Monomorphism

In category theory, a monomorphism is a morphism f: X \to Y between objects X and Y such that for any two morphisms g, h: Z \to X into X, if f \circ g = f \circ h, then g = h; this property is known as left-cancellability.^[1]^[2] This concept generalizes the idea of an injective (one-to-one) function from the category of sets to arbitrary categories, where monomorphisms capture the essence of embeddings that preserve distinct elements without introducing overlaps.^[1]^[3] In concrete categories such as the category of sets (Set), groups (Grp), or modules over a ring, monomorphisms coincide precisely with injective homomorphisms, making the term often synonymous with "injection" outside of abstract category theory.^[1]^[2] For instance, the inclusion of a subgroup H into a group G is a monomorphism in Grp, as it embeds H faithfully without collapsing elements.^[3] Monomorphisms are dual to epimorphisms—the former are left-cancellative while the latter are right-cancellative—and in many well-behaved categories like topoi, a morphism that is both monic and epic is necessarily an isomorphism.^[2]^[3] Key properties of monomorphisms include closure under composition (if f and g are monic with codomain of f matching domain of g, then g \circ f is monic) and preservation under pullbacks, which ensures they behave well in limits.^[2] They are also reflected by faithful functors and preserved by right adjoint functors, highlighting their role in constructing subobjects and embeddings in categorical constructions.^[2] In more specialized settings, such as the category of schemes, a monomorphism corresponds to an immersion where the induced diagonal map is an isomorphism, underscoring applications in algebraic geometry.^[4]

Core Concepts

Definition

In category theory, a category \mathcal{C} consists of a class of objects and, for each pair of objects A and B, a set \mathcal{C}(A, B) of morphisms (or arrows) from A to B, together with identity morphisms \mathrm{id}_A: A \to A for each object A and a composition operation that assigns to each pair of morphisms f: A \to B and g: B \to C a composite morphism g \circ f: A \to C, satisfying the associativity law (h \circ g) \circ f = h \circ (g \circ f) and the unit law f \circ \mathrm{id}_A = f = \mathrm{id}_B \circ f. A morphism f: A \to B in a category \mathcal{C} is a monomorphism, or monic morphism, if it is left-cancellative in the sense that for every object X in \mathcal{C} and every pair of morphisms g, h: X \to A, the equality f \circ g = f \circ h implies g = h. Equivalently, f is a monomorphism if the post-composition map it induces, \mathcal{C}(X, A) \to \mathcal{C}(X, B) defined by g \mapsto f \circ g, is an injective function on hom-sets for every object X in \mathcal{C}. Isomorphisms, which are morphisms possessing two-sided inverses under composition, form a special case of monomorphisms.

Terminology

The term "monomorphism" was originally introduced by the mathematician collective Nicolas Bourbaki in their treatise on abstract algebra during the 1940s and 1950s, serving as shorthand for an injective homomorphism between algebraic structures.^[3] This usage reflected the growing emphasis on structural mappings in modern mathematics, predating its broader adoption in category theory. The word derives from the Greek prefix mono- (μονό-, meaning "single" or "alone") combined with morphē (μορφή, meaning "form" or "shape"), evoking a mapping that embeds a structure uniquely without altering its form, in a manner analogous to "monotone" functions preserving a single directional order. In category theory, the term gained prominence through the work of Samuel Eilenberg and Saunders Mac Lane, who generalized it beyond injectivity to encompass left-cancellative morphisms—those satisfying the condition that if f \circ g_1 = f \circ g_2, then g_1 = g_2. Mac Lane further refined the terminology in his seminal text by introducing the abbreviation "monic" specifically for these categorical monomorphisms, to distinguish them from the underlying set-theoretic injections often implied by "monomorphism" in concrete categories. This distinction arose because, while monomorphisms coincide with injections in familiar settings like the category of sets, they do not in more abstract categories, such as the category of divisible abelian groups, where certain non-injective maps qualify as monic.^[3] Commonly shortened to "mono" in informal discussions and literature, the term "monic" is also employed in algebraic contexts but requires care to avoid confusion with "monic polynomials" in ring theory, which are defined by having a leading coefficient of 1 rather than any injectivity property. Usage variations persist across fields: in some algebraic texts, "injective morphism" is used interchangeably with monomorphism when referring to concrete categories, though this equivalence fails in general categorical settings where the left-cancellative property provides the precise criterion.^[3]

Examples

In the Category of Sets

In the category of sets, denoted Set, monomorphisms are precisely the injective functions between sets.^[2] An injective function f: A \to B maps distinct elements of A to distinct elements of B, ensuring that if f(x) = f(y), then x = y. This property aligns with the categorical definition of a monomorphism, where f is left-cancellative: for any sets C, D and morphisms g, h: C \to A, if f \circ g = f \circ h, then g = h.^[5] To see why injectivity implies this cancellativity, suppose f: A \to B is injective and f \circ g = f \circ h for some g, h: C \to A. For any c \in C, we have f(g(c)) = f(h(c)), so g(c) = h(c) by injectivity of f. Thus, g = h as functions, confirming f is a monomorphism.^[6] Conversely, if f: A \to B is a monomorphism but not injective, there exist distinct a, a' \in A with f(a) = f(a'). Consider the singleton set $1 = \{*\} and the constant maps \alpha, \alpha': 1 \to A defined by \alpha(*) = a and \alpha'(*) = a'. Then f \circ \alpha = f \circ \alpha', but \alpha \neq \alpha', contradicting the monomorphism property. Hence, every monomorphism in Set is injective.^[6] A key consequence is that if f: A \to B is a monomorphism (hence injective), the cardinality of the domain satisfies |A| \leq |B|, as the image of A under f embeds A into B without overlap.^[5] For example, the inclusion map i: \mathbb{N} \to \mathbb{Z} sending natural numbers to positive integers is a monomorphism, since it is injective and preserves distinct elements.^[2] Non-monomorphisms in Set include non-injective functions, such as constant maps from a set with more than one element. For instance, the constant function c: \{1, 2\} \to \{0\} defined by c(1) = c(2) = 0 fails to be a monomorphism: consider maps g, h: \{*\} \to \{1, 2\} with g(*) = 1 and h(*) = 2; then c \circ g = c \circ h, but g \neq h.^[5]

In Algebraic Categories

In algebraic categories, monomorphisms are structure-preserving maps that are left-cancellative, generalizing injectivity while respecting the operations of the algebraic structures involved. These categories, such as those of groups, rings, and vector spaces, are concrete, meaning they have a faithful forgetful functor to the category of sets that identifies monomorphisms with injective maps on underlying sets.^[3] In the category of groups Grp, monomorphisms coincide with injective group homomorphisms, which embed one group as a subgroup while preserving the group operation. A representative example is the inclusion homomorphism i: \mathbb{Z} \to \mathbb{Q}, where \mathbb{Z} is embedded as the subgroup of integers under addition in the additive group of rationals; this map is injective and thus a monomorphism.^[7]^[8] In the category of rings Ring, monomorphisms are injective ring homomorphisms that preserve both addition and multiplication. For instance, the inclusion i: \mathbb{Z} \to \mathbb{Z}, mapping integers to constant polynomials, is a monomorphism, though the presence of zero divisors in more general rings can complicate related concepts like epimorphisms without affecting the injectivity condition for monomorphisms.^[7]^[9] In the category of vector spaces Vect over a field, monomorphisms are precisely the injective linear transformations, which preserve linear combinations and scalar multiplication. In the finite-dimensional case, such a monomorphism from a space of dimension n to one of dimension m (with n \leq m) embeds a basis of the source into a linearly independent subset of the target, establishing key dimensional relationships.^[7] Across these algebraic categories, monomorphisms uniformly preserve the defining operations and induce injections on the underlying sets via the forgetful functor to Set.^[3]

Properties

Relation to Invertibility

In category theory, every isomorphism is a monomorphism, as an invertible morphism is left-cancellative by composing with its inverse.^[10] However, the converse does not hold in general; for instance, the inclusion morphism from the natural numbers \mathbb{N} to the integers \mathbb{Z} in the category of sets (or abelian groups) is a monomorphism but not an isomorphism, since it lacks a right inverse.^[2] A special case where monomorphisms relate closely to invertibility is that of split monomorphisms, which are monomorphisms equipped with a left inverse, known as a retraction. Formally, a morphism f: A \to B is a split monomorphism if there exists a morphism r: B \to A such that r \circ f = \mathrm{id}_A, implying that A is isomorphic to a retract of B.^[11] This structure provides a partial invertibility, as f embeds A into B in a way that allows recovery of A via r, though f itself is not necessarily an isomorphism unless B is also a retract of A.^[12] In abelian categories, split monomorphisms acquire additional significance with respect to invertibility, as they correspond precisely to inclusions of direct summands, where B decomposes as a direct sum A \oplus C for some complement C.^[13] This direct sum decomposition highlights how the left inverse r projects B onto the image of f, reinforcing the invertible-like behavior within the subcategory of summands, while general monomorphisms—such as non-split inclusions—do not yield such decompositions.^[14]

Cancellation and Composition

A monomorphism f: A \to B in a category is characterized by the left cancellation property: whenever f \circ g = f \circ h for morphisms g, h: C \to A, it follows that g = h.^[15] This property distinguishes monomorphisms from general morphisms and generalizes the injectivity of functions in the category of sets. The proof is immediate from the definition, as the equality of composites directly invokes the cancellation condition without requiring additional structure. Monomorphisms are closed under composition: if f: A \to B and g: B \to C are monomorphisms, then their composite g \circ f: A \to C is also a monomorphism.^[15] To see this, suppose (g \circ f) \circ x = (g \circ f) \circ y for morphisms x, y: D \to A. Then g \circ (f \circ x) = g \circ (f \circ y); since g is a monomorphism, this implies f \circ x = f \circ y; and since f is a monomorphism, x = y. This chaining of the cancellation property holds in any category and underscores the stability of monomorphisms under sequential application. No counterexamples exist where this fails, including in the category of posets where monomorphisms are injective order-preserving maps. In categories equipped with pullbacks, monomorphisms are stable under pullback along arbitrary morphisms: if f: A \to B is a monomorphism and p: P \to B is any morphism, then the pullback morphism f': Q \to P in the diagram

\begin{CD} Q @>>> A \\ @V f' VV @VV f V \\ P @>> p > B \end{CD}

is also a monomorphism.^[15] This preservation follows from the universal property of pullbacks combined with left cancellation: equal composites along f' lift to equal composites along f, which cancel to equality by the monicity of f. Such stability is a foundational aspect of monomorphisms in limit-rich categories, facilitating their use in constructions like subobjects.

Advanced Topics

Regular Monomorphisms

In category theory, a regular monomorphism is a monomorphism that arises as the equalizer of some pair of morphisms.^[16] Specifically, given a monomorphism f: A \to B, it is regular if there exist morphisms g, h: B \to C such that f equalizes g and h, meaning g \circ f = h \circ f, and f satisfies the universal property: for any morphism k: D \to B with g \circ k = h \circ k, there exists a unique morphism \ell: D \to A such that f \circ \ell = k.^[16] This structure emphasizes the role of regular monomorphisms in constructing exact sequences and factorizations within categories that admit finite limits.^[17] In categories with kernels, such as abelian categories, a regular monomorphism can equivalently be characterized as the kernel of the cokernel of some morphism. This equivalence highlights their stability under composition and pullbacks, distinguishing them from general monomorphisms.^[17] Representative examples illustrate these properties. In the category of sets, \mathbf{Set}, all monomorphisms—which are precisely the injective functions—are regular monomorphisms, as any injection equalizes the identity on the codomain and a suitable characteristic function distinguishing the image.^[16] In the category of groups, \mathbf{Grp}, all monomorphisms—which are inclusions of subgroups—are regular monomorphisms.^[18] In particular, inclusions of normal subgroups serve as kernels of quotient homomorphisms onto factor groups.^[19] However, not all monomorphisms are regular in every category. For instance, in the category of topological spaces, \mathbf{Top}, monomorphisms are continuous injections, but regular monomorphisms are specifically the closed embeddings. Dense embeddings, such as the inclusion of the rationals with the subspace topology into the reals, are monomorphisms but not regular, as they fail to equalize any nontrivial pair of continuous maps on the codomain.^[16] This distinction underscores the refinement provided by regularity in categories without sufficient free objects or exactness properties.^[16]

Monomorphisms in Universal Algebra

In universal algebra, monomorphisms within the category of algebras belonging to a variety are exactly the injective homomorphisms between those algebras. These maps preserve all finitary operations of the signature defining the variety and induce an isomorphism between the domain algebra and its image, which is thereby a subalgebra of the codomain. This embedding property ensures that the structure of the domain is fully preserved in the subalgebra it generates.^[20] A key result is that every monomorphism in a variety of algebras is regular, meaning it arises as the equalizer of a pair of parallel morphisms. This regularity stems from the fact that the category of algebras in any variety is itself a regular category, a structural feature tied to Birkhoff's variety theorem, which characterizes varieties as equationally defined classes closed under the formation of homomorphic images (H), subalgebras (S), and arbitrary products (P). The theorem's implications guarantee that injective homomorphisms factor appropriately in kernel pairs, confirming their status as regular monomorphisms.^[21]^[20] In specific varieties, such as that of lattices, monomorphisms act as order-embeddings: they are injective lattice homomorphisms that reflect the order, satisfying a \leq b if and only if f(a) \leq f(b). A parallel situation holds in the variety of Boolean algebras, where monomorphisms embed the algebra as a subalgebra preserving all Boolean operations, including complements. These examples highlight how monomorphisms enforce strict structural preservation in algebraic settings. In contrast, non-varietal structures, such as the category of posets under order-preserving maps, admit injective homomorphisms that fail to reflect order relations, thereby not qualifying as embeddings in an algebraic sense.^[22]^[20] Monomorphisms also influence subalgebra generation in varieties: the image of a monomorphism directly forms a subalgebra isomorphic to the domain, and in varieties admitting free algebras on sets, such embeddings facilitate the construction of free subalgebras within larger algebras when the domain consists of free generators satisfying the variety's equations. This property underscores the role of monomorphisms in extending algebraic structures while maintaining freeness conditions.^[20]