Functor-Induced Isomorphisms

• Functor-Induced Isomorphisms are defined as isomorphisms arising when functors transfer object-level equivalences to structured diagrammatic or invariant levels.
• Key frameworks, such as the larder method and G-matrix techniques, leverage properties like colimit preservation and projectability to systematically lift isomorphisms.
• These approaches find practical applications in representation theory, localization in derived categories, and six-functor formalisms, ensuring robust transfer of invariants and categorical rigidity.

A functor-induced isomorphism formally refers to an isomorphism in a target category or structure that arises from, and is “encoded” by, a functor or system of functors acting on an initial object, morphism, or entire diagrammatic structure. The core mathematical problem is to establish precise conditions and frameworks under which isomorphisms at the level of objects can be canonically promoted to isomorphisms on more complex or structured data: diagrams, categories of modules/representations, Grothendieck groups, derived categories, or invariants such as K-theory or Hochschild homology. The paper of functor-induced isomorphisms is central both in abstract category theory and in concrete applications throughout representation theory, algebraic geometry, algebraic topology, and mathematical physics.

1. General Frameworks for Lifting Object Isomorphisms to Diagrammatic Isomorphisms

A foundational categorical approach is articulated in "From objects to diagrams for ranges of functors" (Gillibert et al., 2010). Here, the "larder framework" is developed to precisely characterize when an isomorphism at the object level between functor images can be extended to an entire diagram indexed by a poset. The critical data consist of three categories—$\mathcal{A}$ (source), $\mathcal{B}$ (target/witness), and $\mathcal{S}$ (ambient)—with functors $\Phi \colon \mathcal{A} \to \mathcal{S}$, $\Psi \colon \mathcal{B} \to \mathcal{S}$ possessing favorable continuity properties. Under assumptions such as preservation of directed colimits and products, and projectability (existence of "projectability witnesses"), the main theorem—the Condensate Lifting Lemma—proves that an isomorphism at the ambient category level lifts functorially to a diagrammatic isomorphism indexed by a poset $P$ (with specified combinatorial conditions).

For example, a "double arrow" $$\chi : \Psi(B) \Rightarrow \Phi(F(X) \otimes \vec{A})$$ can be promoted to a natural transformation between the functor images of diagrams, provided that $P$ is an almost join-semilattice and the underlying Boolean algebra constructions ($F(X)$) admit suitable scaling and tensor properties. Explicit projectability and norm-covering constructions are integral, as are combinatorial properties of $P$. Applications include functorial extensions of the Grätzer–Schmidt theorem to diagrams, critical point theorems, and compactness results. The technique systematically links the category theory of diagrams with logic, lattice theory, and universal algebra.

2. Functor-Induced Isomorphisms and $G$-Matrices in Representation Theory

Recent work ("Functor-induced isomorphisms and $G$-matrices" (Geng, 22 Sep 2025)) demonstrates that, in the context of tilting theory, silting theory, and the associated module categories, functor-induced isomorphisms are explicitly “realized” by $G$-matrices. For a basic tilting $A$-module $T = T_1 \oplus \dots \oplus T_n$, the map $K_0(A) \to K_0(B)$ (with $B = \mathrm{End}_A(T)$) induced by the tilting functor is represented by the transpose of the $G$-matrix $G_T$: $$G_T^{\rm t} [N] = [\operatorname{Hom}_A(T, N)] - [\operatorname{Ext}_A^1(T, N)] \qquad \forall N \in \text{mod-}A.$$ This construction generalizes for $2$-term silting complexes $P^*$; again, the functor-induced isomorphism on the Grothendieck group $K_0(A)$ is given by $G_{P^*}^{\rm t}$ acting as a linear operator.

Moreover, key autoequivalences and endofunctors—such as the Coxeter transformation, Nakayama functor, and Auslander–Reiten translation—are represented by products of (possibly transposed or inverted) $G$-matrices of specific (dual) tilting modules. For instance, writing $DA$ for the dual of $A$, the Coxeter transformation $\Phi_A$ satisfies $\Phi_A = - ({G_{DA}^{-1}})^{\rm t}$. There is a full realization of symmetric and Weyl group elements as transposed $G$-matrices of tilting or support $\tau$-tilting modules in various settings (e.g., $A$ the Auslander algebra of $k[x]/(x^n)$).

In all these cases, the functorial action on $K_0$—the categorical “shadow” of the module category—is essentially the combinatorial data recorded by these $G$-matrices.

3. Explicit Structures: Replacement Theory, Localization, and Derived Categories

Functor-induced isomorphisms appear in localization frameworks, particularly in homotopy theory and derived categories. The characterization due to Thomas (Thomas, 2018) hinges on the notion of "replacement functors" (e.g., projective or cofibrant replacements). For a functor $F: \mathcal{C} \to \mathcal{D}$ between categories with denominator sets (e.g., weak equivalences or quasi-isomorphisms), the induced functor on Gabriel–Zisman localizations $GZ(F): GZ(\mathcal{C}) \to GZ(\mathcal{D})$ is an equivalence if $F$ is $S$-dense (enough replacements), $S$-full, and $S$-faithful (defined in terms of $S$-2-arrows constructed out of denominators). The explicit isomorphism inverse is then constructed via a replacement functor, generalizing dense–full–faithful equivalence conditions.

In derived algebra, functor-induced isomorphisms likewise appear in the context of recollement of derived categories, as with the functor $j^*: D(A) \to D(C)$ in a recollement setting (Qin, 2018). Here, $j^*$ is called an "eventually homological isomorphism" if, after a finite degree, $\operatorname{Hom}$ spaces agree and induce isomorphisms for all large shifts. Such properties ensure that deep invariants—singularity categories, Gorensteinness, or finite generation of Hochschild cohomology—are transferred between $A$ and $C$.

4. Canonical Isomorphisms in Six-Functor Formalisms

The canonical isomorphisms—such as those between direct and exceptional direct/exceptional inverse images—arising in Grothendieck six-functor formalisms are functor-induced isomorphisms in a highly structured setting (Dauser et al., 20 Dec 2024). The uniqueness theorem (confirming Scholze’s conjecture) asserts that, for a class of morphisms factoring as a composite of “cohomologically proper” and “cohomologically étale” morphisms, the entire six-functor formalism is uniquely determined by its tensor product and inverse image functors. The core is that, for morphisms in the “proper” or “étale” classes, the canonical isomorphisms

$f_+ \cong f_*, \quad f^* \cong f^!$

arise via Beck–Chevalley conditions and explicit adjunction and base-change diagrams. The uniqueness (and rigidity) of six-functor formalisms in these settings is ultimately governed by the induced functorial isomorphisms, with the failure of uniqueness in the nontruncated situation measured by algebraic $K$-theory.

5. Applications: Representation Theory, Geometry, and Physics

Functor-induced isomorphisms are ubiquitous in concrete settings. In the representation theory of rational Cherednik algebras, induction and restriction functors with seemingly distinct definitions are shown to be isomorphic via explicit functorial constructions and completion isomorphisms (Losev, 2010). In noncommutative topology and $C^*$-algebra K-theory, isomorphisms in groupoid homology, established via functors and spectral sequences, are lifted to K-theoretic isomorphisms of operator algebras (Miller, 30 Jan 2024). In constructions of Tannakian categories, the existence and uniqueness of fiber functors up to local isomorphism—again a functor-induced statement—are the foundation of the Tannaka duality framework (Schäppi, 2018).

Analogous techniques underlie the nature of equivalences in derived functor categories (e.g., pushforward functors between matrix factorization categories inducing explicit maps on Hochschild homology, described via the Hochschild–Kostant–Rosenberg theorem and connecting maps involving Todd classes (Nordstrom, 15 Feb 2024)), the functorial naturality of the Freed-Hopkins-Teleman isomorphism in twisted $K$-theory (Takata, 2015), and the structure of topological field theories via barycentric subdivision and groupoid isomorphisms (Sommer-Simpson, 2015).

6. Summary

Functor-induced isomorphisms manifest in multiple guises: linear transformations on Grothendieck groups represented by $G$-matrices, canonical diagrammatic natural isomorphisms controlled by larder frameworks and condensate constructions, explicit isomorphisms between modules or algebras induced by replacement procedures or localization, and universal categorical constructions (six-functor formalisms, derivator enhancements) encoding geometric or topological invariants. At their core, they formalize the passage from local or object-wise equivalence to global, coherent, or invariant equivalence across algebraic, topological, and categorical structures, with deep consequences for the transfer and computation of complex invariants and for the rigidity of mathematical frameworks.