Infinity Categorical Localization

• Infinity categorical localization is a formalism that constructs ∞-categories by universally inverting designated morphisms up to coherent higher homotopy.
• It underpins symplectic topology by organizing Liouville sectors and their stabilized forms, linking them with wrapped Fukaya categories via a symmetric monoidal structure.
• The framework rigorously encodes coherence, automorphism groups, and higher representations, streamlining Floer-theoretic invariants and sectorial moduli.

Infinity categorical localization is a rigorous formalism for constructing higher categories in which a specified class of morphisms becomes invertible up to coherent higher homotopy. In the context of symplectic topology and sectorial geometry, infinity categorical localization underpins the modern approach to organizing Liouville sectors, their automorphisms, and the associated functoriality of wrapped Fukaya categories. This notion allows one to systematically extract homotopy-invariant information by universal inversion, enabling precise control of coherence and symmetric monoidal structures. The concept is central in relating sectorial topoi, automorphism groupoids, and monoidal operations to the algebraic theory of infinity-categories, as exemplified in the recent breakthrough identifying the infinity-category of stabilized Liouville sectors as a localization of an ordinary category and the subsequent symmetric monoidal enhancement (Lazarev et al., 2021).

1. Definition and Construction of Infinity Categorical Localization

For a (small) ordinary category $\mathcal{C}$ and a distinguished class of morphisms $W \subseteq \text{Mor}(\mathcal{C})$, the localization $\mathcal{C}[W^{-1}]$ formally inverts the morphisms in $W$, resulting in a new category where all morphisms in $W$ become isomorphisms. In the setting of higher category theory, the analogous process produces an $\infty$-category, or more specifically a quasi-category, in which the morphisms of $W$ become equivalences up to coherent higher homotopy.

The process requires meticulous encoding of higher structure:

• Model: The localization of $\mathcal{C}$ at $W$ is constructed as a simplicial nerve or as a homotopy-coherent diagram, e.g., via the hammock localization or the categorical calculus of fractions.
• Universal Property: Given any $\infty$-category $\mathcal{D}$, functors $F:\mathcal{C}\to\mathcal{D}$ that send $W$ to equivalences factor uniquely (up to equivalence) through $\mathcal{C}[W^{-1}]$.

In the geometric context of Liouville sectors, the category $\mathcal{C}$ consists of sectors and strict sectorial embeddings, and $W$ is the subcategory of those embeddings that are equivalences up to isotopy. The localization procedure produces an $\infty$-category of stabilized Liouville sectors, $\text{Liou}^{\text{stab}}[W^{-1}]$, encoding all homotopy and isotopy data relevant for symplectic and Floer-theoretic invariants (Lazarev et al., 2021).

2. Liouville Sectors, Stabilization, and the Role of Localization

Liouville sectors are exact symplectic manifolds with corners, provided with compatible collar data inducing splitting into fibered and cotangent directions at each corner stratum (Oh, 2021, Oh, 2021, Lazarev et al., 2021). The stabilization functor $M \mapsto M \times T^*[0,1]$ allows passage to a category $\text{Liou}^{\text{stab}}$ where cotangent directions have been stabilized to control homotopical complexity.

Localization at sectorial equivalences is essential because:

• Finite-Type Invariance: Many wrapped Fukaya and Floer-theoretic invariants are only invariant under equivalence up to stabilization and isotopy.
• Mapping Spaces: The mapping space in the localized $\infty$-category $\text{Liou}^{\text{stab}}[W^{-1}]$ captures the space of stable sectorial embeddings modulo isotopies, encoding the topology relevant for families, moduli, and automorphism structures.
• Equivalence with Semisimplicial Models: The construction admits comparison with explicit semisimplicial sets parameterizing compatible diagrams of embeddings, further solidifying the topological and categorical universalities at play.

3. Symmetric Monoidal Structures and Universal Properties

A remarkable advance is the identification of a symmetric monoidal structure on the localized $\infty$-category of stabilized Liouville sectors:

• Tensor Product: The tensor is given by Cartesian product of sectors, $(M,k)\otimes(N,l)=(M\times N,k+l)$, reflecting monoidality at both the level of geometry and associated Fukaya-type invariants (Lazarev et al., 2021, Oh, 2021).
• Unit and Invertibility: The point sector $(\mathbb{R}^0,0)$ is the tensor unit. The stabilized cotangent factor $T^*[0,1]$ becomes invertible, with all necessary isotopy relations coherently encoded in the localization.
• Universal Property: $\text{Liou}^{\text{stab}}[W^{-1}]$ is the initial symmetric monoidal $\infty$-category under the ordinary category of Liouville sectors with $T^*[0,1]$ invertible, conferring an optimal categorical foundation for multiplicative structures and module categories.

This symmetric monoidal enhancement is crucial for the multiplicative compatibility of invariants such as the wrapped Fukaya category, Lagrangian cobordism modules, and factorization homology (Lazarev et al., 2021).

4. Applications: Functoriality and Wrapped Fukaya Categories

Infinity categorical localization realizes highly nontrivial coherence properties for sectorial invariants:

• Fully Coherent Action: The assignment $M \mapsto \mathcal{W}(M\times T^*[0,1]^\infty)$ (the wrapped Fukaya category of a stably stabilized sector) extends to a functor from $\text{Liou}^{\text{stab}}[W^{-1}]$ to the $\infty$-category of $A_\infty$-categories, so that every $k$-simplex, i.e., family of stable sectorial embeddings parametrized by a simplex, induces a family of $A_\infty$-functors (Lazarev et al., 2021).
• Mapping Spaces and Moduli: The mapping spaces in the localized $\infty$-category compute derived mapping spaces between sectors and correspond, under this functor, to mapping spaces of $A_\infty$-categories, respecting composition and homotopy equivalence.
• Families and Higher Representations: Sector bundles and principal automorphism-bundles over a base $B$ produce smooth fiber bundles of sectors, with associated higher Seidel representations in Floer theory.
• Monoidality and Künneth Functor: The monoidal structure on Liouville $\sigma$-sectors guarantees, via the Künneth theorem, a monoidal functoriality for wrapped Fukaya categories:

$$\mathcal{W}(M) \otimes \mathcal{W}(M') \longrightarrow \mathcal{W}(M \times M')$$

with the monoidal structure induced by categorical localization on sectors (Oh, 2021).

5. Sectorial Automorphisms, Bundles, and Higher Algebraic Structures

Automorphism groups and principal bundles of Liouville sectors are naturally captured in the localizing $\infty$-categorical framework:

• Automorphism Group: Liouville automorphisms are diffeomorphisms of the stratified boundary-corner manifold preserving all sectorial structure up to compactly supported exact forms. In the localized $\infty$-category, these assemble to group-objects acting on functorial invariants (Oh, 2021).
• Bundles of Sectors: Principal bundles with fiber $(M,\lambda)$ and structure group $\text{Aut}(M,\lambda)$ yield smooth fiber bundles of Liouville sectors. Their classifying maps induce actions on Floer-theoretic invariants, and their higher moduli are naturally understood in terms of mapping spaces in $\text{Liou}^{\text{stab}}[W^{-1}]$.
• Factorization Homology: The symmetric monoidal structure allows the application of factorization homology, ensuring that for sectors $M_1, M_2$, one obtains $$\mathcal{W}(M_1)\otimes\mathcal{W}(M_2)\simeq\mathcal{W}(M_1\times M_2)$$, with coherence inherited from the localized structure (Lazarev et al., 2021).

6. Impact, Examples, and Connections to High-Dimensional Topology

The techniques of infinity categorical localization:

• Give a foundation for functorial and multiplicative invariants in Floer/sectorial/symplectic topology.
• Streamline coherence arguments in the construction of wrapped Fukaya categories, Lagrangian cobordism modules, and Seidel/higher representation spaces.
• Provide explicit models for spaces of sectorial embeddings, automorphisms, and geometric structures through the semisimplicial and Kan complex models for mapping spaces.
• Permit extension to related tangential structures, such as gradings and spin, via modifications of the base topological category (Lazarev et al., 2021).

A crucial conceptual implication is that geometric and algebraic invariants of Liouville sectors are most naturally interpreted not as set-based or category-based objects, but rather as $\infty$-categorical or higher stack-theoretic entities, with all isotopy, moduli, and monoidal data encoded at the correct level of homotopical abstraction. This paradigm is increasingly standard in factorization homology, topological field theory, and categorical symplectic topology.

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Key references:

• "The infinity-category of stabilized Liouville sectors" (Lazarev et al., 2021)
• "Presymplectic geometry and Liouville sectors with corners and its monoidality" (Oh, 2021)
• "Geometry of Liouville sectors and the maximum principle" (Oh, 2021)