Contents

Hom FunctorEdit

In category theory, the Hom functor is one of the most fundamental constructions for organizing and studying the maps between objects. For any category C and any pair of objects A and B in C, the set Hom_C(A,B) collects all arrows from A to B. More than just a static set, this collection can be organized into a functor, yielding a powerful way to translate questions about arrows into questions about functors and natural transformations.

The Hom construction behaves differently depending on which argument is held fixed. If you keep A fixed and vary B, you get a covariant functor Hom_C(A,-): C → Set that sends each object B to the set of arrows from A to B and each morphism f: B → C to the map that composes on the right with f. If you fix B and vary A, you obtain a contravariant functor Hom_C(-,B): C^op → Set that sends A to the set of arrows from A to B and each morphism g: A' → A to the precomposition with g. Collectively, these two directions show how the Hom construction encodes morphisms as functorial data.

In many familiar categories, the Hom sets reflect concrete structure. For the category of sets, Hom_Set(X,Y) is precisely the set of functions from X to Y. In the category of modules over a ring R, Hom_R(M,N) is the set (or, in many contexts, the R-module) of R-linear maps from M to N. In the category of abelian groups, Hom_A(G,H) is the group of group homomorphisms from G to H. Across these settings, the same categorical pattern emerges: composition of arrows corresponds to composition of the corresponding functions or maps, and the Hom construction respects the categorical structure in a natural way.

Overview

  • The Hom functor is a bifunctor: it is contravariant in its first argument and covariant in its second, giving a single framework to discuss how morphisms compose.
  • Representability plays a central role: a functor F: C^op → Set is representable if it is naturally isomorphic to Hom_C(-,A) for some object A; representable functors connect objects to all tests against them via morphisms.
  • The Yoneda lemma provides a deep bridge between objects and functors: natural transformations from Hom_C(-,A) to any F correspond to elements of F(A). This is the keystone of many categorical arguments and constructions.

Definition and construction

Given a category C, the Hom functor is built from the hom-sets Hom_C(A,B) for objects A,B ∈ C. The two standard variants are:

  • Covariant Hom in the second argument: For a fixed A, the functor Hom_C(A,-): C → Set assigns to each B the set Hom_C(A,B) and to each morphism f: B → C the function Hom_C(A,B) → Hom_C(A,C) given by precomposing with f on the right.
  • Contravariant Hom in the first argument: For a fixed B, the functor Hom_C(-,B): C^op → Set assigns to each A the set Hom_C(A,B) and to each morphism g: A' → A the function Hom_C(A,B) → Hom_C(A',B) given by postcomposing with g on the left.

When you consider both arguments together, Hom_C(-,-) is a bifunctor C^op × C → Set. This bifunctorial view encodes, in a single object, how every arrow A → B interacts with arrows to and from other objects, and it serves as a natural testing ground for many universal properties.

Examples and intuition

  • In Set, Hom_Set(X,Y) is the set of all functions X → Y. If X is a one-point set, Hom_Set(1,Y) is naturally isomorphic to Y itself, reflecting the idea that a function from a point is just the choice of an element of Y.
  • In the category of modules over a ring R, Hom_R(M,N) consists of all R-linear maps from M to N. If M = R as a left module, then Hom_R(R,N) ≅ N, since an R-linear map is determined by the image of 1 ∈ R.
  • In the category of abelian groups, Hom_G(H,K) collects all group homomorphisms H → K. When H is a cyclic group generated by one element, Hom_G(H,K) often reduces to a familiar subset of K (for example, endomorphisms of a finite cyclic group relate to multiplication by integers modulo the group's order).

These examples illustrate how the Hom construction specializes to concrete function spaces in familiar settings, while retaining a uniform categorical life across different contexts.

Properties and naturality

  • Functoriality in both arguments means that Hom_C(-,-) respects composition of arrows in each variable. This gives a systematic way to track how maps between objects induce maps between their sets of morphisms.
  • Natural evaluation: there is a canonical evaluation map ev_{A,B}: Hom_C(A,B) × A → B that sends (f,a) to f(a). This evaluation is natural in A and B and plays a crucial role in many constructions, such as adjunctions and representability.
  • Representability and adjunctions: a representable functor is one that is naturally isomorphic to a Hom functor. Representability often gives a hands-on handle on abstract properties. The Yoneda lemma formalizes the basic principle that knowing all maps into A from every other object is precisely knowing A up to canonical identifications.

Yoneda viewpoint and consequences

The Yoneda lemma states that, for any object A and any functor F: C^op → Set, there is a natural isomorphism: Natural transformations Hom_C(-,A) → F ≅ F(A).

This means that A is completely described by its relationships to every other object, as witnessed by morphisms into A. The lemma formalizes the intuition that studying how A is tested by all arrows into it already captures the essence of A within the category. It underpins the idea of representable functors and explains why Hom-centered data is so powerful in algebra, topology, and geometry.

Applications and connections

  • Representable functors and reconstruction: by identifying objects with representable Hom-functors, one can reconstruct objects from how they relate to others, a viewpoint that is central in many areas of algebraic geometry and algebraic topology.
  • Dualities and evaluation maps: the Hom construction interacts with dualities in various categories, enabling the passage between objects and their function spaces, which is a common theme in linear algebra, topology, and geometry.
  • Enriched categories: when the target of Hom is not Set but another category (e.g., abelian groups, modules), the Hom construction leads to enriched category theory, where morphism sets carry additional structure and functorial behavior reflects that structure.
  • Concrete algebraic settings: in representation theory and homological algebra, Hom functors measure maps between representations or complexes, while exact sequences and derived functors arise from studying how Hom interfaces with limits, colimits, and resolutions.

See also