We study the beginning of the Taylor tower, supplied by manifold calculus of functors, for the space of r-immersions, which are immersions without r-fold self-intersections. We describe the first r layers of the tower and discuss the connectivities of the associated maps. We also prove several results about r-immersions that are of independent interest.
The goal of this short paper is to study the convergence of the Taylor tower of the identity functor in the context of operadic algebras in spectra. Specifically, we show that if $A$ is a $(-1)$-connected $\mathcal{O}$-algebra with $0$-connected $\mathsf{TQ}$-homology spectrum $\mathsf{TQ}(A)$, then there is a natural weak equivalence $P_\infty$(id)$A\simeq A_\mathsf{TQ}^\wedge$ between the limit of the Taylor tower of the identity functor evaluated on $A$ and the $\mathsf{TQ}$-completion of $A$. Since, in this context, the identity functor is only known to be $0$-analytic, this result extends…
The orthogonal and unitary calculi give a method to study functors from the category of real or complex inner product spaces to the category of based topological spaces. We construct functors between the calculi from the complexification-realification adjunction between real and complex inner product spaces. These allow for movement between the versions of calculi, and comparisons between the Taylor towers produced by both calculi. We show that when the inputted orthogonal functor is weakly polynomial, the Taylor tower of the functor restricted through realification and the restricted Taylor t…
We construct a calculus of functors in the spirit of orthogonal calculus, which is designed to study "functors with reality" such as the Real classifying space functor, $BU_\mathbb{R}(-)$. The calculus produces a Taylor tower, the $n$-th layer of which is classified by a spectrum with an action of $C_2 \ltimes U(n)$. We further give model categorical considerations, producing a zig-zag of Quillen equivalences between spectra with an action of $C_2 \ltimes U(n)$ and a model structure on the category of input functors which captures the homotopy theory of the $n$-th layer of the Taylor tower.
We show Goodwillie's calculus of functors and $n$-geometric $D^{-}$-stacks share similar features by starting to focus on the convergence of Taylor towers for homotopy functors and the fact that $\mathbb{R} F(A) \cong \text{holim} \mathbb{R} F(A_{\leq n})$ for geometric stacks, where $\{ A_{\leq n} \}$ provides a Postnikov tower of some given $A \in s \text{k-Alg}$. From there we show parallel results, such as similar homotopy fibers of connecting maps in towers, as well as polynomial approximations, pointwise approximations and reconstruction theorems for towers.
The block structure spaces of real projective spaces and orthogonal calculus of functors
给定一个紧致流形X，X x Δ^k上相对于边界的简单流形结构的集合可以看作空间同伦群。这个空间称为X的块结构空间。研究了实射影空间的块结构空间。推广Wall的并构式，证明了存在一个从有限维内积实向量空间范畴到点空间范畴的函子，它把向量空间V送到V的射影空间的块结构空间。我们证明了它是正交微积分意义下的次数≤ 1的多项式。这一结果表明，利用正交微积分的泰勒塔，可以很好地描述无限维实射影空间的块结构空间。该空间被定义为有限维实向量空间的投射空间的块结构空间的余极限，并与实射影空间的某些自同构空间密切相关。
Directional derivatives and higher order chain rules for abelian functor calculus
本文研究了由第二作者和McCarthy建立的阿贝尔范畴的函子演算--阿贝尔函子演算。我们仔细地构造了一个阿贝尔范畴和适当同伦定义的函子范畴，并证明了这个范畴，配备了方向导数，在Blute，Cockett和Seely的意义下是一个Carnival微分范畴.这提供了一个抽象的框架，使经典和函子演算之间的某些类比明确。受Huang，Marcantognini和Young关于函数的高阶方向导数的链定则的启发，我们定义了阿贝尔范畴的函子的高阶方向导数.我们证明了我们的高阶方向导数与第二作者和McCarthy的迭代偏方向导数通过Faà di Bruno式公式相关。我们得到了一个高阶链式规则，我们的方向导数使用的Carnival微分范畴结构的一个功能，并与此提供了一个配方的$n$第th层的泰勒塔的函子组合$F\circ G$在衍生物和方向衍生物的$F$和$G$，让人想起类似的配方的空间或频谱的函子由Arone和Ching。在整个过程中，我们提供明确的链同伦等价，收紧先前建立的拟同构的阿贝尔函子演算的属性。
