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Higher-categorical combinatorics of configuration spaces of Euclidean space

Anna Cepek

Abstract

We examine configurations of finite subsets of manifolds within the homotopy-theoretic context of $\infty$-categories by way of stratified spaces. Through these higher categorical means, we identify the homotopy types of such configuration spaces in the case of n-dimensional Euclidean space in terms of the category $\mathbfΘ_n$.

Higher-categorical combinatorics of configuration spaces of Euclidean space

Abstract

We examine configurations of finite subsets of manifolds within the homotopy-theoretic context of -categories by way of stratified spaces. Through these higher categorical means, we identify the homotopy types of such configuration spaces in the case of n-dimensional Euclidean space in terms of the category .

Paper Structure

This paper contains 33 sections, 35 theorems, 190 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Finite subsets of Euclidean space and trees
  3. Exit-paths in the unital Ran space
  4. Lg is an Lg-category
  5. Statement of main result
  6. Exit-paths in the Fox-Neuwirth unital Ran space of Lg
  7. Lg is an Lg-category
  8. Part 1 of the proof of the main result
  9. Part 2 of the proof of the main result
  10. Identifying the space of objects of the localization
  11. The space of configurations of $r$ unordered points in $\mathbb{R}^n$
  12. Identifying the space of morphisms of the localization
  13. Cartesian fibrations
  14. The fibers of $\text{ev}_0$
  15. The final lemma
  16. ...and 18 more sections

Key Result

Theorem 2

For each $n \geq 1$, there is a localization of $\infty$-categories from the subcategory of $\mathbf{\Theta}_n$ consisting of active morphisms (Def. act) to the exit-path $\infty$-category of the unital Ran space of $\mathbb{R}^n$.

Figures (7)

  • Figure 1:
  • Figure 2:
  • Figure 3: The values of a $[0]$-point and a $[1]$-point in $\mathop{\mathrm{\mathcal{R}{\sf ef}}}\nolimits^0(\mathbb{R})$ under $F_2$.
  • Figure 4: A $[0]$-point and a $[1]$-point in $\widetilde{\mathop{\mathrm{\mathcal{R}{\sf ef}}}\nolimits}\left(\mathbb{R}^2\right)$.
  • Figure 5: A $[0]$-point and a $[1]$-point in $\widetilde{\mathop{\mathrm{\mathcal{R}{\sf ef}}}\nolimits}\left(\underline{\mathbb{R}}^2\right)$.
  • ...and 2 more figures

Theorems & Definitions (122)

  • Definition 1: Defs. 1 & 2, AH
  • Theorem 2: Thm. \ref{['loc']}
  • Corollary 3: Cor. \ref{['locexit']}
  • Corollary 4: Cor. \ref{['cor 2']}
  • Conjecture 5
  • Conjecture 6
  • Conjecture 7
  • Remark 8
  • Definition 1.0.1: Def. 6.6.12, AFR
  • Example 1.0.2
  • ...and 112 more