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In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Although many examples of morphisms are structure-preserving maps, morphisms need not be maps, but they can be composed in a way that is similar to function composition.

Morphisms and objects are constituents of a category. Morphisms, also called maps or arrows, relate two objects called the source and the target of the morphism. There is a partial operation, called composition, on the morphisms of a category that is defined if the target of the first morphism equals the source of the second morphism. The composition of morphisms behaves like function composition (associativity of composition when it is defined, and existence of an identity morphism for every object).

Morphisms and categories recur in much of contemporary mathematics. Originally, they were introduced for homological algebra and algebraic topology. They belong to the foundational tools of Grothendieck's scheme theory, a generalization of algebraic geometry that applies also to algebraic number theory.

Definition

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A category C consists of two classes, one of objects and the other of morphisms. There are two objects that are associated to every morphism, the source and the target. A morphism f from X to Y is a morphism with source X and target Y; it is commonly written as f : X → Y or X f→ Y the latter form being better suited for commutative diagrams.

For many common categories, an object is a set (often with some additional structure) and a morphism is a function from an object to another object. Therefore, the source and the target of a morphism are often called domain and codomain respectively.

Morphisms are equipped with a partial binary operation, called composition (partial because the composition is not necessarily defined over every pair of morphisms of a category). The composition of two morphisms f and g is defined precisely when the target of f is the source of g, and is denoted g ∘ f (or sometimes simply gf). The source of g ∘ f is the source of f, and the target of g ∘ f is the target of g. The composition satisfies two axioms:

Identity
For every object X, there exists a morphism idX : X → X called the identity morphism on X, such that for every morphism f : A → B we have idB ∘ f = f = f ∘ idA.
Associativity
h ∘ (g ∘ f) = (h ∘ g) ∘ f whenever all the compositions are defined, i.e. when the target of f is the source of g, and the target of g is the source of h.

For a concrete category (a category in which the objects are sets, possibly with additional structure, and the morphisms are structure-preserving functions), the identity morphism is just the identity function, and composition is just ordinary composition of functions.

The composition of morphisms is often represented by a commutative diagram. For example,

The collection of all morphisms from X to Y is denoted HomC(X, Y) or simply Hom(X, Y) and called the hom-set between X and Y. Some authors write MorC(X, Y), Mor(X, Y) or C(X, Y). The term hom-set is something of a misnomer, as the collection of morphisms is not required to be a set; a category where Hom(X, Y) is a set for all objects X and Y is called locally small. Because hom-sets may not be sets, some people prefer to use the term "hom-class".

The domain and codomain are in fact part of the information determining a morphism. For example, in the category of sets, where morphisms are functions, two functions may be identical as sets of ordered pairs, while having different codomains. The two functions are distinct from the viewpoint of category theory. Many authors require that the hom-classes Hom(X, Y) be disjoint. In practice, this is not a problem because if this disjointness does not hold, it can be assured by appending the domain and codomain to the morphisms (say, as the second and third components of an ordered triple).

Some special morphisms

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Monomorphisms and epimorphisms

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A morphism f : X → Y is called a monomorphism if f ∘ g1 = f ∘ g2 implies g1 = g2 for all morphisms g1, g2 : Z → X. A monomorphism can be called a mono for short, and we can use monic as an adjective.[1] A morphism f has a left inverse or is a split monomorphism if there is a morphism g : Y → X such that g ∘ f = idX. Thus f ∘ g : Y → Y is idempotent; that is, (f ∘ g)2 = f ∘ (g ∘ f) ∘ g = f ∘ g. The left inverse g is also called a retraction of f.[1]

Morphisms with left inverses are always monomorphisms (f-1l ∘ f ∘ g1 = f-1l ∘ f ∘ g2 implies g1 = g2, where f-1l is the left inverse of f), but the converse is not true in general; a monomorphism may fail to have a left inverse. In concrete categories, where morphisms are functions, a morphism that has a left inverse is injective, and a morphism that is injective is a monomorphism. In concrete categories, monomorphisms are often, but not always, injective; thus the condition of being an injection is stronger than that of being a monomorphism, but weaker than that of being a split monomorphism.

Dually to monomorphisms, a morphism f : X → Y is called an epimorphism if g1 ∘ f = g2 ∘ f implies g1 = g2 for all morphisms g1, g2 : Y → Z. An epimorphism can be called an epi for short, and we can use epic as an adjective.[1] A morphism f has a right inverse or is a split epimorphism if there is a morphism g : Y → X such that f ∘ g = idY. The right inverse g is also called a section of f.[1] Morphisms having a right inverse are always epimorphisms (g1 ∘ f ∘ f-1r = g2 ∘ f ∘ f-1r implies g1 = g2 where f-1r is the right inverse of f), but the converse is not true in general, as an epimorphism may fail to have a right inverse.

If a monomorphism f splits with left inverse g, then g is a split epimorphism with right inverse f. In concrete categories, a function that has a right inverse is surjective.[2] Thus, in concrete categories, epimorphisms are often, but not always, surjective. The condition of being a surjection is stronger than that of being an epimorphism, but weaker than that of being a split epimorphism. In the category of sets, the statement that every surjection has a section is equivalent to the axiom of choice.

A morphism that is both an epimorphism and a monomorphism is called a bimorphism.

For example, in the category of vector spaces over a fixed field, injective morphisms, monomorphisms and split homomorphisms are the same, as well as surjective morphisms, epimorphisms and split epimorphisms.

In the category of commutative rings, monomorphisms and injective morphisms are the same, while the injection from ⁠⁠ into ⁠⁠ is an epimorphism that is not surjective; it is neither a split epimorphism nor a split monomorphism. (See Homomorphism#Special homomorphisms for more details and proofs.)

Isomorphisms

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A morphism f : X → Y is called an isomorphism if there exists a morphism g : Y → X such that f ∘ g = idY and g ∘ f = idX. If a morphism has both left-inverse and right-inverse, then the two inverses are equal, so f is an isomorphism, and g is called simply the inverse of f. Inverse morphisms, if they exist, are unique. The inverse g is also an isomorphism, with inverse f. Two objects with an isomorphism between them are said to be isomorphic or equivalent.

While every isomorphism is a bimorphism, a bimorphism is not necessarily an isomorphism. For example, in the category of commutative rings the inclusion Z → Q is a bimorphism that is not an isomorphism. However, any morphism that is both an epimorphism and a split monomorphism, or both a monomorphism and a split epimorphism, must be an isomorphism. A category, such as a Set, in which every bimorphism is an isomorphism is known as a balanced category.

Endomorphisms and automorphisms

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A morphism f : X → X (that is, a morphism with identical source and target) is an endomorphism of X. A split endomorphism is an idempotent endomorphism f if f admits a decomposition f = h ∘ g with g ∘ h = id. In particular, the Karoubi envelope of a category splits every idempotent morphism.

An automorphism is a morphism that is both an endomorphism and an isomorphism. In every category, the automorphisms of an object always form a group, called the automorphism group of the object.

Examples

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For more examples, see Category theory.

See also

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Notes

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Morphism

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In , a morphism is an arrow or structure-preserving mapping from one object to another within a category, abstracting the notion of a that preserves operations between algebraic structures. Introduced by mathematicians and in 1945, the concept forms the foundational relation in categories, enabling the study of mathematical structures through their interconnections rather than intrinsic properties alone. A category consists of a collection of objects and a collection of morphisms between them, with morphisms satisfying key axioms: they can be composed associatively (if f:A→Bf: A \to B and g:B→Cg: B \to C, then g∘f:A→Cg \circ f: A \to C), and each object has an identity morphism idA:A→A\mathrm{id}_A: A \to A that acts as a neutral element for composition. In concrete categories like the (Set), morphisms are ordinary functions; in the (Top), they are continuous functions; and in the category of vector spaces (Vec), they are linear transformations. These examples illustrate how morphisms generalize mappings across diverse mathematical domains, emphasizing relational properties over absolute ones. Morphisms admit special types that highlight injectivity, surjectivity, and bijectivity in categorical terms: a is left-cancellative (injective-like), an is right-cancellative (surjective-like), and an is a morphism with an inverse, establishing equivalence between objects. Endomorphisms map an object to itself, while automorphisms are isomorphisms from an object to itself, playing crucial roles in and studies. Beyond basic categories, the notion extends to higher-dimensional settings like 2-categories, where 2-morphisms relate morphisms, underpinning advanced areas such as and .

Fundamentals

Definition

In , a morphism is an connecting two objects in a category, typically denoted as f:A→Bf: A \to B, where AA and BB are objects, indicating that ff maps from AA to BB. Morphisms are required to respect the underlying structure of the category; for instance, in concrete categories like the , they correspond to functions that preserve the specified relations or operations between objects, with each morphism having a well-defined domain (the source object) and (the target object). The term "morphism" was introduced by and in 1945 to describe these arrows within the framework of . A simple morphism can be illustrated diagrammatically as follows:

A ──f──> B

A ──f──> B

where the arrow ff represents the morphism from object AA to object BB.

Role in Categories

In , a category is formally defined as a consisting of a class of objects and a class of morphisms between those objects, equipped with operations of composition and identity that satisfy certain axioms. The morphisms, often depicted as directed arrows from a source object to a target object, serve as the fundamental relations that connect objects within the category, allowing for the abstraction of structural similarities across diverse mathematical domains such as sets, groups, and topological spaces. This relational role enables to unify concepts by focusing on how objects interact via morphisms rather than their internal details, facilitating comparisons and transformations between seemingly unrelated . The key axioms governing categories ensure that morphisms behave consistently under composition and identity. Specifically, composition of morphisms is associative, meaning that for any composable morphisms f:A→Bf: A \to B, g:B→Cg: B \to C, and h:C→Dh: C \to D, the equation (h∘g)∘f=h∘(g∘f)(h \circ g) \circ f = h \circ (g \circ f) holds. Additionally, every object AA has an identity morphism idA:A→A\mathrm{id}_A: A \to A such that for any morphism f:A→Bf: A \to B, idB∘f=f=f∘idA\mathrm{id}_B \circ f = f = f \circ \mathrm{id}_A. These axioms, introduced by Eilenberg and Mac Lane, provide the minimal framework for morphisms to form a coherent system of interconnections, abstracting away set-theoretic foundations to emphasize relational patterns. Categories are distinguished as concrete or abstract based on their relationship to the category of sets, Set\mathbf{Set}. A concrete category is one equipped with a faithful forgetful functor to Set\mathbf{Set}, where objects are structured sets and morphisms are structure-preserving functions, allowing explicit set-theoretic realizations. In contrast, abstract categories may lack such a direct embedding into Set\mathbf{Set}, existing purely as collections of objects and morphisms without underlying sets, which highlights the generality of the categorical framework in capturing relational structures beyond concrete models. This distinction underscores the role of morphisms in enabling both tangible and purely conceptual interconnections within categories.

Special Types

Monomorphisms and Epimorphisms

In , a , or mono, is a morphism m:A→Bm: A \to B in a category C\mathcal{C} that satisfies the following : for any object XX in C\mathcal{C} and any pair of morphisms f,g:X→Af, g: X \to A, if m∘f=m∘gm \circ f = m \circ g, then f=gf = g. This condition means that mm is left-cancellative with respect to composition, ensuring that it embeds AA into BB in a way that distinguishes distinct morphisms into AA. In categories—those equipped with a faithful to the —monomorphisms often coincide with injective morphisms on the underlying sets. Dually, an , or epi, is a morphism e:A→Be: A \to B such that for any object XX in C\mathcal{C} and any pair of morphisms f,g:B→Xf, g: B \to X, if f∘e=g∘ef \circ e = g \circ e, then f=gf = g. This right-cancellative property captures a universal notion of ee "covering" BB sufficiently to equate any subsequent morphisms that agree after composition with ee. Like monomorphisms, epimorphisms align with surjective morphisms in many concrete categories, such as the (Set\mathbf{Set}) and the category of groups (Grp\mathbf{Grp}). However, this alignment does not hold universally across all categories. For instance, in the category of rings (Ring\mathbf{Ring}), the inclusion morphism Z→Q\mathbb{Z} \to \mathbb{Q} is an epimorphism because any two ring homomorphisms from Q\mathbb{Q} to another ring that agree on Z\mathbb{Z} must be equal, yet it is not surjective as a function on underlying sets. Similarly, while monomorphisms are typically injective in standard concrete categories like Set\mathbf{Set}, Grp\mathbf{Grp}, and the category of topological spaces (Top\mathbf{Top}), there exist categories where monomorphisms fail to be injective on underlying sets. These examples illustrate how the abstract universal properties of monomorphisms and epimorphisms generalize injectivity and surjectivity without relying on set-theoretic assumptions.

Isomorphisms

In , an isomorphism is a morphism i:A→Bi: A \to B that admits a two-sided inverse morphism i−1:B→Ai^{-1}: B \to A such that the compositions satisfy i∘i−1=idBi \circ i^{-1} = \mathrm{id}_B and i−1∘i=idAi^{-1} \circ i = \mathrm{id}_A. This invertibility ensures that the morphism establishes a perfect structural correspondence between the objects AA and BB. A morphism qualifies as an if it is both a and an , and possesses this two-sided inverse, distinguishing it from one-sided injectivity or surjectivity alone. The inverse property guarantees that the morphism is bijective in the categorical sense, preserving all relevant structure without loss or redundancy. Isomorphic objects AA and BB are indistinguishable within their category, meaning they share identical categorical properties up to relabeling via the ; any or relation involving one can be equivalently translated to the other. This on objects underscores the abstraction in , where isomorphisms capture "sameness" without requiring identical implementations. Objects related by an isomorphism are commonly denoted A≅BA \cong B, with the symbol ≅\cong indicating this structural equivalence. The collection of all isomorphisms forms a key subclass of morphisms, closed under composition and inversion.

Endomorphisms and Automorphisms

In category theory, an endomorphism of an object AA in a category C\mathcal{C} is a morphism e:A→Ae: A \to A. The collection of all endomorphisms of AA, denoted EndC(A)\mathrm{End}_{\mathcal{C}}(A), equipped with the operation of composition and the identity morphism as the unit element, forms a monoid known as the endomorphism monoid. This structure arises naturally because composition of morphisms is associative and the identity acts as a neutral element for self-maps. An is a special case of an that is also an , meaning it admits an inverse morphism within the category. Thus, an α:A→A\alpha: A \to A is an invertible self-map preserving the categorical structure. The set of all automorphisms of AA, denoted AutC(A)\mathrm{Aut}_{\mathcal{C}}(A), forms a group under composition, with the identity as the neutral element and inverses provided by the categorical inverses. This group structure highlights the invertible symmetries inherent to the object AA. Automorphisms can be viewed as a subgroup of the endomorphism monoid, consisting precisely of its units. In enriched categories, particularly those enriched over abelian groups, the endomorphism monoid acquires additional structure to become a ring, called the endomorphism ring. For instance, in the category of left modules over a ring RR, the endomorphism ring EndR(M)\mathrm{End}_R(M) of a module MM consists of all RR-linear maps M→MM \to M, with ring operations defined by pointwise and composition as multiplication. This ring plays a central role in module theory, as it governs the actions and decompositions of modules; for example, if MM is a projective module, EndR(M)\mathrm{End}_R(M) often exhibits properties like being a or having specific ideals corresponding to submodules. The structure of such rings can vary widely, from commutative rings in simple cases to non-commutative division rings in others, reflecting the complexity of the underlying module. Automorphisms are fundamentally linked to the notion of in mathematical structures, as the Aut(A)\mathrm{Aut}(A) encodes all ways to map AA to itself while preserving its internal relations and operations. In algebraic contexts, such as groups or vector spaces, elements of Aut(A)\mathrm{Aut}(A) represent rigid motions or transformations that leave the object invariant, providing a quantitative measure of its ; for example, the of a captures its inner and outer symmetries through conjugations and beyond. This perspective extends the categorical to applications, where Aut(A)\mathrm{Aut}(A) often determines classes or rigidity properties of AA.

Properties and Operations

Composition

In category theory, the composition of morphisms is a partial binary operation that combines compatible morphisms between objects in a category. For a category C\mathcal{C}, given objects A,B,C∈CA, B, C \in \mathcal{C} and morphisms f:A→Bf: A \to B, g:B→Cg: B \to C, the composite g∘f:A→Cg \circ f: A \to C exists and is uniquely determined by the category's structure. This operation formalizes the chaining of arrows, where the output of the first morphism feeds into the input of the second. Composition requires precise matching of domains and codomains: it is defined only when the codomain of the first morphism equals the domain of the second, ensuring compatibility in the sequence. This partiality reflects the category's focus on typed arrows, preventing invalid juxtapositions. Without this requirement, the operation would not preserve the intended relational structure between objects. A key property of composition is its associativity. For composable morphisms f:A→Bf: A \to B, g:B→Cg: B \to C, and h:C→Dh: C \to D, the equation (h∘g)∘f=h∘(g∘f)(h \circ g) \circ f = h \circ (g \circ f) holds, allowing unambiguous definition of longer chains without parentheses. This axiom, alongside the existence of identity morphisms as neutral elements, ensures that each endomorphism set End(A) = Hom(A,A) forms a monoid under composition. Functoriality extends composition across categories: a F:C→DF: \mathcal{C} \to \mathcal{D} preserves it via F(g∘f)=F(g)∘F(f)F(g \circ f) = F(g) \circ F(f) for all composable f,gf, g in C\mathcal{C}. This ensures that mappings between categories respect the sequential nature of morphisms, facilitating comparisons and transformations while maintaining structural integrity.

Identity Morphisms

In , for each object AA in a category C\mathcal{C}, the identity morphism \idA:A→A\id_A: A \to A (often denoted 1A1_A) is defined as the morphism satisfying f∘\idA=ff \circ \id_A = f and \idB∘f=f\id_B \circ f = f for every morphism f:A→Bf: A \to B. This property positions the identity morphism as the neutral element with respect to composition in the category. Identity morphisms are unique: if i:A→Ai: A \to A and j:A→Aj: A \to A both satisfy the identity axioms, then i=ji = j. The standard notation \idA\id_A or 1A1_A emphasizes this role, distinguishing it from other endomorphisms on AA. Identity morphisms play a key role in characterizing certain universal objects. In the coslice category A/CA / \mathcal{C}, the identity morphism \idA:A→A\id_A: A \to A serves as the initial object, with a unique morphism from it to any other object in the coslice. Dually, in the slice category C/A\mathcal{C} / A, \idA\id_A is the terminal object. In pointed categories, which possess a zero object that is both initial and terminal, identity morphisms contribute to the definition of zero morphisms. The zero morphism from any object XX to YY is the composite X→0→YX \to 0 \to Y via the unique morphisms to and from the zero object 00; notably, the identity \id0:0→0\id_0: 0 \to 0 coincides with this zero morphism on 00.

Examples in Mathematical Structures

Sets and Functions

In the category Set, the objects are sets and the morphisms are functions between those sets, with no additional structure required beyond the basic mapping from elements of one set to elements of another. This category exemplifies the abstract notion of a morphism in a setting, where functions serve as arrows that preserve the inherent equality of elements without imposing axioms like continuity or properties. Within Set, monomorphisms correspond to injective functions, which embed one set into another without collapsing distinct elements; epimorphisms are surjective functions that cover the entire ; and isomorphisms are bijective functions that establish a one-to-one correspondence between sets. These align with the general categorical definitions, where injectivity ensures left-cancellability, surjectivity ensures right-cancellability, and bijectivity provides invertible arrows. Composition of morphisms in Set follows the standard rule for : for functions f:A→Bf: A \to B and g:B→Cg: B \to C, the composite g∘f:A→Cg \circ f: A \to C is defined by (g∘f)(x)=g(f(x))(g \circ f)(x) = g(f(x)) for all x∈Ax \in A. This operation is associative, reflecting the category's structure, and enables the formation of chains of mappings that maintain the set-theoretic relations. Examples of endomorphisms in Set include constant functions, which map every element of a set SS to a fixed element in SS, such as the function c:S→Sc: S \to S where c(x)=ac(x) = a for some fixed a∈Sa \in S and all x∈Sx \in S. The identity morphism on a set SS is the idS:S→S\mathrm{id}_S: S \to S defined by idS(x)=x\mathrm{id}_S(x) = x for all x∈Sx \in S, serving as the neutral element for composition.

Groups and Homomorphisms

In the category Grp, the objects are groups and the are . A ϕ:(G,⋅)→(H,∗)\phi: (G, \cdot) \to (H, *) is a function that preserves the group operation, satisfying ϕ(g1⋅g2)=ϕ(g1)∗ϕ(g2)\phi(g_1 \cdot g_2) = \phi(g_1) * \phi(g_2) for all g1,g2∈Gg_1, g_2 \in G. This ensures that the is maintained under the mapping, distinguishing from mere set functions by requiring compatibility with the . Within Grp, monomorphisms correspond to injective homomorphisms, epimorphisms to surjective homomorphisms, and isomorphisms to bijective homomorphisms. These align with the categorical notions, where injectivity provides left-cancellability, surjectivity right-cancellability, and bijectivity invertibility while preserving the group operation. The kernel of a homomorphism ϕ:G→H\phi: G \to H, denoted ker⁡(ϕ)={g∈G∣ϕ(g)=eH}\ker(\phi) = \{g \in G \mid \phi(g) = e_H\}, forms a of GG. The image im⁡(ϕ)={ϕ(g)∣g∈G}\operatorname{im}(\phi) = \{\phi(g) \mid g \in G\} is a of HH. These structures enable key results, such as the first isomorphism theorem, which states that if ϕ:G→H\phi: G \to H is a group homomorphism, then G/ker⁡(ϕ)≅im⁡(ϕ)G / \ker(\phi) \cong \operatorname{im}(\phi). Automorphisms in Grp are group isomorphisms ϕ:G→G\phi: G \to G, which are bijective homomorphisms with inverses that are also homomorphisms. A prominent example is , generated by conjugation: for g∈Gg \in G, the map ϕg:h↦ghg−1\phi_g: h \mapsto g h g^{-1} is an automorphism, and the inner automorphism group Inn⁡(G)\operatorname{Inn}(G) consists of all such maps. Endomorphisms of a group GG are ϕ:G→G\phi: G \to G, and under composition, they form the endomorphism monoid End⁡(G)\operatorname{End}(G), which includes the identity as the unit element. Composition of homomorphisms yields another homomorphism, supporting the monoidal structure.

Topological Spaces

In the category Top, the objects are s, and the morphisms are continuous functions between them. A function f:X→Yf: X \to Y between topological spaces XX and YY is continuous if, for every V⊆YV \subseteq Y, the preimage f−1(V)f^{-1}(V) is an in XX. This definition ensures that the morphism preserves the topological structure by mapping "nearby" points in XX to "nearby" points in YY in a way compatible with the open sets defining the topologies. Composition of continuous functions is again continuous, and the on any topological space serves as the identity morphism. Homeomorphisms in Top are the isomorphisms, consisting of continuous bijections f:X→Yf: X \to Y whose inverses f−1:Y→Xf^{-1}: Y \to X are also continuous. Two topological spaces related by a homeomorphism are called homeomorphic and are considered topologically equivalent, meaning they share all topological properties such as connectedness, , and Hausdorff separation. For instance, the identity map idX:X→X\mathrm{id}_X: X \to X is always a homeomorphism, as it is continuous, bijective, and self-inverse. This equivalence captures the idea that homeomorphic spaces can be continuously deformed into one another without tearing or gluing. Embeddings provide a way to include one as a subspace of another while preserving its . An is an injective continuous f:X→Yf: X \to Y such that ff is a from XX onto its image f(X)f(X), where f(X)f(X) is equipped with the induced from YY. In Top, embeddings coincide with the regular monomorphisms, which are monomorphisms that are also the equalizers of some pair of parallel arrows; however, general monomorphisms are merely the injective continuous maps, which may not induce the subspace topology on their images if the topology on XX is coarser than expected. Constant maps offer simple examples of morphisms in Top: for any topological spaces XX and YY with a fixed point y0∈Yy_0 \in Y, the function c:X→Yc: X \to Y defined by c(x)=y0c(x) = y_0 for all x∈Xx \in X is continuous, as the preimage of any V⊆YV \subseteq Y is either empty (if y0∉Vy_0 \notin V) or all of XX (if y0∈Vy_0 \in V), both of which are open. In non-Hausdorff spaces, such maps illustrate how morphisms can collapse distinct points without violating continuity, highlighting the flexibility of topological structure preservation beyond metric constraints.
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