Hom functor

Hom functor
Name	Hom functor
Field	Category theory, Homological algebra
Related	Category, Functor, Natural transformation, Adjunction

Hom functor

The Hom functor is a basic construction in Category theory and Homological algebra that assigns to a pair of objects the set of morphisms between them; it underlies concepts in Algebraic topology, Algebraic geometry, Representation theory, Functional analysis, and Model theory. Originating in work related to David Hilbert's foundations and formalized in the 1940s and 1950s alongside advances by Saunders Mac Lane and Samuel Eilenberg, the Hom functor organizes morphism-sets into contravariant and covariant functors and appears in constructions such as Ext functor, Tor functor, and the Yoneda lemma. It provides a language connecting classical computations in Galois theory, structural results in Noetherian ring theory, and categorical dualities like Pontryagin duality.

Definition and basic properties

For objects A and B in a category C, the set Hom_C(A,B) consists of morphisms from A to B; this assignment yields functors Hom_C(A,–): C → Set and Hom_C(–,B): C^op → Set. Fundamental properties include composition-induced maps, identity morphisms acting as units, and the representability condition that an object R represents a functor F when F ≅ Hom_C(R,–), as in the role of representing objects in Hilbert space theory or representing functors in Grothendieck's approach to schemes. The Hom sets are often endowed with additional structure: abelian group objects in Abelian categorys, modules over endomorphism rings in Ring theory, or topologies in Banach space and Frechet space contexts.

Variants (Hom sets, Hom functors, Hom bifunctor)

Variants include the plain Hom sets Hom_C(A,B), the covariant Hom functor Hom_C(A,–), the contravariant Hom functor Hom_C(–,B), and the Hom bifunctor Hom_C(–,–): C^op × C → Set. In enriched contexts one considers Hom objects in a monoidal category, such as Hom_V(X,Y) in enrichment over Monoidal categorys like Vect over a field or Top spaces; these generalizations are essential in settings studied by Jean-Pierre Serre, Alexander Grothendieck, and Michael Artin.

Functoriality and natural transformations

Functoriality of Hom yields natural transformations arising from pre- and post-composition; given f: A' → A and g: B → B' one has induced maps Hom_C(A,B) → Hom_C(A',B') by h ↦ g ∘ h ∘ f. Natural transformations between Hom functors are central to the Yoneda lemma, which links elements of Hom sets to natural transformations from representable functors; this principle is a cornerstone in theories developed by Yoneda, Grothendieck and applied in contexts such as Deligne's work on categories of sheaves and Beilinson's derived categories.

Adjointness and tensor-Hom adjunctions

Adjoint functor relationships frequently involve Hom: for a monoidal category with tensor ⊗, one often has natural isomorphisms Hom(X ⊗ Y, Z) ≅ Hom(X, Hom(Y,Z)), encoding internal Hom adjunctions that appear in Tannaka–Krein duality, Serre duality, and constructions in Derived categorys used by Verdier and Grothendieck. In module categories over a ring R, the standard adjunction between tensor and Hom underlies dualities in Morita theory and appears in Eilenberg–Watts theorem contexts; adjoints play a key role in the existence results proven in the Adjoint functor theorem by Freyd.

Exactness and Hom in homological algebra

The behavior of Hom with respect to exact sequences is measured by its left or right exactness: Hom_C(A,–) is left exact in abelian categories while Hom_C(–,B) is left exact in the first variable; failure of exactness gives rise to derived functors Ext^n and Ext^1, central to Cartan–Eilenberg homological methods and computations in Group cohomology, Sheaf cohomology developed by Godement and Cartan, and spectral sequences such as the Grothendieck spectral sequence. These derived constructions connect to classification results like those studied by Noether, Artin, and Serre.

Examples and computations

Concrete computations include Hom in categories: Hom_Z(Z/nZ,Z/mZ) ≅ Z/gcd(n,m)Z in Number theory flavored module calculations; Hom_k(V,W) as linear maps between vector spaces V and W over a field k in Linear algebra; Hom_C(X,Y) as continuous maps in Topology or bounded linear operators in Functional analysis between Banach spaces; and Hom in category of sheaves on a Scheme computed via global section functors and Čech methods used by Grothendieck and Serre. Representation-theoretic examples include Hom between Lie algebra modules and intertwining operators in the theory of Harish-Chandra modules.

Enriched and derived Hom constructions

Enriched Hom replaces Hom sets by objects in a base monoidal category (for example, Hom in SSet for simplicial enrichment or internal Homs in Monoidal categorys), employed in Quillen model category theory and Lurie's higher category frameworks. Derived Hom, RHom, lives in derived categories and model categories and encodes higher Ext groups; it is fundamental in Derived algebraic geometry developed by Toën and Vezzosi and in duality theories from Grothendieck duality to Serre duality and applications in Mirror symmetry investigated by Kontsevich.

Category theory Homological algebra Algebraic topology Algebraic geometry Functional analysis Representation theory Yoneda lemma Grothendieck Eilenberg Mac Lane Freyd Serre Noether Artin Deligne Beilinson Verdier Quillen Lurie Kontsevich Toën Vezzosi Cartan Eilenberg–Mac Lane Godement Pontryagin duality Tannaka–Krein duality Morita theory Hilbert space Banach space SSet Vect Ring theory Scheme Čech cohomology Derived category Ext functor Tor functor Spectral sequence Grothendieck spectral sequence Group cohomology Harish-Chandra Lie algebra Noetherian ring Adjoint functor theorem Eilenberg–Watts theorem Yoneda David Hilbert Jean-Pierre Serre Alexander Grothendieck Samuel Eilenberg Saunders Mac Lane Michael Artin Alexander Grothendieck Jean-Louis Verdier Maxime Kontsevich

Category:Category theory