Restriction and Induction Functors

• Restriction and induction functors are foundational tools defined as adjoint pairs that transfer structures between global and local contexts in algebra, topology, and category theory.
• They enable precise gluing of local data into global structures, supporting analysis in representation theory, perverse schobers, and cluster-tilting frameworks.
• Their applications span group representations, Hopf algebras, and higher categories, providing critical methods for establishing exact, spherical, and Mackey-type properties.

Restriction and induction functors are foundational tools for transferring algebraic, categorical, or geometric structures along embeddings or coverings, with crucial applications across representation theory, categorical topology, and higher algebra. They formalize the processes of “restricting” objects from a larger context to a subcontext and “inducing” objects back up, providing adjoint pairs that encode rich structural and compatibilty properties, including the ability to assemble local data into global structures and analyze decompositions, symmetries, and cohomological features.

1. General Definition and Paradigm Examples

Given a morphism of algebraic, categorical, or topological data (such as a group homomorphism, an inclusion of categories, or a covering map), the restriction functor generally “pulls back” from a theory over a large object to a subobject, while the induction functor provides a way to “push forward” from the subobject back to the larger context. These functors often arise as an adjoint pair, with induction left (or sometimes right) adjoint to restriction, and play a central role in formulating transfer principles, gluing local structures, and establishing dualities and Mackey-type decompositions.

Classical examples include:

• The induction and restriction of group representations: For an inclusion $H \hookrightarrow G$, $\mathrm{Res}^G_H$ is restriction and $\mathrm{Ind}_H^G$ is induction, giving adjoint functors between module categories.
• In the context of module categories over Hopf algebras, restriction of scalars along a Hopf algebra morphism $\varphi: K \to H$ has both induction ($H \otimes_K -$) and coinduction ($\mathrm{Hom}_K(H,-)$) adjoints, enabling module-theoretic transfer with precise monoidal control (Flake et al., 15 Feb 2024).

2. Restriction and Induction in Higher Categories and Perverse Schobers

In the modern framework of stable $\infty$-categories and perverse schobers, restriction and induction functors are used to control the passage between global and local sections. Let $\Sigma$ be an oriented surface with boundary, $M$ a set of marked boundary points, and $\Gamma$ a spanning ribbon graph. A $\Gamma$-parametrized perverse schober is a functor $F: \mathrm{Exit}(\Gamma) \to \mathrm{St}$ to small stable $\infty$-categories.

• The category of global sections $H(\Sigma,F)$ is the limit of $F$ over the exit-path category.
• The restriction (boundary evaluation) functor $\mathrm{ev}_x : H(\Sigma,F) \to F(x)$, for vertices or edges $x$, extracts the value at $x$ and, for subsurfaces $U$ corresponding to a full subgraph $\Gamma'$, restriction along the inclusion $$i : \mathrm{Exit}(\Gamma') \hookrightarrow \mathrm{Exit}(\Gamma)$$ induces $Res_U := i^*$.
• These restriction functors admit both left and right adjoints, denoted $Ind^L_U$, $Ind^R_U$, constructed by explicit gluing mechanisms reflecting the combinatorics of half-edges and trajectories on the surface (Christ, 1 Sep 2025).

Adjunctions are realized via explicit Hom-formulas:

$$\mathrm{Hom}_{H(\Sigma,F)}\big(ind^L_x(A), X\big) \cong \mathrm{Hom}_{F(x)}\big(A, ev_x(X)\big)$$

and similarly for the right adjoint. The boundary evaluations along all external edges assemble into a perverse schober on a multi-spider and are spherical functors, imparting Frobenius exact structures to the global sections.

3. Structural Impact: Gluing and Cluster-Tilting

Restriction and induction functors are the key tools for gluing local cluster-tilting subcategories to global ones in the category of global sections, especially within higher categorical settings. In a Frobenius exact $\infty$-category $C$, a subcategory $T$ is cluster-tilting if it is rigid (i.e., $\operatorname{Ext}^1_{C}(T,T) = 0$) and every object admits two-term resolutions by $T$ on both sides.

Given local rigid $T_v \subset F(v)$ at each vertex, the induction functor yields objects $ind_v(T_v)$ in $H(\Sigma,F)$, satisfying:

• $$\operatorname{Ext}^1(ind_v(T_v), ind_v(T_v)) \cong \operatorname{Ext}^1_{F(v)}(T_v, T_v)$$ for each $v$,
• vanishing $\operatorname{Ext}^1$ for $v \ne v'$, so their union remains rigid. Gluing two-term resolutions across vertices, using pullback/pushout diagrams equipped with adjoints, ensures the resolution property globally, yielding a glued cluster-tilting subcategory:

$$T_{glob} = \mathrm{add}\left\{ ind_v(T_v) \right\}_{v\in\Gamma_0}$$

which is cluster-tilting in $H(\Sigma,F)$ (Christ, 1 Sep 2025).

Induction furthermore realizes the familiar amalgamation of endomorphism algebras (ice‐quivers) along matched frozen parts, paralleling constructions in cluster algebras.

4. Adjunctions, Sphericality, and Exactness

Restriction and induction functors frequently form biadjoint pairs—both left and right adjoints exist and are compatible. In higher representation theory, these functors are often constructed to be (bi)spherical or exact in the sense of exact functors of abelian or exact $\infty$-categories. Explicitly:

• In the perverse schober context, the product of boundary evaluations and the product of edge-induction functors form a spherical pair, endowing the category of global sections with a Frobenius exact structure (Christ, 1 Sep 2025).
• The spherical property ensures that sequences of functors satisfy certain unit/counit decomposition formulas essential for the interplay between "restriction then induction" and "induction then restriction" (cf. unit/counit splitting and Mackey formulas).

This framework allows for a precise manipulation of fiber and cofiber sequences, fundamental for rigidity and the analysis of derived categories.

5. Concrete Examples in Topological and Categorical Settings

Restriction and induction mechanisms are concretely illustrated in topological Fukaya categories and cluster categories:

• For unpunctured surfaces with 1-periodic topological Fukaya categories, local categories are of the form $D^{perf}(k[t^{\pm 1}])$ and cluster-tilting objects are described combinatorially via triangulations. Induction transports local stalk objects around edges, while restriction forgets external contributions. Gluing recovers the classical bijection between ideal triangulations and cluster-tilting objects (Christ, 1 Sep 2025).
• In the once-punctured digon, with a spherical perverse schober, local categories at 2-valent vertices support cluster-tilting objects bijecting the tagged triangulations, and induction and restriction correspond to reconnecting endpoints and evaluating on arcs.

These examples demonstrate how restriction/induction functors analytically realize the combinatorics of arcs, triangulations, and their amalgamations within the global category.

6. Universality and Broader Context

The fundamental pattern observed in the categorical and geometric frameworks above recurs across mathematical disciplines:

• In the context of Mackey and Green 2-functors, restriction and induction functors are incorporated into higher-categorical structures equipped with additivity, ambidextrous adjunctions, and Mackey base-change formulas, providing a universal setting for such interplays (Dell'Ambrogio, 2023).
• In module categories for Hopf algebras and multifusion categories, restriction/induction pairs are tightly linked to monoidal adjunctions and notions such as the Drinfeld center, with projection formulas governing induced functors on central objects (Flake et al., 15 Feb 2024).
• In derived and perverse sheaf theory, restriction and induction functors give rise to exact, adjoint pairs that are central in the study of character sheaves and categorification of quantum groups (Fang et al., 2021, Bezrukavnikov et al., 2018).

Restriction and induction functors, through their adjunction, exactness, and compatibility with geometric and algebraic structures, are thus indispensable for understanding gluing phenomena, the construction of cluster-tilting objects, and the categorification of combinatorial and representation-theoretic invariants.