1 Basic definition

1.1 Homomorphisms and notation

In algebra, a Hom space typically denotes the collection of all morphisms of a specified kind from one algebraic object to another, equipped with a structure that reflects the surrounding context. For algebraic objects \(A\) and \(B\), the notation \[ \mathrm{Hom}(A,B) \] stands for the set of all structure-preserving maps \(f:A\to B\) that respect the defining operations (for example, addition, multiplication, or scalar action) according to the rules of the chosen setting.

1.2 Domain and codomain

The domain \(A\) determines the type of inputs the maps accept, while the codomain \(B\) determines the outputs and the constraints on how operations in \(A\) must be matched with operations in \(B\). Changing either object usually changes not only the elements of \(\mathrm{Hom}(A,B)\) but also the natural algebraic structure that can be placed on this collection (such as addition or scalar multiplication).

1.3 Examples in common algebraic settings

Common instances include:

  • In linear algebra, \(\mathrm{Hom}(V,W)\) is the space of all linear maps \(V\to W\).
  • In abelian group theory, \(\mathrm{Hom}(A,B)\) is the set of all group homomorphisms \(A\to B\).
  • In module theory, \(\mathrm{Hom}_R(M,N)\) denotes the module homomorphisms between \(R\)-modules \(M\) and \(N\).
  • In ring theory, \(\mathrm{Hom}(R,S)\) often refers to ring homomorphisms \(R\to S\) satisfying the relevant conventions (e.g., preserving identity if required).

These examples share the same organizing principle: morphisms are turned into a mathematical object in their own right.

2 Algebraic structure on Hom spaces

2.1 Hom as a set

At the most basic level, \(\mathrm{Hom}(A,B)\) is merely a set whose elements are the eligible morphisms from \(A\) to \(B\). This viewpoint is the most general and is naturally aligned with category theory, where morphisms between objects form a set (or sometimes a class) without assuming extra operations.

2.2 Hom as an abelian group

When the objects carry abelian-group structures and the morphisms are required to respect addition, \(\mathrm{Hom}(A,B)\) becomes an abelian group under pointwise addition: \[ (f+g)(a)=f(a)+g(a). \] The zero map serves as the identity element, and inverses are given by \( (-f)(a)=-f(a)\). This construction is standard, for instance, when \(A\) and \(B\) are abelian groups and morphisms are group homomorphisms.

2.3 Hom as a vector space

If \(A\) and \(B\) are vector spaces over a field \(k\) and morphisms are required to be \(k\)-linear, then \(\mathrm{Hom}(V,W)\) is a vector space over \(k\) with the usual pointwise operations: \[ (\lambda f)(v)=\lambda\, f(v). \] Thus linear maps can be studied using linear-algebraic tools, such as basis expansions and linear independence arguments.

2.4 Hom as a module

For modules over a ring \(R\), \(\mathrm{Hom}_R(M,N)\) often carries the structure of an \(R\)-module (or at least an abelian group), depending on the side on which scalars act and on the module conventions. In many standard cases—e.g., right \(R\)-modules mapping into an appropriate module—scalar multiplication is defined compatibly with the structure on \(N\), producing a module structure on homomorphisms.

3 Hom spaces in linear algebra

3.1 Spaces of linear maps

Let \(V\) and \(W\) be vector spaces over a field \(k\). The set of all linear maps from \(V\) to \(W\) forms a vector space: \[ \mathrm{Hom}(V,W)=\{f:V\to W \mid f \text{ is linear}\}. \] Elements of this space represent linear transformations, and the operations of addition and scalar multiplication are performed pointwise on outputs in \(W\).

3.2 Matrix representation

Given bases for \(V\) and \(W\), each linear map \(f:V\to W\) corresponds to a matrix. If \(\dim V=n\) and \(\dim W=m\), then matrices are \(m\times n\), and the correspondence between linear maps and matrices is an isomorphism of vector spaces (after fixing chosen bases). This provides a computational model for \(\mathrm{Hom}(V,W)\) and connects abstract linear maps to explicit arithmetic.

3.3 Dimension of Hom spaces

If \(\dim V=n\) and \(\dim W=m\), then \[ \dim \mathrm{Hom}(V,W)=mn, \] assuming \(V\) and \(W\) are finite-dimensional. The count arises because a linear map is determined by its values on a basis of \(V\), each value lying in \(W\), giving \(m\) degrees of freedom for each of \(n\) basis vectors.

3.4 Dual spaces as Hom spaces

A central example is the dual space \(V^*\), defined as the set of linear functionals \(V\to k\). This is precisely \[ V^*=\mathrm{Hom}(V,k). \] Duality therefore fits the general pattern of Hom constructions, and many properties of dual spaces can be rephrased in terms of Hom spaces.

4 Hom spaces in group theory

4.1 Group homomorphisms

For groups \(G\) and \(H\), \(\mathrm{Hom}(G,H)\) denotes the set of all group homomorphisms \(G\to H\). Depending on whether additional conditions are imposed (such as preserving a specified structure or identity conventions), the elements of \(\mathrm{Hom}(G,H)\) are the maps that respect the group operation: \[ f(gg')=f(g)f(g'). \]

4.2 Endomorphism and automorphism sets

Special cases include:

  • Endomorphisms: \(\mathrm{Hom}(G,G)\), the set of all homomorphisms from \(G\) to itself.
  • Automorphisms: those endomorphisms that are bijective, forming a group under composition.

The endomorphisms are not automatically a group unless additional structure is present, but they are always closed under composition (forming a monoid).

4.3 Hom sets between cyclic groups

When \(G\) is cyclic, homomorphisms are determined by the image of a generator. For example, if \(G=\mathbb{Z}/n\mathbb{Z}\) and \(H\) is a group, a homomorphism corresponds to choosing an element \(h\in H\) such that \(h^n=e\), where \(e\) is the identity in \(H\). This provides concrete classification in many elementary cases and illustrates how \(\mathrm{Hom}(G,H)\) can often be described in terms of internal constraints on the codomain.

5 Hom spaces in module theory

5.1 Module homomorphisms

Let \(R\) be a ring and let \(M,N\) be \(R\)-modules. The set \(\mathrm{Hom}_R(M,N)\) consists of \(R\)-linear maps: \[ f(am+bn)=af(m)+bf(n), \] with the exact scalar and additive compatibility depending on whether the modules are left or right modules and on the conventions chosen. This set carries a natural additive structure and often an \(R\)-module structure as described earlier.

5.2 Hom functor

In categorical language, the assignment \[ N\mapsto \mathrm{Hom}_R(M,N) \] defines a functor from \(R\)-modules to abelian groups when \(M\) is fixed. Similarly, \(M\mapsto \mathrm{Hom}_R(M,N)\) defines a dual-type functor (typically contravariant in the first argument). These functors are fundamental because they convert algebraic problems about modules into problems about maps.

5.3 Exactness properties

Hom functors interact with exact sequences in a way that is described using the language of exactness. In many settings:

  • \(\mathrm{Hom}_R(M,-)\) is left exact, meaning it preserves exactness at the earlier terms of a short exact sequence.
  • Its behavior on the right end is controlled by derived functors, such as \(\mathrm{Ext}\).

These properties make Hom spaces a key tool in tracking how module morphisms behave under quotienting and extension.

5.4 Projective and injective modules

Certain module classes are defined by how Hom functors preserve exactness:

  • A module \(P\) is projective if \(\mathrm{Hom}_R(P,-)\) is exact (not only left exact).
  • A module \(I\) is injective if \(\mathrm{Hom}_R(-,I)\) is exact (again, full exactness).

Thus, projectivity and injectivity can be characterized through the structure of Hom spaces and the way they respond to exact sequences.

6 Hom spaces in ring theory

6.1 Ring homomorphisms

For rings \(R\) and \(S\), \(\mathrm{Hom}(R,S)\) commonly denotes ring homomorphisms \(R\to S\). Depending on conventions, such homomorphisms may be required to preserve identity elements, additive structure, and multiplication. When identity preservation is demanded, maps typically satisfy \(f(1_R)=1_S\).

6.2 Hom sets of ideals and modules

Rings often interact with their ideals through module structures. If \(I\) and \(J\) are ideals viewed as modules over \(R\), then \(\mathrm{Hom}_R(I,J)\) refers to module homomorphisms between these ideal-modules. This formulation is used in deformation questions, extension problems, and in studying how ring maps restrict to substructures.

6.3 Algebra homomorphisms

If \(A\) and \(B\) are algebras over a field \(k\) (or over a commutative ring), an algebra homomorphism is typically a ring homomorphism that also respects the scalar structure. Hom spaces of algebra homomorphisms are then studied similarly, with the added requirement that maps be compatible with the algebra framework, not merely with the underlying rings.

7 Category-theoretic viewpoint

7.1 Hom-sets in a category

In a category \(\mathcal{C}\), the notation \(\mathrm{Hom}_{\mathcal{C}}(A,B)\) refers to the Hom set of morphisms from object \(A\) to object \(B\). The category axioms ensure that these Hom sets fit together coherently with composition and identities, and many abstract statements about algebra reduce to properties of Hom sets.

7.2 Composition of morphisms

Given morphisms \(f:A\to B\) and \(g:B\to C\), category theory supplies a composition operation producing \(g\circ f:A\to C\). This composition is associative: \[ (h\circ g)\circ f=h\circ(g\circ f), \] which allows Hom sets to be organized into a structured system rather than isolated collections of maps.

7.3 Identity morphisms

Each object \(A\) has an identity morphism \(\mathrm{id}_A:A\to A\) such that for any \(f:A\to B\) and \(g:C\to A\), \[ f\circ \mathrm{id}_A=f,\quad \mathrm{id}_A\circ g=g. \] The existence of identities is what makes Hom sets compatible with the categorical framework.

7.4 Functorial behavior of Hom

Hom constructions behave systematically under functors. Roughly, given a functor between categories, morphisms correspond in a way that preserves composition and identities, inducing relationships among Hom sets. This functoriality is essential in abstract dualities and in the formulation of universal properties.

8.1 Endomorphism spaces

The endomorphism space \(\mathrm{End}(A)=\mathrm{Hom}(A,A)\) collects self-maps of \(A\). In many contexts it carries additional structure: for vector spaces it is an algebra under composition; for modules it forms a ring under addition and composition; for groups it forms a monoid under composition.

8.2 Automorphism groups

The subset of invertible elements inside \(\mathrm{End}(A)\) yields the automorphism group \(\mathrm{Aut}(A)\). Automorphisms capture the symmetries of \(A\) that preserve the relevant operations. They are widely used in classification questions and in describing how objects relate under isomorphism.

8.3 Tensor-Hom adjunction

A standard relationship in linear settings is the tensor-Hom adjunction, linking tensor products and Hom spaces. Conceptually, it expresses a natural correspondence between maps from a tensor product and maps into a Hom space, allowing one to shift between “input-combining” and “output-assigning” viewpoints. This adjunction is a recurring theme in algebra and category theory.

8.4 Internal Hom

In categories with enough structure (such as closed monoidal categories), one can form an internal Hom object \(\underline{\mathrm{Hom}}(A,B)\) rather than a mere external set of morphisms. Internal Hom packages morphisms into the category itself, enabling richer constructions such as currying and generalized duality.

9 Applications

9.1 Representation theory

Representation theory studies how groups or algebras act on vector spaces (or modules). Hom spaces arise naturally because intertwiners between representations—linear maps commuting with the action—form Hom spaces. These spaces help measure when representations are equivalent and how they decompose into simpler constituents.

9.2 Homological algebra

In homological algebra, Hom spaces are used to define cochains, chain maps, and derived invariants. The behavior of Hom functors on exact sequences leads to tools such as Ext and Tor, which quantify extensions and obstructions that ordinary Hom sets alone cannot capture.

9.3 Universal properties

Universal properties often characterize objects by the existence and uniqueness of morphisms into or out of them. Since such morphisms live in Hom spaces, the language of Hom sets becomes a central organizing mechanism for statements like representing functors or expressing adjunctions in a universal way.

9.4 Classification problems

Many classification tasks reduce to analyzing Hom spaces: for example, describing all maps satisfying certain constraints, determining isomorphism classes via invariants built from morphisms, or classifying extensions by examining Hom and related functors. In each case, turning morphisms into structured objects provides a systematic pathway to classification.