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Structured flow categories and twisted presheaves

Alice Hedenlund, Trygve Poppe Oldervoll

Abstract

An orientation theory for flow categories without bubbling is determined by a functor of $\infty$-categories $μ\colon \mathcal{C} \to U/O$. For any such functor, we construct a stable $\infty$-category $\mathcal{F}low^μ$ of $μ$-structured flow categories and bimodules. We also construct the expected functors between such $\infty$-categories, giving a tractable framework for manipulating orientations, local systems, and filtrations in exact Floer homotopy theory. Classifying spaces for certain bordism theories determined by $μ$ appear as mapping spaces in $\mathcal{F}low^μ$, and we use a Pontrjagin--Thom construction to naturally identify $\mathcal{F}low^μ$ with the $\infty$-category of $μ$-twisted presheaves on $\mathcal{C}$.

Structured flow categories and twisted presheaves

Abstract

An orientation theory for flow categories without bubbling is determined by a functor of -categories . For any such functor, we construct a stable -category of -structured flow categories and bimodules. We also construct the expected functors between such -categories, giving a tractable framework for manipulating orientations, local systems, and filtrations in exact Floer homotopy theory. Classifying spaces for certain bordism theories determined by appear as mapping spaces in , and we use a Pontrjagin--Thom construction to naturally identify with the -category of -twisted presheaves on .

Paper Structure

This paper contains 33 sections, 86 theorems, 417 equations.

Table of Contents

  1. Categorical Preliminaries
  2. $\infty$-categories and their models
  3. Fibrations and straightening-unstraightening
  4. Monoidal and double $\infty$-categories
  5. Enriched categories
  6. Completeness for enriched categories
  7. Slice double categories
  8. Monoids and delooping
  9. Inner Kan spaces and companions
  10. Structured Flow Categories
  11. Virtual vector bundles
  12. Models and stratified manifolds with corners
  13. Structured manifolds with corners
  14. Double categories of structured manifolds with corners
  15. Categories of directed arcs
  16. ...and 18 more sections

Key Result

Theorem 1

There is a natural equivalence of functors $\mathrm{Cat}\newline_{\infty /(U/O)} \longrightarrow \mathrm{Pr}^{\mathrm{L}}_{\mathop{\mathrm{St}}\nolimits,\omega}$

Theorems & Definitions (335)

  • Theorem 1
  • Definition 2
  • Definition 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Example 7
  • Example 8
  • Example 9
  • Example 10
  • ...and 325 more