Structured flow categories and twisted presheaves

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 $\mathcal{C}$.

• The paper constructs a stable ∞-category low^μ that unifies μ-structured flow categories with twisted spectral presheaves, enabling refined Floer invariants.
• It employs enriched double ∞-categories and combinatorial arc models to systematically encode Morse and Floer-theoretic constructions.
• The framework accommodates tangential, local, and filtered structures, paving the way for extending and applying Floer homotopy theory.

Structured Flow Categories and Twisted Presheaves

Overview

The paper "Structured Flow Categories and Twisted Presheaves" (2603.29576) formulates a general theory of flow categories endowed with additional tangential, local system, and filtration structure, providing a unifying $\infty$-categorical framework for exact Floer homotopy theory. Building on the orientation-theoretic perspective, the authors construct a stable, presentable $\infty$-category $low^\mu$ of $\mu$-structured flow categories and bimodules for any $\infty$-functor $\mu : C \to U/O$, where $U/O$ parametrizes stable tangential structures. The analysis centers on identifying $low^\mu$ with $\mu$-twisted spectral presheaves, which subsume essential flavors of coefficient, local system, and filtered structures prevalent in modern Floer theory.

Main Construction and Theoretical Contributions

A key contribution is the construction of a stable $\infty$-category $\infty$0 whose objects are flow categories equipped with $\infty$1-structure—that is, compatibly assigned objects in a base $\infty$2-category $\infty$3 (encoding reference points for moduli spaces and their parametrizations) and control data for the tangential/framing structure of moduli spaces via $\infty$4. Morphisms are encoded as bimodules with compatible $\infty$5-structure. The authors leverage enriched $\infty$6-category theory, constructing a double $\infty$7-category $\infty$8 encoding $\infty$9-structured stratified manifolds with corners, and use a recursive combinatorial model of directed arcs to encode the “flow” combinatorics necessary for Morse and Floer-theoretic constructions.

A central technical result is the establishment of an equivalence

$low^\mu$0

as functors $low^\mu$1 (where $low^\mu$2 denotes $low^\mu$3-twisted presheaves of spectra), i.e., every $low^\mu$4-structured flow category corresponds to a twisted spectral presheaf, and every twisted presheaf arises from such a flow category. The identification is canonical and natural in the tangential functor $low^\mu$5 and the underlying combinatorics of the base $low^\mu$6.

The authors prove that $low^\mu$7 is a compactly generated, stable, presentable $low^\mu$8-category, and construct canonical functors between various such categories as induced by morphisms between orientation data.

Technical Framework

The approach utilizes several advanced categorical and topological constructions:

• Enriched Double $low^\mu$9-Categories: The authors employ double $\mu$0-categories as in Haugseng [GH], encoding both horizontal compositions (combinatorial composition of flow lines/bimodules) and vertical structure (homotopical coherence of $\mu$1-structure).
• Semi-simplicial Spaces and Inner Kan Complexes: The structure of $\mu$2-structured flow categories and their bimodules is encoded as a semi-simplicial space $\mu$3, shown to be a quasi-unital inner Kan space, and so to present an $\mu$4-category.
• Companion Functors and Colimits: By carefully encoding the combinatorics of flow categories, the authors employ companion constructions to control universal properties including coproducts, pushouts, and mapping cones. These in turn are crucial in characterizing the stability and presentability of $\mu$5.
• Pontrjagin–Thom Collapse and Bordism: The mapping spaces in $\mu$6 recover classifying spaces for $\mu$7-structured bordism, and the authors construct a natural Pontrjagin–Thom equivalence between the one-object flow category subcategory $\mu$8 and spectra valued on the Thom spectrum of $\mu$9.

Classification of Bordism, Orientations, and Local Systems

A strength of the framework is its flexibility in capturing a broad spectrum of geometric and topological data:

• Tangential Structures: The general formalism accommodates classical framing, $\infty$0-orientations (modules over $\infty$1), and situations where the moduli spaces only admit weaker tangential reduction. The authors’ construction of $\infty$2 naturally interpolates between these settings and recovers the expected classifying spectra.
• Local Systems and Spectral Coefficient Systems: Reference path data and induced local systems can be encoded, including both classical group ring and spectral local systems (modules over $\infty$3), unifying Morse and Floer-theoretic approaches.
• Filtrations: The method generalizes to flow categories with additional ordering or grading, i.e., filtered Floer theory, by replacing the base poset as needed; the framework then identifies the appropriate spectral diagrams (e.g., functors $\infty$4 for a poset $\infty$5).
• Twists and Non-frameable Examples: For cases with non-trivial index bundles (as in exact Lagrangian Floer theory, where $\infty$6 is not frameable), the formalism produces the correct coefficient twisting at the level of spectra.

Examples and Comparisons

Multiple explicit examples illustrate the theory:

• Morse Homotopy Types: For the Morse flow category equipped with the zero tangential structure, the category $\infty$7 recovers spectral local systems, and the standard Morse flow category corresponds canonically to the constant local system.
• Lagrangian Floer Homotopy Theory: The construction interprets the index and orientation data in terms of Thom spectra and spectral local systems, offering a context to handle filtered and twisted coefficients beyond the frameable case, and relating the algebraic models to those of recent work by Abouzaid–Blumberg, Porcelli, and others.
• Filtration by a Poset: The general analysis of $\infty$8-structured flow categories recovers diagram categories $\infty$9, and the functoriality of the mapping spaces aligns with expected restriction and inclusion maps for filtered objects.

Implications and Future Directions

This work substantially systematizes and generalizes the foundations of Floer (and Morse) homotopy theory, providing an $\mu : C \to U/O$0-categorical architecture that subsumes tangential, local system, and filtration data. By identifying structured flow categories with twisted spectral presheaves, the authors unlock multiple avenues for both practical and conceptual advances:

• Stable Homotopical Invariants: The framework establishes the groundwork for defining and computing more refined stable invariants in Floer theory, even when classical framing is unavailable.
• Functoriality and Manipulation of Structures: The naturality of the equivalence $\mu : C \to U/O$1, and the induced adjoints between structure categories, promise new techniques for controlling and comparing invariants with varying tangential data.
• Extension to Further Twists: The generality of the formalism suggests straightforward adaptability to further settings, e.g., situations with bundle-twisted, equivariant, or sheaf-theoretic structure, provided the basic orientation data can be coherently functorially packaged as input.
• Connections with Higher Symplectic and Topological Field Theories: The identification with twisted presheaf categories relates the theory to global perspectives in the study of extended field theories, categorical characterizations of bordism, and categorification in homotopy theory.

The authors’ method opens the door to further analyses of the interplay between geometric, algebraic, and categorical ingredients in Floer and Donaldson-type theories.

Conclusion

The paper delivers a comprehensive construction of structured flow category $\mu : C \to U/O$2-categories and situates them within the larger context of twisted presheaves of spectra. The methodology unifies disparate orientations, local systems, and filtrations, while ensuring the crucial stable and presentable properties necessary for modern homotopy theory. The established natural equivalence between $\mu : C \to U/O$3 and twisted presheaf categories not only recasts the foundations of Floer homotopy theory but also provides robust technical tools for subsequent advances in symplectic topology, Morse theory, and related domains.