1 Definition and Basic Notation

1.1 Morphisms between objects

In a category \(\mathcal{C}\), objects are abstract entities and morphisms (also called arrows) describe structure-preserving maps between objects. A morphism \(f\) from an object \(A\) to an object \(B\) is often written as \(f:A\to B\). The key organizational idea is that for any fixed pair \((A,B)\), one collects all morphisms with that domain and codomain into a single mathematical object.

1.2 Hom-set notation \(\mathrm{Hom}(A,B)\)

The collection of all morphisms \(A\to B\) is called the hom-set and is denoted by \(\mathrm{Hom}(A,B)\) (or simply \(\mathrm{Hom}(A,B)\) when the ambient category is understood). Elements of \(\mathrm{Hom}(A,B)\) are the morphisms from \(A\) to \(B\).

1.3 Variants: \(\mathrm{Hom}_\mathcal{C}(A,B)\) and \(\mathrm{Hom}(A,B)\)

When there is potential ambiguity about the category, one writes \(\mathrm{Hom}_\mathcal{C}(A,B)\) to emphasize that the morphisms are taken in the category \(\mathcal{C}\). In many contexts—especially when a single category is being studied throughout—the shorter notation \(\mathrm{Hom}(A,B)\) is used for brevity.

2 Role in the Axioms of a Category

2.1 Composition as a function of hom-sets

A category’s axioms are most directly expressed in terms of how morphisms from different hom-sets combine. Composition takes a morphism landing in \(B\) and a morphism starting at \(A\) and produces a new morphism from \(A\) to the eventual codomain.

2.1.1 Composition map \(\mathrm{Hom}(B,C)\times \mathrm{Hom}(A,B)\to \mathrm{Hom}(A,C)\)

Formally, there is a composition operation that, for all objects \(A,B,C\), provides a function \[ \mathrm{Hom}(B,C)\times \mathrm{Hom}(A,B)\to \mathrm{Hom}(A,C). \] If \(g\in \mathrm{Hom}(B,C)\) and \(f\in \mathrm{Hom}(A,B)\), then the composite corresponds to \(g\circ f\in \mathrm{Hom}(A,C)\). This makes the “gluing” of processes explicit: morphisms are composable precisely when the codomain of the first matches the domain of the second.

2.2 Associativity of composition

The composition must satisfy associativity. If \(f\in \mathrm{Hom}(A,B)\), \(g\in \mathrm{Hom}(B,C)\), and \(h\in \mathrm{Hom}(C,D)\), then \[ (h\circ g)\circ f = h\circ (g\circ f) \] as elements of \(\mathrm{Hom}(A,D)\). Associativity ensures that nested composition expressions produce the same morphism regardless of how parentheses are placed.

2.3 Identity morphisms as hom-set elements

For every object \(A\), there is a distinguished morphism \(\mathrm{id}_A\in \mathrm{Hom}(A,A)\) that acts as a neutral element for composition. Concretely, for any \(f\in \mathrm{Hom}(A,B)\) and \(g\in \mathrm{Hom}(C,A)\), \[ f\circ \mathrm{id}_A = f,\qquad \mathrm{id}_A\circ g = g. \] Thus, identity morphisms live inside the hom-sets \(\mathrm{Hom}(A,A)\) and determine how these sets participate in composition.

3 Examples of Hom-sets

3.1 Set and functions category \(\mathbf{Set}\)

In the category \(\mathbf{Set}\), objects are sets and morphisms are functions. Therefore, \[ \mathrm{Hom}_{\mathbf{Set}}(A,B) \] is exactly the set of all functions from \(A\) to \(B\). Composition corresponds to ordinary function composition.

3.2 Group homomorphisms

In the category of groups (often written \(\mathbf{Grp}\)), objects are groups and morphisms are group homomorphisms. For groups \(G\) and \(H\), \[ \mathrm{Hom}_{\mathbf{Grp}}(G,H) \] consists of all homomorphisms \(G\to H\). Composition is the usual composition of homomorphisms, and identity morphisms are group identity maps.

3.3 Vector spaces and linear maps

In the category \(\mathbf{Vect}\) (over a fixed field), objects are vector spaces and morphisms are linear transformations. The hom-set \[ \mathrm{Hom}_{\mathbf{Vect}}(V,W) \] is the collection of all linear maps \(V\to W\). Unlike in \(\mathbf{Set}\), these hom-sets typically carry additional algebraic structure, such as being vector spaces themselves (in standard formulations).

3.4 Topological spaces and continuous maps

In the category \(\mathbf{Top}\), objects are topological spaces and morphisms are continuous maps. For spaces \(X\) and \(Y\), \[ \mathrm{Hom}_{\mathbf{Top}}(X,Y) \] is the set of all continuous functions from \(X\) to \(Y\). Here, the constraints of continuity shape which functions belong to the hom-set.

4 Hom-sets as Mathematical Structures

4.1 When hom-sets are sets vs. additional structure

The “set” in hom-set suggests that one is dealing with an ordinary set of morphisms. However, in many categories the hom-collection has extra structure compatible with composition. For instance, hom-sets in linear categories may form vector spaces, and in algebraic contexts they may form abelian groups. The distinction matters: a category can be defined so that hom-sets are merely sets, or it can be defined with hom-sets enriched by operations (addition, scalar multiplication, ordering, norms, and so on).

4.2 Categories enriched over algebraic objects

In enriched category theory, the hom-objects are taken in some ambient monoidal category rather than in the category of sets. Instead of \(\mathrm{Hom}(A,B)\) being just a set, one may have hom-objects carrying the structure of groups, modules, topological spaces, or spectra. Composition is then required to be compatible with the enrichment. This viewpoint refines the basic idea of hom-sets by keeping track of additional internal information about morphisms.

4.3 Hom-sets in abelian and additive categories

In additive categories, hom-sets are equipped with an addition operation, turning each \(\mathrm{Hom}(A,B)\) into an abelian group, and composition is bilinear with respect to that addition. In abelian categories, the existence of kernels and cokernels interacts with this additive structure. As a result, hom-sets are not only repositories of morphisms but also algebraic objects on which one can perform arithmetic operations.

5.1 Covariant hom-functor \(\mathrm{Hom}(A,-)\)

Fix an object \(A\). The construction \(X\mapsto \mathrm{Hom}(A,X)\) defines a functor that is covariant in the second variable. A morphism \(u:X\to Y\) induces a map \[ \mathrm{Hom}(A,X)\to \mathrm{Hom}(A,Y),\qquad f\mapsto u\circ f. \] Thus the hom-functor translates “post-composition by \(u\)” into a function between hom-sets.

5.2 Contravariant hom-functor \(\mathrm{Hom}(-,B)\)

Dually, fixing \(B\) and considering \(X\mapsto \mathrm{Hom}(X,B)\) yields a contravariant functor in the first variable. A morphism \(u:X\to Y\) gives an induced map \[ \mathrm{Hom}(Y,B)\to \mathrm{Hom}(X,B),\qquad g\mapsto g\circ u. \] This reflects “pre-composition by \(u\),” producing arrows with the same codomain \(B\) but changing the domain.

5.3 Currying intuition via hom-sets

Hom-sets provide a conceptual bridge to currying. In many settings, a morphism \(A\times B\to C\) (or a map depending on two inputs) can correspond to a morphism \(A\to (B\to C)\). While this requires extra structure to make precise (such as cartesian closedness), hom-sets are the starting point: they organize morphisms so that one can reformulate multi-argument maps as morphisms into function-like objects.

5.4 Functoriality and action on morphisms

The defining feature of these hom-functors is functoriality: the induced maps respect identities and composition. For example, for covariant \(\mathrm{Hom}(A,-)\), post-composing by an identity map leaves a hom-set element unchanged, and post-composing by a composite morphism corresponds to composing the induced maps. This ensures that the hom-functor behaves consistently with the categorical structure.

6 Representability and Universal Properties

6.1 Representable functors

A functor \(F:\mathcal{C}\to\mathbf{Set}\) is representable if there exists an object \(R\) such that \(F\) is naturally isomorphic to \(\mathrm{Hom}(R,-)\) (covariant representability) or \(\mathrm{Hom}(-,R)\) (contravariant representability). Representation means that values of the functor can be recovered by looking at morphisms out of (or into) a single object.

6.2 The Yoneda perspective on hom-sets

The Yoneda lemma formalizes the central idea that hom-functors capture an object’s behavior with respect to all other objects. One standard phrasing is that natural transformations from \(\mathrm{Hom}(A,-)\) correspond to elements of \(\mathrm{Hom}(A,B)\) for the appropriate target object. In practical terms, Yoneda explains why hom-sets are not merely collections of maps: they determine how objects interact.

6.3 Universal mapping properties stated using hom-sets

Many universal constructions (products, coproducts, limits, colimits, free objects) can be described using hom-set bijections. The core pattern is: an object defined by a universal property is characterized by the fact that it makes certain maps correspond uniquely to morphisms into or out of it. Expressing those correspondences as bijections between hom-sets is often the most direct way to state and prove universal properties.

7 Hom-sets in Special Categories

7.1 Preorders as thin categories

A preorder can be viewed as a category in which there is at most one morphism between any two objects. If the preorder relation is \(\le\), then one includes a morphism \(A\to B\) exactly when \(A\le B\). Consequently, \(\mathrm{Hom}(A,B)\) is either empty or a singleton, making the hom-set structure “thin.” Despite this simplicity, composition and identities mirror the transitivity and reflexivity of the preorder.

7.2 Groupoids and hom-sets with invertibility

In a groupoid, every morphism is invertible. For objects \(A\) and \(B\), any morphism \(f\in \mathrm{Hom}(A,B)\) has an inverse \(f^{-1}\in \mathrm{Hom}(B,A)\). As a result, hom-sets reflect symmetries: the existence and size of \(\mathrm{Hom}(A,B)\) are tightly constrained by the connectivity of objects within the groupoid.

7.3 Categories with zero morphisms and kernels

In categories that support a notion of zero morphisms (common in pointed or additive contexts), there is a distinguished “zero” element in each relevant hom-set. Kernels and cokernels then involve universal properties described via hom-sets: for a morphism \(f:A\to B\), the kernel object \(K\) is defined so that maps into \(K\) correspond exactly to maps into \(A\) that factor through \(f\) in the appropriate zero way. The hom-set formulation captures the exact “factorization behavior” that defines kernels.

7.4 Topological and metric-category variants

In categories of spaces with extra structure, hom-collections may be restricted or enhanced. For example, one may consider continuous maps, Lipschitz maps, or maps satisfying additional constraints. In metric-enriched contexts, morphism spaces may carry a metric or seminorm structure, influencing how hom-sets behave under limits and convergence. The hom-set viewpoint remains central: it specifies precisely which maps are allowed and how they can be composed.

8 Advanced Uses and Derived Concepts

8.1 Natural transformations between hom-functors

Hom-functors are fundamental test objects for naturality. A natural transformation between functors built from hom-sets reflects a systematic way to transform morphisms while respecting composition across all objects. Such transformations often correspond to morphisms in the underlying category via Yoneda-type arguments, turning a problem about families of functions into a problem about a single categorical arrow.

8.2 Ends and coends (high-level connection)

Ends and coends generalize universal constructions by taking limits or colimits over functor categories, frequently involving hom-functors. In many presentations, expressions built from hom-sets can be assembled into end/coend objects, which then encode universal “integrations” of morphism data. While technical details vary by context, the guiding idea is that hom-sets provide the raw components that ends and coends combine.

8.3 Adjunctions expressed via hom-sets

Adjunctions are frequently characterized by bijections between hom-sets. If \(L:\mathcal{C}\to\mathcal{D}\) and \(R:\mathcal{D}\to\mathcal{C}\) form an adjoint pair \(L\dashv R\), then there are natural bijections \[ \mathrm{Hom}_{\mathcal{D}}(L A, B)\cong \mathrm{Hom}_{\mathcal{C}}(A, R B) \] for all objects \(A\in\mathcal{C}\) and \(B\in\mathcal{D}\). This relationship explains how mapping properties in one category correspond to mapping properties in the other.

8.3.1 Hom-set bijections in adjunctions

The bijection above is not arbitrary: it is natural in both variables, meaning it is compatible with how morphisms act in each category. This naturality ensures that the correspondence between maps respects composition on both sides, giving adjunctions their powerful “transfer” principle across categorical contexts.

9 Common Confusions and Technical Caveats

9.1 Small vs. large categories (size issues)

In foundational treatments, categories may be “large,” meaning that the total collection of objects or morphisms may not form a set. Hom-sets themselves are typically required to be sets, but this can conflict with naive size assumptions. One often distinguishes between small categories (where all hom-sets and the object class fit within a chosen set-theoretic universe) and large categories that require careful foundational control.

9.2 Hom-sets vs. hom-classes

In some settings, one cannot guarantee that \(\mathrm{Hom}(A,B)\) is a set; instead it may be a proper class. Authors may then use “hom-class” terminology to reflect that the collection of morphisms is too large to be a set. Many categorical results assume set-sized hom-sets to keep standard constructions well-behaved.

9.3 Distinguishing morphisms from elements of objects

Hom-sets consist of morphisms, not elements of objects. Even though in many familiar categories (like \(\mathbf{Set}\)) morphisms are functions and elements of a set can be represented via maps from a singleton, the general categorical viewpoint does not treat elements of objects as primitive. Confusing morphisms with elements leads to incorrect interpretations of what \(\mathrm{Hom}(A,B)\) is measuring.

9.4 Duality: \(\mathrm{Hom}(A,B)\) vs. \(\mathrm{Hom}(B,A)\)

Duality in category theory swaps the direction of morphisms by passing to the opposite category. As a result, hom-sets can change from \(\mathrm{Hom}(A,B)\) to \(\mathrm{Hom}(B,A)\), reflecting different mapping directions. Many constructions have dual versions: what is described as a universal property in one direction corresponds to a colimit-style property in the opposite direction.