Homotopy Functors Research Articles

Some Cardinal and Geometric Properties of the Space of Permutation Degree

This paper is devoted to the investigation of cardinal invariants such as the hereditary density, hereditary weak density, and hereditary Lindelöf number. The relation between the spread and the extent of the space SP2(R,τ(A)) of permutation degree of the Hattori space is discussed. In particular, it is shown that the space SP2(R,τS) contains a closed discrete subset of cardinality c. Moreover, it is shown that the functor SPGn preserves the homotopy and the retraction of topological spaces. In addition, we prove that if the spaces X and Y are homotopically equivalent, then the spaces SPGnX and SPGnY are also homotopically equivalent. As a result, it has been proved that the functor SPGn is a covariant homotopy functor.

THH and TC are (very) far from being homotopy functors

We compute the A 1 -localization of several invariants of schemes namely, topological Hochschild homology ( THH ), topological cyclic homology ( TC ) and topological periodic cyclic homology ( TP ). This procedure is quite brutal and kills the completed versions of most of these invariants. The main ingredient for the vanishing statements is the vanishing of A 1 -localization of de Rham cohomology (and, eventually, crystalline cohomology) in positive characteristics.

On the fundamental groups of commutative algebraic groups

Consider the abelian category ${\mathcal C}$ of commutative group schemes of finite type over a field $k$, its full subcategory ${\mathcal F}$ of finite group schemes, and the associated pro category ${\rm Pro}({\mathcal C})$ (resp. ${\rm Pro}({\mathcal F})$) of pro-algebraic (resp. profinite) group schemes. When $k$ is perfect, we show that the profinite fundamental group $\varpi_1 : {\rm Pro}({\mathcal C}) \to {\rm Pro}({\mathcal F})$ is left exact and commutes with base change under algebraic field extensions; as a consequence, the higher profinite homotopy functors $\varpi_i$ vanish for $i \geq 2$. Along the way, we describe the indecomposable projective objects of ${\rm Pro}({\mathcal C})$ over an arbitrary field $k$.

EMBEDDINGS, NORMAL INVARIANTS AND FUNCTOR CALCULUS

This paper investigates the space of codimension zero embeddings of a Poincaré duality space in a disk. One of our main results exhibits a tower that interpolates from the space of Poincaré immersions to a certain space of “unlinked” Poincaré embeddings. The layers of this tower are described in terms of the coefficient spectra of the identity appearing in Goodwillie’s homotopy functor calculus. We also answer a question posed to us by Sylvain Cappell. The appendix proposes a conjectural relationship between our tower and the manifold calculus tower for the smooth embedding space.

Cross-effects and the classification of Taylor towers

Let F be a homotopy functor with values in the category of spectra. We show that partially stabilized cross-effects of F have an action of a certain operad. For functors from based spaces to spectra, it is the Koszul dual of the little discs operad. For functors from spectra to spectra it is a desuspension of the commutative operad. It follows that the Goodwillie derivatives of F are a right module over a certain `pro-operad'. For functors from spaces to spectra, the pro-operad is a resolution of the topological Lie operad. For functors from spectra to spectra, it is a resolution of the trivial operad. We show that the Taylor tower of the functor F can be reconstructed from this structure on the derivatives.

Comparing the orthogonal and homotopy functor calculi

Goodwillie's homotopy functor calculus constructs a Taylor tower of approximations to F, often a functor from spaces to spaces. Weiss's orthogonal calculus provides a Taylor tower for functors from vector spaces to spaces. In particular, there is a Weiss tower associated to the functor V↦F(SV), where SV is the one-point compactification of V.In this paper, we give a comparison of these two towers and show that when F is analytic the towers agree up to weak equivalence. We include two main applications, one of which gives as a corollary the convergence of the Weiss Taylor tower of BO. We also lift the homotopy level tower comparison to a commutative diagram of Quillen functors, relating model categories for Goodwillie calculus and model categories for the orthogonal calculus.

Homotopy theory of G–diagrams and equivariant excision

Let $G$ be a finite group acting on a small category $I$. We study functors $X \colon I \to \mathscr{C}$ equipped with families of compatible natural transformations that give a kind of generalized $G$-action on $X$. Such objects are called $G$-diagrams. When $\mathscr{C}$ is a sufficiently nice model category we define a model structure on the category of $G$-diagrams in $\mathscr{C}$. There are natural $G$-actions on Bousfield-Kan style homotopy limits and colimits of $G$-diagrams. We prove that weak equivalences between point-wise (co)fibrant $G$-diagrams induce weak $G$-equivalences on homotopy (co)limits. A case of particular interest is when the indexing category is a cube. We use homotopy limits and colimits over such diagrams to produce loop and suspension spaces with respect to permutation representations of $G$. We go on to develop a theory of enriched equivariant homotopy functors and give an equivariant "linearity" condition in terms of cubical $G$-diagrams. In the case of $G$-topological spaces we prove that this condition is equivalent to Blumberg's notion of $G$-linearity. In particular we show that the Wirthm\"{u}ller isomorphism theorem is a direct consequence of the equivariant linearity of the identity functor on $G$-spectra.

A classification of small homotopy functors from spectra to spectra

We show that every small homotopy functor from spectra to spectra is weakly equivalent to a filtered colimit of representable functors represented in cofibrant spectra. Moreover, we present this classification as a Quillen equivalence of the category of small functors from spectra to spectra equipped with the homotopy model structure and the opposite of the pro-category of spectra with the strict model structure.

Comparison of categorical characteristic classes of a transitive Lie algebroid with the Chern-Weil homomorphism

This paper is devoted to analyzing two approaches to characteristic classes of transitive Lie algebroids. The first approach is due to Kubarski [5] and is a version of the Chern-Weil homomorphism. The second approach is related to the so-called categorical characteristic classes (see, e.g., [6]). The construction of transitive Lie algebroids due to Mackenzie [1] can be considered as a homotopy functor T LAg from the category of smooth manifolds to the transitive Lie algebroids. The functor T LAg assigns to every smooth manifold M the set T LAg(M) of all transitive algebroids with a chosen structural finite-dimensional Lie algebra g. Hence, one can construct [2, 3] a classifying space Bg such that the family of all transitive Lie algebroids with the chosen Lie algebra g over the manifold M is in one-to-one correspondence with the family of homotopy classes of continuous maps [M, Bg]: T LAg(M) ≈ [M, Bg]. This enables us to describe characteristic classes of transitive Lie algebroids from the point of view of a natural transformation of functors similar to the classical abstract characteristic classes for vector bundles and to compare them with those derived from the Chern-Weil type homomorphism by Kubarski [5]. As a matter of fact, we show that the Chern-Weil type homomorphism by Kubarski does not cover all characteristic classes from the categorical point of view.

EQUIVARIANT CALCULUS OF FUNCTORS AND-ANALYTICITY OF REAL ALGEBRAIC -THEORY

We define a theory of Goodwillie calculus for enriched functors from finite pointed simplicial $G$-sets to symmetric $G$-spectra, where $G$ is a finite group. We extend a notion of $G$-linearity suggested by Blumberg to define stably excisive and ${\it\rho}$-analytic homotopy functors, as well as a $G$-differential, in this equivariant context. A main result of the paper is that analytic functors with trivial derivatives send highly connected $G$-maps to $G$-equivalences. It is analogous to the classical result of Goodwillie that ‘functors with zero derivative are locally constant’. As the main example, we show that Hesselholt and Madsen’s Real algebraic $K$-theory of a split square zero extension of Wall antistructures defines an analytic functor in the $\mathbb{Z}/2$-equivariant setting. We further show that the equivariant derivative of this Real $K$-theory functor is $\mathbb{Z}/2$-equivalent to Real MacLane homology.

Cosimplicial models for the limit of the Goodwillie tower

We call attention to the intermediate constructions $\T_n F$ in Goodwillie's Calculus of homotopy functors, giving a new model which naturally gives rise to a family of towers filtering the Taylor Tower of a functor. We also establish a surprising equivalence between the homotopy inverse limits of these towers and the homotopy inverse limits of certain cosimplicial resolutions. This equivalence gives a greatly simplified construction for the homotopy inverse limit of the Taylor tower of a functor $F$ under general assumptions.

Homotopical Presentations and Calculations of Algebraic $K_0$-Groups for Rings of Continuous Functions

Let K_0(C_{\Bbb F}(X)) = K_0\circ C_{\Bbb F}(X) be the K_0 -group of the ring C_{\Bbb F}(X) of {\Bbb F} -valued continuous functions on a topological space X , where {\Bbb F} is the field of real or complex numbers or the quaternion algebra. It is known that the functor K_0\circ C_{\Bbb F} is representable on the category of compact Hausdorff spaces. It is a homotopy functor which is not representable on the category of topological spaces. Making use of the compactly-bounded homotopy set, which is a variant of the homotopy set, the functor K_0\circ C_{\Bbb F} has a homotopical presentation by the product of the ring of integers {\Bbb Z} and the infinite Grassmannian G_{\infty}(\Bbb F ) . This presentation makes it possible to calculate the groups K_0(C_{\Bbb F}(X)) explicitly for some infinite dimensional complexes X by use of the results of H. Miller on Sullivan conjecture.

The induced homology and homotopy functors on the coarse shape category

The induced homology and homotopy functors on the coarse shape category

Bordism groups of solutions to differential relations

In terms of category theory, the Gromov homotopy principle for a set valued functor $F$ asserts that the functor $F$ can be induced from a homotopy functor. Similarly, we say that the bordism principle for an abelian group valued functor $F$ holds if the functor $F$ can be induced from a (co)homology functor. We examine the bordism principle in the case of functors given by (co)bordism groups of maps with prescribed singularities. Our main result implies that if a family $R$ of prescribed singularity types satisfies certain mild conditions, then there exists an infinite loop space $B(R)$ such that for each smooth manifold $N$ the cobordism group of maps into $N$ with only $R$-singularities is isomorphic to the group of homotopy classes of maps $[N, B(R)]$.

Locally definable homotopy

In [E. Baro, M. Otero, On o-minimal homotopy, Quart. J. Math. (2009) 15pp, in press ( doi:10.1093/qmath/hap011)] o-minimal homotopy was developed for the definable category, proving o-minimal versions of the Hurewicz theorems and the Whitehead theorem. Here, we extend these results to the category of locally definable spaces, for which we introduce homology and homotopy functors. We also study the concept of connectedness in ⋁ -definable groups — which are examples of locally definable spaces. We show that the various concepts of connectedness associated to these groups, which have appeared in the literature, are non-equivalent.

Loop structures in Taylor towers

We study spaces of natural transformations between homogeneous functors in Goodwillie’s calculus of homotopy functors and in Weiss’s orthogonal calculus. We give a description of such spaces of natural transformations in terms of the homotopy fixed point construction. Our main application uses this description in combination with the Segal Conjecture to obtain a delooping theorem for connecting maps in the Goodwillie tower of the identity and in the Weiss tower of BU.V/. The interest in such deloopings stems from conjectures made by the first and the third author [4] that these towers provide a source of contracting homotopies for certain projective chain complexes of spectra. 55P65; 55P47, 18G55 1 Introduction and notation In this paper, we study homogeneous functors in the sense of Weiss’s orthogonal calculus [13]. More precisely, we calculate the space of natural transformations between such functors, and we give a few examples and applications. Our main application is a delooping theorem for connecting maps in the Goodwillie tower of the identity functor for pointed spaces and in the Weiss tower of the functor V7!BU.V/. The motivation for this delooping theorem will be discussed later in this introduction. To state our results, we summarize the usual notation for this context. Let F be a functor from pointed spaces or from finite-dimensional real or complex vector spaces to pointed spaces or spectra. Goodwillie and Weiss calculus assign to such a functor F a “Taylor” tower of functors ! Pn F! Pn 1 F! ; together with a natural map from F to the homotopy inverse limit of the tower that is often a weak homotopy equivalence. The homotopy fiber of the map Pn F! Pn 1 F is customarily denoted Dn F and is referred to as “the nth homogeneous layer of F ,” while Pn F is referred to as “the nth Taylor polynomial of F .” Depending on whether

Filtered spectra arising from permutative categories

Given a special Γ-category 𝒞 satisfying some mild hypotheses, we construct a sequence of spectra interpolating between the spectrum associated to 𝒞 and the Eilenberg-Mac Lane spectrum Hℤ. Examples of categories to which our construction applies are: the category of finite sets, the category of finite-dimensional vector spaces, and the category of finitely-generated free modules over a reasonable ring. In the case of finite sets, our construction recovers the filtration of Hℤ by symmetric powers of the sphere spectrum. In the case of finite-dimensional complex vector spaces, we obtain an apparently new sequence of spectra, {Am}, that interpolate between bu and . We think of Am as a “bu-analogue” of Spm(S) and describe far-reaching formal similarities between the two sequences of spectra. For instance, in both cases the mth subquotient is contractible unless m is a power of a prime, and in vk-periodic homotopy the filtration has only k + 2 non-trivial terms. There is an intriguing relationship between the bu-analogues of symmetric powers and Weiss's orthogonal calculus, parallel to the not yet completely understood relationship between the symmetric powers of spheres and the Goodwillie calculus of homotopy functors. We conjecture that the sequence {Am}, when rewritten in a suitable chain complex form, gives rise to a minimal projective resolution of the connected cover of bu. This conjecture is the bu-analogue of a theorem of Kuhn and Priddy about the symmetric power filtration. The calculus of functors provides substantial supporting evidence for the conjecture.

Exactness of homotopy functors of spaces

We will provide an analysis of the generalized Atiyah–Hirzebruch spectral sequence (GAHSS), which was introduced by Hakim-Hashemi and Kahn. To do so, we introduce a new class of functors, called n - exact functors, which are analogous to Goodwillie’s n -excisive functors. In the study of these functors, we introduce a new spectral sequence, the homological Barratt–Goerss spectral sequence (HBGSS), which has properties similar to those of the classical Barratt–Goerss Spectral Sequence on homotopy. We close by giving an identification of the E 2 term of the GAHSS in the case of 2-exact functors on Moore spaces.

Homotopic Theory of Bundles whose Fibers Are Matrix Algebras

In this paper, we consider homotopic properties of bundles of algebras, matrix Grassmanians, and homotopy functors connected with bundles of algebras.

Tate cohomology and periodic localization of polynomial functors

In this paper, we show that Goodwillie calculus, as applied to functors from stable homotopy to itself, interacts in striking ways with chromatic aspects of the stable category. Localized at a fixed prime p, let T(n) be the telescope of a v_n self map of a finite S--module of type n. The Periodicity Theorem of Hopkins and Smith implies that the Bousfield localization functor associated to T(n) is independent of choices. Goodwillie's general theory says that to any homotopy functor F from S--modules to S--modules, there is an associated tower under F, {P_dF}, such that F --> P_dF is the universal arrow to a d--excisive functor. Our first theorem says that P_dF --> P_{d-1}F always admits a homotopy section after localization with respect to T(n) (and so also after localization with respect to Morava K--theory K(n)). Thus, after periodic localization, polynomial functors split as the product of their homogeneous factors. This theorem follows from our second theorem which is equivalent to the following: for any finite group G, the Tate spectrum t_G(T(n)) is weakly contractible. This strengthens and extends previous theorems of Greenlees--Sadofsky, Hovey--Sadofsky, and Mahowald--Shick. The Periodicity Theorem is used in an essential way in our proof. The connection between the two theorems is via a reformulation of a result of McCarthy on dual calculus.