On the abstract wrapped Floer setups

Abstract: The wrapped Fukaya category of a Liouville sector is defined via an axiomatic construction from the associated abstract wrapped Floer setup. In this paper, we propose a modified axiomatic construction, removing the irrelevant choices and the factorization axiom from the abstract wrapped Floer setup. Based on our modification, we reformulate and then prove a conjecture which essentially claims that the wrapped Fukaya category is obtained as the infinity categorical localization along continuation maps. This amounts to compare the two axiomatic constructions.

• The paper demonstrates that the wrapped Fukaya category is explicitly obtained via ∞-categorical localization along continuation maps.
• It introduces a weak abstract Floer setup by removing unnecessary choices and the factorization axiom to streamline previous constructions.
• Canonical equivalences between the new method and the original construction validate the approach and offer practical computational benefits.

Abstract Wrapped Floer Setups: Minimal Axiomatization and $\infty$-Localization

Introduction and Context

The paper addresses the foundational formalism underlying the wrapped Fukaya category, specifically for Liouville sectors, by seeking a minimal and more conceptual axiomatic framework. The original construction of Ganatra–Pardon–Shende (GPS) introduces an "abstract wrapped Floer setup," which depends on a choice of envelope and wrapping categories, as well as a technical factorization axiom. This paper eliminates extraneous choices and the factorization requirement, reformulates the corresponding conjecture from GPS, and establishes that the wrapped Fukaya category can be described as an explicit $\infty$-categorical localization along continuation maps.

This work sits at the confluence of symplectic topology, categorical localization theory, and homological mirror symmetry (HMS), providing categorical tools that refine the GPS functorial construction and unpick the role of continuation maps and the geometric meaning of localization in $A_\infty$-categories.

Summary of Main Contributions

Modified Axiomatization: From "Strong" to "Weak" Abstract Setups

The central technical advance is a modified definition of the abstract wrapped Floer setup:

• All irrelevant choices (envelope, wrapping categories) and the factorization axiom are removed.
• The data are distilled to a "weak abstract Floer setup" $w S$, augmented by a set $C$ of continuation maps satisfying right multiplicativity and a locality property.
• The distinction that remains is minimal: $w S$ encodes composability via cohomologically consistent Floer data (closed under permutation and with pairwise distinct Lagrangians), and $C$ encodes morphisms inverted under localization.

$\infty$-Localization and Canonical Equivalences

The wrapped Fukaya category is constructed as the strict $\infty$-categorical localization of a universal envelope $F^{\text{univ}}_{w S}$ at the system of continuation maps $C_{\text{univ}}$ canonically determined by $C$. This sharpens the informal GPS claim that the wrapped Fukaya category should be the localization at "continuation maps": this work provides an explicit and minimal construction.

Main Theorem

There exists an $\infty$-functorial assignment

$$(w S, C) \mapsto W_{w S, C} := F^{\text{univ}}_{w S}[C_{\text{univ}}^{-1}]$$

where $F^{\text{univ}}_{w S}$ is a cofibrant strictly unital $A_\infty$-category depending functorially (up to unique equivalence) only on $w S$, and $C_{\text{univ}}$ is determined uniquely (up to unique isomorphism) by $C$. This assignment is universal and compatible with all the geometric and functorial structures of the theory.

Additionally, there are canonical equivalences between the cohomology of this construction and the original wrapped Donaldson–Fukaya-type category, justifying that the stripped-down version recovers every "geometric" wrapped Fukaya category.

Removal of the Factorization Axiom

A significant technical observation is that the factorization axiom from GPS—an artifact needed for some combinatorial gluing constructions—can be omitted. The colimit over decorated semisimplicial sets, combined with the properties ascribed to the right multiplicative system $C$, suffices to guarantee all essential categorical properties, including existence of cofinal wrapping categories and right locality.

Reformulation and Proof of the $\infty$-Localization Conjecture

The paper proves (in a comparative theorem) that the original and weak axiomatic constructions yield quasi-equivalent $\infty$-categories; the essential image of the corresponding localization functors is characterized as those functors sending continuation maps to isomorphisms/equivalences. This pins down the GPS conjecture precisely.

Categorical and Homotopical Refinement

The construction proceeds entirely within the framework of strictly unital, cofibrant $A_\infty$-categories, and their $\infty$-localizations and quotients, ensuring functoriality and universal mapping properties in the strongest homotopical sense.

Key Technical Components

• Canonical Envelopes and Universal Envelope: The universal envelope $F^{\text{univ}}_{w S}$ arises as a homotopy colimit over "entanglements" (poset diagrams encoding all choices of Floer data), ensuring canonical mapping properties and minimal dependence on arbitrary data.
• Continuation Maps as Multiplicative Systems: The system $C$ is axiomatized to ensure right-multiplicativity, closure under composition, and sufficient cofinality for wrapping, mirroring the role these maps play geometrically (as Hamiltonian isotopies).
• Cofibrant and Unital Categories: All constructions take place in the strictly unital, cofibrant setting; thus, the results of Oh–Tanaka, Canonaco–Ornaghi–Stellari, and Tanaka guarantee existence and uniqueness up to equivalence for all localization and colimit operations required.

Implications and Outlook

This work solidifies the categorical underpinnings of wrapped Floer theory, aligning symplectic geometry constructions with modern $\infty$-category theory. On the practical side, it enables the construction of the wrapped Fukaya category without having to specify extraneous data, and ensures that computations (e.g., via sectorial descent or Viterbo restriction) depend only on essential geometric and categorical data.

Theoretically, it confirms that the role of continuation maps in the wrapped Fukaya category is precisely as the morphisms to be inverted in the $\infty$-categorical localization, strengthening the analogy (and technical foundation) for the use of such localizations in categorical symplectic topology, and clarifying the bridge to corresponding $B$-side categories in HMS.

A direct avenue for further research is the construction and study of analogous "canonical envelopes" and continuation morphisms on the $B$-side (coherent or constructible sheaves), which would facilitate even more direct formulations of homological mirror symmetry. The modifications here also set the stage for functoriality and comparison results beyond the context of Liouville sectors.

Conclusion

The paper provides a minimal and fully functorial axiomatic approach to wrapped Floer setups, proving that all extraneous data and the factorization axiom are unnecessary for the existence and $\infty$-categorical description of the wrapped Fukaya category. The wrapped Fukaya category is thereby realized canonically as the localization along continuation maps, offering a clear and robust foundation for both computation and further theoretical advances in symplectic geometry and mirror symmetry.