Spaces of smooth embeddings and configuration categories

Asphericity structures, smooth functors, and fibrations

The aim of this paper is to generalize Grothendieck's theory of smooth functors in order to include within this framework the theory of fibered categories. We obtain in particular a new characterization of fibered categories.

关于∞-范畴Whitehead定理与拟范畴在预导子中的嵌入

On the $\infty$-categorical Whitehead theorem and the embedding of quasicategories in prederivators

本文证明小拟范畴可单纯及2-范畴地嵌入任意小范畴上的预导子，使其成为(∞,1)-范畴模型，并证明预导子可检测任意大型拟范畴等价的Whitehead定理。

Derived smooth stacks and prequantum categories

The Weil-Kostant integrality theorem states that given a smooth manifold endowed with an integral complex closed 2-form, then there exists a line bundle with connection on this manifold with curvature the given 2-form. It also characterises the moduli space of line bundles with connection that arise in this way. This theorem was extended to the case of p-forms by Gajer in [Ga]. In this paper we provide a generalization of this theorem where we replace the original manifold by a derived smooth Artin stack. Our derived Artin stacks are geometric stacks on the étale (\infty,1)-site of affine deri…

无穷操作算子基础下的嵌入计算

Infinity-operadic foundations for embedding calculus

本文基于无穷操作算子构建嵌入计算框架，扩展了Goodwillie-Weiss的嵌入计算到 bordism 类别，并提出新的嵌入计算版本，如基于BTop(d)的拓扑嵌入计算和基于BAut(E_d)的配置类别计算。

Equivariant Cohomology of Rationally Smooth Group Embeddings

We describe the equivariant cohomology ring of rationally smooth projective embeddings of reductive groups. These embeddings are the projectivizations of reductive monoids. Our main result describes their equivariant cohomology in terms of roots, idempotents, and underlying monoid data. Also, we characterize those embeddings whose equivariant cohomology ring is obtained via restriction to the associated toric variety. Such characterization is given in terms of the closed orbits.

Parametrized cobordism categories and the Dwyer-Weiss-Williams index theorem

We define parametrized cobordism categories and study their formal properties as bivariant theories. Bivariant transformations to a strongly excisive bivariant theory give rise to characteristic classes of smooth bundles with strong additivity properties. In the case of cobordisms between manifolds with boundary, we prove that such a bivariant transformation is uniquely determined by its value at the universal disk bundle. This description of bivariant transformations yields a short proof of the Dwyer-Weiss-Williams family index theorem for the parametrized A-theory Euler characteristic of a s…

A 2Cat-inspired model structure for double categories

We construct a model structure on the category $\mathrm{DblCat}$ of double categories and double functors. Unlike previous model structures for double categories, it recovers the homotopy theory of 2-categories through the horizontal embedding $\mathbb{H}\colon2\mathrm{Cat}\to\mathrm{DblCat}$, which is both left and right Quillen, and homotopically fully faithful. Furthermore, we show that Lack's model structure on $2\mathrm{Cat}$ is both left- and right-induced along $\mathbb{H}$ from our model structure on $\mathrm{DblCat}$. In addition, we obtain a $2\mathrm{Cat}$-enrichment of our model st…

在 simplicial 类型理论中的 Yoneda 入射

The Yoneda embedding in simplicial type theory

本文在 simplicial 类型理论中构建 Yoneda 入射，发展预层范畴理论，并证明 Quillen 定理 A。

Whitney categories and the Tangle Hypothesis

We propose a new notion of `n-category with duals', which we call a Whitney n-category. There are two motivations. The first is that Baez and Dolan's Tangle Hypothesis is (almost) tautological when interpreted as a statement about Whitney categories. The second is that we can functorially construct `fundamental Whitney n-categories' from each smooth stratified space X. These are obtained by considering the homotopy theory of smooth maps into X which are transversal to all strata. This makes concrete another idea of Baez and Dolan's which is that a suitable version of homotopy theory for strati…

Embedding, compression and fiberwise homotopy theory

Given Poincare spaces M and X, we study the possibility of compressing embeddings of M x I in X x I down to embeddings of M in X. This results in a new approach to embedding in the metastable range both in the smooth and Poincare duality categories.

Classification of smooth embeddings of 4-manifolds in 7-space, I

We work in the smooth category. Let N be a closed connected n-manifold and assume that m>n+2. Denote by E^m(N) the set of embeddings N -> R^m up to isotopy. The group E^m(S^n) acts on E^m(N) by embedded connected sum of a manifold and a sphere. If E^m(S^n) is non-zero (which often happens for 2m<3n+4) then no results on this action and no complete description of E^m(N) were known. Our main results are examples of the triviality and the effectiveness of this action, and a complete isotopy classification of embeddings into R^7 for certain 4-manifolds N. The proofs are based on the Kreck modifica…

Classification of embeddings below the metastable dimension

We develop a new approach to the classical problem on isotopy classification of embeddings of manifolds into Euclidean spaces. This approach involves studying of a new embedding invariant, of almost-embeddings and of smoothing, as well as explicit constructions of embeddings. Using this approach we obtain complete concrete classification results below the metastable dimension range, i.e. where the configuration spaces invariant of Haefliger-Wu is incomplete. Note that all known complete concrete classification results, except for the Haefliger classification of links and smooth knots, can be o…

Microlocal sheaf categories and the $J$-homomorphism

Let $X$ be a smooth manifold and $\mathbf{k}$ be a commutative (or at least $\mathbb{E}_2$) ring spectrum. Given a smooth exact Lagrangian $L\hookrightarrow T^*X$, the microlocal sheaf theory (following Kashiwara--Schapira) naturally assigns a locally constant sheaf of categories on $L$ with fiber equivalent to the category of $\mathbf{k}$-spectra $\mathrm{Mod}(\mathbf{k})$. We show that the classifying map for the local system of categories factors through the stable Gauss map $L\rightarrow U/O$ and the delooping of the $J$-homomorphism $U/O\rightarrow B\mathrm{Pic}(\mathbf{S})$. As an applic…