Homotopy pull-back squares up to localization

Brown representability for space-valued functors

In this paper we prove two theorems which resemble the classical cohomological and homological Brown representability theorems. The main difference is that our results classify small contravariant functors from spaces to spaces up to weak equivalence of functors. In more detail, we show that every small contravariant functor from spaces to spaces which takes coproducts to products up to homotopy and takes homotopy pushouts to homotopy pullbacks is naturally weekly equivalent to a representable functor. The second representability theorem states: every contravariant continuous functor from the …

Bar Complexes and Formality of Pull-backs

We prove a result concerning formality of the pull-back of a fibration. Our approach is to use bar complexes in the category of commutative differential graded algebras. As an application, we generalize an old result of Baum and Smith.

Finite Homotopy Limits of Nerves of Categories

Let $I$ be a small category with finite dimensional nerve, and $X\colon I\to Cat$ a diagram of small categories. We show that, under a "Reedy quasi-fibrancy condition", the homotopy limit of the geometric realization of $X$ is itself the geometric realization of a category. This categorical model for the homotopy limit is defined explicitly, as a category of natural transformations of diagrams. For the poset $\bullet\to\bullet\leftarrow\bullet$ we recover the model for homotopy pullbacks provided by Quillen's Theorem $B$ (specifically Barwick and Kan's version of Quillen's Theorem $B_2$). For …

Geometric versus homotopy theoretic equivariant bordism

By results of Loeffler and Comezana, the Pontrjagin-Thom map from geometric G-equivariant bordism to homotopy theoretic equivariant bordism is injective for compact abelian G. If G = S^1 x ... x S^1, we prove that the associated fixed point square is a pull back square, thus confirming a recent conjecture of D. Sinha. This is used in order to determine the image of the Pontrjagin-Thom map for toral G.

Fibration theorems for TQ-completion of structured ring spectra

The aim of this short paper is to establish a spectral algebra analog of the Bousfield-Kan "fibration lemma" under appropriate conditions. We work in the context of algebraic structures that can be described as algebras over an operad $\mathcal{O}$ in symmetric spectra. Our main result is that completion with respect to topological Quillen homology (or TQ-completion, for short) preserves homotopy fibration sequences provided that the base and total $\mathcal{O}$-algebras are connected. Our argument essentially boils down to proving that the natural map from the homotopy fiber to its TQ-complet…

The zeroth P^1-stable homotopy sheaf of a motivic space

We establish a kind of "degree zero Freudenthal Gm-suspension theorem" in motivic homotopy theory. From this we deduce results about the conservativity of the P^1-stabilization functor. In order to establish these results, we show how to compute certain pullbacks in the cohomology of a strictly homotopy invariant sheaf in terms of the Rost--Schmid complex. This establishes the main conjecture of [BY18], which easily implies the aforementioned results.

持续同调编码的拉回几何

Pull-back Geometry of Persistent Homology Encodings

本文提出利用PH编码在数据流形上诱导的拉回几何分析持续同调，通过诱导度量谱、特征向量和切向量范数揭示其信息重要性、扰动敏感性及下游任务性能。

Some characterizations of acyclic maps

We discuss two categorical characterizations of the class of acyclic maps between (path-connected) spaces. The first one is in terms of the higher categorical notion of an epimorphism. The second one employs the notion of a balanced map, that is, a map whose homotopy pullbacks define also homotopy pushouts. We also identify the modality in the homotopy theory of spaces that is defined by the class of acyclic maps, and discuss the content of the generalized Blakers-Massey theorem for this modality.

Equivariant Gysin maps and pulling back fixed points

We develop a new approach to the pulling back fixed point theorem of W. Browder and use it in order to prove various generalizations of this result.

Continuous functors as a model for the equivariant stable homotopy category

In this paper, we investigate the properties of the category of equivariant diagram spectra indexed on the category W_G of based G-spaces homeomorphic to finite G-CW-complexes for a compact Lie group G. Using the machinery of Mandell, May, Schwede, and Shipley, we show that there is a "stable model structure" on this category of diagram spectra which admits a monoidal Quillen equivalence to the category of orthogonal G-spectra. We construct a second "absolute stable model structure" which is Quillen equivalent to the "stable model structure". Our main result is a concrete identification of the…

Center conditions: pull back of differential equations

The space of polynomial differential equations of a fixed degree with a center singularity has many irreducible components. We prove that pull back differential equations form an irreducible component of such a space. The method used in this article is inspired by Ilyashenko and Movasati s method. The main concepts are the Picard Lefschetz theory of a polynomial in two variables with complex coefficients, the Dynkin diagram of the polynomial and the iterated integral.

Representations of the homotopy surface category of a simply connected space

We introduce the homotopy surface category of a space which generalizes the 1+1-dimensional cobordism category of circles and surfaces to the situation where one introduces a background space. We explain how for a simply connected background space, monoidal functors from this category to vector spaces can be interpreted in terms of Frobenius algebras with additional structure.

2-categorical opfibrations, Quillen's Theorem B, and $S^{-1}S$

In this paper we show that the strict and lax pullbacks of a 2-categorical opfibration along an arbitrary 2-functor are homotopy equivalent. We give two applications. First, we show that the strict fibers of an opfibration model the homotopy fibers. This is a version of Quillen's Theorem B amenable to applications. Second, we compute the $E^2$ page of a homology spectral sequence associated to an opfibration and apply this machinery to a 2-categorical construction of $S^{-1}S$. We show that if $S$ is a symmetric monoidal 2-groupoid with faithful translations then $S^{-1}S$ models the group com…