路径空间的范畴模型

Categorical models for equivariant classifying spaces

Starting categorically, we give simple and precise models of equivariant classifying spaces. We need these models for work in progress in equivariant infinite loop space theory and equivariant algebraic K-theory, but the models are of independent interest in equivariant bundle theory and especially equivariant covering space theory.

Menger (Nobeling) manifolds versus Hilbert cube (space) manifolds - a categorical comparison

We show that the n-homotopy category of connected (n+1)-dimensional Menger manifolds is isomorphic to the homotopy category of connected Hilbert cube manifolds whose k-dimensional homotopy groups are trivial for each k > n.

欧几里得空间构型空间的高阶范畴组合学

Higher-categorical combinatorics of configuration spaces of Euclidean space

本文在∞-范畴同伦理论背景下，通过分层空间考察流形有限子集的构型，并利用范畴Θ_n确定了n维欧几里得空间此类构型空间的同伦类型。

Rigidification of higher categorical structures

Given a limit sketch in which the cones have a finite connected base, we show that a model structure of "up to homotopy" models for this limit sketch in a suitable model category can be transferred to a Quillen equivalent model structure on the category of strict models. As a corollary of our general result, we obtain a rigidification theorem which asserts in particular that any $Θ_n$-space in the sense of Rezk is levelwise equivalent to one that satisfies the Segal conditions on the nose. There are similar results for dendroidal spaces and $n$-fold Segal spaces.

Categorical Properties Of Smooth Unfoldings On Stratified Spaces

In a previous work we proved the uniqueness and functoriality of primary unfoldings on simple Thom-Mather spaces, which is a functor to the category of smooth manifolds. In this article we extend these results for any stratified Thom-Mather pseudomanifold with arbitary finite length, through a new kind of intermediate desingularizations, the unbendings, which coincide with primary unfoldings in the simple case.

$(\infty,1)$-范畴的双范畴模型

Double categorical model of $(\infty,1)$-categories

基于已有工作，本文在双范畴上构造了四种模型结构，分别建模单纯空间、Segal空间、$(\infty,1)$-范畴和$\infty$-群胚，并利用Grothendieck构造给出了计算同伦余极限的显式公式。

Two-Categorical Bundles and Their Classifying Spaces

For a 2-category 2C we associate a notion of a principal 2C-bundle. In case of the 2-category of 2-vector spaces in the sense of M.M. Kapranov and V.A. Voevodsky this gives the the 2-vector bundles of N.A. Baas, B.I. Dundas and J. Rognes. Our main result says that the geometric nerve of a good 2-category is a classifying space for the associated principal 2-bundles. In the process of proving this we develop a lot of powerful machinery which may be useful in further studies of 2-categorical topology. As a corollary we get a new proof of the classification of principal bundles. A calculation bas…

On the Existence of Categorical Universal Coverings

In this paper, we study necessary and sufficient conditions for the existence of categorical universal coverings using open covers of a given space $X$. As some applications, first we present a generalized version of the Shelah Theorem (Mycielski's conjecture: If $X$ is a Peano continuum, then $π_1(X,x)$ is uncountable or $X$ has a simply connected universal covering) which states that a first countable Peano space has a categorical universal covering or has an uncountable fundamental group. Second, we prove that the one point union $X_1\vee X_2=\frac{{X_1}\cup {X_2}}{{x_1}\sim {x_2}}$ has a c…

具有 $L_{\infty}[1]$-代数的 Kuranishi 空间的范畴结构

Categorical structures of Kuranishi spaces with $L_{\infty}[1]$-algebras

本文通过将每个图表与 Kuranishi 截面零点邻域上的 $L_{\infty}[1]$-代数关联，引入 $L_{\infty}$-Kuranishi 空间，并证明它们构成一个范畴，光滑流形范畴自然嵌入其中。

高阶范畴的同调

Homology of higher categories

本文基于Eilenberg-Steenrod公理定义高阶范畴的同调，通过范畴论类比公理建立Dold-Kan对应，利用Street神经网络作为奇异复形，推导出具有乘法结构的高阶同调计算。

The Mukai pairing, I: a categorical approach

We study the Hochschild homology of smooth spaces, emphasizing the importance of a pairing which generalizes Mukai's pairing on the cohomology of K3 surfaces. We show that integral transforms between derived categories of spaces induce, functorially, linear maps on homology. Adjoint functors induce adjoint linear maps with respect to the Mukai pairing. We define a Chern character with values in Hochschild homology, and we discuss analogues of the Hirzebruch-Riemann-Roch theorem and the Cardy Condition from physics. This is done in the context of a 2-category which has spaces as its objects and…

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.

Convergence between Categorical Representations of Reeb Space and Mapper

The Reeb space, which generalizes the notion of a Reeb graph, is one of the few tools in topological data analysis and visualization suitable for the study of multivariate scientific datasets. First introduced by Edelsbrunner et al., it compresses the components of the level sets of a multivariate mapping and obtains a summary representation of their relationships. A related construction called mapper, and a special case of the mapper construction called the Joint Contour Net have been shown to be effective in visual analytics. Mapper and JCN are intuitively regarded as discrete approximations…