similar to Rost's cycle modules. We close by providing some sample applications of the spectral sequence and some hints at future work.
As a step towards understanding the $\mathrm{tmf}$-based Adams spectral sequence, we compute the $K(1)$-local homotopy of $\mathrm{tmf} \wedge \mathrm{tmf}$, using a small presentation of $L_{K(1)}\mathrm{tmf}$ due to Hopkins. We also describe the $K(1)$-local $\mathrm{tmf}$-based Adams spectral sequence.
We study a version of the James model for the loop space of a suspension in unstable ${\mathbb A}^1$-homotopy theory. We use this model to establish an analog of G.W. Whitehead's classical refinement of the Freudenthal suspension theorem in ${\mathbb A}^1$-homotopy theory: our result refines F. Morel's ${\mathbb A}^1$-simplicial suspension theorem. We then describe some $E_1$-differentials in the EHP sequence in ${\mathbb A}^1$-homotopy theory. These results are analogous to classical results of G.W. Whitehead's. Using these tools, we deduce some new results about unstable ${\mathbb A}^1$-homo…
This paper contains a complete computation of the homotopy ring of the spectrum of topological modular forms constructed by Hopkins and Miller. The computation is done away from 6, and at the (interesting) primes 2 and 3 separately, and in each of the latter two cases, a sequence of algebraic Bockstein spectral sequences is used to compute the E_2 term of the elliptic Adams-Novikov spectral sequence from the elliptic curve Hopf algebroid. In a further step, all the differentials in the latter spectral sequence are determined. The result of this computation is originally due to Hopkins and Maho…
$RO(G)$-graded homotopy fixed point spectral sequence for height $2$ Morava $E$-theory
本文利用现代等变同伦论技术，完全计算了素数 2 处高度 2 Morava E-理论关于四个有限稳定子群的同伦不动点谱序列，并给出附加分次情形。
Obstruction sequences to homotopy equivalences
研究完备微分分次李代数中规范等价存在性的阻碍理论，用于刻画特定微分分次代数间同伦等价，应用此理论在代数拓扑和几何中建立新结果，尤其关注高连通簇极小模型。
Towards the $p=3$ Kervaire Invariant Problem: The $E_2$-page for the homotopy fixed points spectral sequence computing $π_*({E_6}^{hC_9})$
针对 $p=3$ 的 Kervaire 不变量问题，本文证明了相关检测定理，并利用 $E_6$ 上 $C_9$ 作用的猜想形式，计算了 $π_* E_6^{hC_9}$ 同伦不动点谱序列在非零同调度处的 $E_2$ 页。
Let f: X -> Y be a based map of simply connected spaces. The corresponding evaluation map w: map(X,Y;f) -> Y induces a homomorphism of homotopy groups whose image in pi_n(Y) is called the nth evaluation subgroup of f. The nth Gottlieb group of X occurs as the special case in which Y = X and f = 1_X. We identify the homomorphism induced on rational homotopy groups by this evaluation map, in terms of a map of complexes of derivations constructed using Sullivan minimal models. Our identification allows for the characterization of the rationalization of the nth evaluation subgroup of f. It also al…
Global homotopy theory via spectral Mackey functors
该研究证明Hausmann的对称谱整体稳定同伦理论模型等价于特定整体有效Burnside范畴上的谱Mackey函子∞-范畴，并给出Schwede超交换幺半群作为空间值整体Mackey函子的类似描述。
Homotopy groups of cubical sets
本文研究立方集的同调群，证明其等价性并提供经典定理的组合证明，应用于简单图的同调理论研究。
We prove that every stable, combinatorial model category has a natural enrichment by symmetric spectra (or more precisely, a natural equivalence class of enrichments). This in some sense generalizes the simplicial enrichments of model categories provided by the Dwyer-Kan hammock localization. As one particular application, we are able to associate to every object of a stable, combinatorial model category a "homotopy endomorphism ring spectrum". The homotopy types of these ring spectra are preserved by Quillen equivalences, and so are an invariant for stable model categories.
We establish a differential $d_2(D_1)=h_0^2h_3g_2$ in the $51$-stem of the Adams spectral sequence at the prime $2$, which gives the first correct calculation of the stable 51 and 52 stems. This differential is remarkable since we know of no way to prove it without recourse to the motivic Adams spectral sequence. It is the last undetermined differential in the range of the first author's detailed calculations of the $n$-stems for $n<60$ [6]. This note advertises the use of the motivic Adams spectral sequence to obtain information about classical stable homotopy groups.
Nested homotopy models of finite metric spaces and their spectral homology
该研究定义拟度量空间的r-同伦等价与r-极小模型，将有向图的幅度-路径谱序列分解为谱序列直和，并构造推广多种同调不变量的r-同伦不变量谱同调。
We answer a question of John Baez, on the relationship between the classical Euler characteristic and the Baez-Dolan homotopy cardinality, by constructing a unique additive common generalization after restriction to an odd prime p. This is achieved by ell-adically extrapolating to height n = -1 the sequence of Euler characteristics associated with the Morava K(n) cohomology theories for (any) ell | p-1. We compute this sequence explicitly in several cases and incorporate in the theory some folklore heuristic comparisons between the Euler characteristic and the homotopy cardinality involving su…
The James fibrations give rise to the geometric EHP sequences of homotopy groups of spheres. Using techniques from the Lambda algebra, \cite{BCKQRS66} shows that there are similar long exact sequences of Ext groups defining the $E_{2}-$page of the Bousfield-Kan spectral sequence (also known as the unstable Adams spectral sequence) computing homotopy groups of spheres. In this paper, we give another proof for this phenomenon.
For X a simply-connected finite H-space, there is a Bousfield-Kan spectral sequence which converges to the homotopy of its K-completion. When X=Spin(2n+1), we expect that these homotopy groups equal the v1-periodic homotopy groups in dimension greater than n^2. In this paper, we accomplish two things. (1) We prove that, for any X, the 1-line of this spectral sequence is determined in an explicit way from K-theory and Adams operations. (2) For X=Spin(2n+1), we make an explicit computation of this 1-line.
Exact sequences are a well known notion in homological algebra. We investigate here the more vague properties of 'homotopical exactness', appearing for instance in the fibre or cofibre sequence of a map. Such notions of exactness can be given for very general 'categories with homotopies' having homotopy kernels and cokernels, but become more interesting under suitable stability hypotheses, satisfied - in particular - by chain complexes. It is then possible to measure the default of homotopical exactness of a sequence by the homotopy type of a certain object, a sort of 'homotopical homology'.
We give sufficient conditions for the existence of a Quillen model structure on small categories enriched in a given monoidal model category. This yields a unified treatment for the known model structures on simplicial, topological, dg- and spectral categories. Our proof is mainly based on a fundamental property of cofibrant enriched categories on two objects, stated below as the Interval Cofibrancy Theorem.
Adams spectral sequences and Franke's algebraicity conjecture
本文通过为表现良好的同调理论构造导出 ∞-范畴来编码 Adams 谱序列，并以此证明 Franke 代数性猜想及 Adams 滤过的同伦相干幺半性。
Toen has interpreted the schematization problem as originally imagined by Grothendieck in "Pursuing Stacks" in such a way that solution(s) to this problem could be given. As he pointed out, there are many solutions available, and he gave two constructions solving this problem. What we do in the present work is reconsider what Grothendieck initially had in mind and develop a formalism that provides a concept of "schematization" and corresponding homotopy groups of "schematized" homotopy types. In our view this can be realized if we generalize topological spaces to symmetric spectra, and the map…
We present a detailed analysis of 2-complete stable homotopy groups, both in the classical context and in the motivic context over C. We use the motivic May spectral sequence to compute the cohomology of the motivic Steenrod algebra over C through the 70-stem. We then use the motivic Adams spectral sequence to obtain motivic stable homotopy groups through the 59-stem. In addition to finding all Adams differentials in this range, we also resolve all hidden extensions by 2, eta, and nu, except for a few carefully enumerated exceptions that remain unknown. The analogous classical stable homotopy …
Given a suitable functor T:C -> D between model categories, we define a long exact sequence relating the homotopy groups of any X in C with those of TX, and use this to describe an obstruction theory for lifting an object G in D to C. Examples include finding spaces with given homology or homotopy groups.
Homotopy similarity of maps
本文在带基点胞腔空间（X紧致）的带基点同伦集合上定义了一列愈发精细的等价关系。
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