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In the context of
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
or
universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, ...
, a monomorphism is an
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, a monomorphism (also called a monic morphism or a mono) is a
left-cancellative In mathematics, the notion of cancellative is a generalization of the notion of invertible. An element ''a'' in a magma has the left cancellation property (or is left-cancellative) if for all ''b'' and ''c'' in ''M'', always implies that . A ...
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
. That is, an arrow such that for all objects and all morphisms , : f \circ g_1 = f \circ g_2 \implies g_1 = g_2. Monomorphisms are a categorical generalization of
injective function In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
s (also called "one-to-one functions"); in some categories the notions coincide, but monomorphisms are more general, as in the examples below. The
categorical dual In category theory, a branch of mathematics, duality is a correspondence between the properties of a category ''C'' and the dual properties of the opposite category ''C''op. Given a statement regarding the category ''C'', by interchanging the sou ...
of a monomorphism is an
epimorphism In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f \ ...
, that is, a monomorphism in a category ''C'' is an epimorphism in the
dual category In category theory, a branch of mathematics, the opposite category or dual category ''C''op of a given category ''C'' is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields t ...
''C''op. Every
section Section, Sectioning or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section sign ...
is a monomorphism, and every retraction is an epimorphism.


Relation to invertibility

Left-invertible morphisms are necessarily monic: if ''l'' is a left inverse for ''f'' (meaning ''l'' is a morphism and l \circ f = \operatorname_), then ''f'' is monic, as : f \circ g_1 = f \circ g_2 \Rightarrow l\circ f\circ g_1 = l\circ f\circ g_2 \Rightarrow g_1 = g_2. A left-invertible morphism is called a split mono or a section. However, a monomorphism need not be left-invertible. For example, in the category Group of all
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...
and
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) wh ...
s among them, if ''H'' is a subgroup of ''G'' then the inclusion is always a monomorphism; but ''f'' has a left inverse in the category if and only if ''H'' has a normal complement in ''G''. A morphism is monic if and only if the induced map , defined by for all morphisms , is
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
for all objects ''Z''.


Examples

Every morphism in a
concrete category In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets (or sometimes to another category, ''see Relative concreteness below''). This functor makes it possible to think of the objects of ...
whose underlying
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
is injective is a monomorphism; in other words, if morphisms are actually functions between sets, then any morphism which is a one-to-one function will necessarily be a monomorphism in the categorical sense. In the
category of sets In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition of m ...
the converse also holds, so the monomorphisms are exactly the
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
morphisms. The converse also holds in most naturally occurring categories of algebras because of the existence of a
free object In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. Informally, a free object over a set ''A'' can be thought of as being a "generic" algebraic structure over ''A'': the only equations that hold between ele ...
on one generator. In particular, it is true in the categories of all groups, of all
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
s, and in any
abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ab ...
. It is not true in general, however, that all monomorphisms must be injective in other categories; that is, there are settings in which the morphisms are functions between sets, but one can have a function that is not injective and yet is a monomorphism in the categorical sense. For example, in the category Div of
divisible In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
(abelian) groups and
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) wh ...
s between them there are monomorphisms that are not injective: consider, for example, the quotient map , where Q is the rationals under addition, Z the integers (also considered a group under addition), and Q/Z is the corresponding
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For examp ...
. This is not an injective map, as for example every integer is mapped to 0. Nevertheless, it is a monomorphism in this category. This follows from the implication , which we will now prove. If , where ''G'' is some divisible group, and , then . Now fix some . Without loss of generality, we may assume that (otherwise, choose −''x'' instead). Then, letting , since ''G'' is a divisible group, there exists some such that , so . From this, and , it follows that :0 \leq \frac = h(y) < 1 Since , it follows that , and thus . This says that , as desired. To go from that implication to the fact that ''q'' is a monomorphism, assume that for some morphisms , where ''G'' is some divisible group. Then , where . (Since , and , it follows that ). From the implication just proved, . Hence ''q'' is a monomorphism, as claimed.


Properties

*In a
topos In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notion ...
, every mono is an equalizer, and any map that is both monic and
epic Epic commonly refers to: * Epic poetry, a long narrative poem celebrating heroic deeds and events significant to a culture or nation * Epic film, a genre of film with heroic elements Epic or EPIC may also refer to: Arts, entertainment, and medi ...
is an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
. *Every isomorphism is monic.


Related concepts

There are also useful concepts of ''regular monomorphism'', ''extremal monomorphism'', ''immediate monomorphism'', ''strong monomorphism'', and ''split monomorphism''. * A monomorphism is said to be regular if it is an equalizer of some pair of parallel morphisms. * A monomorphism \mu is said to be extremal if in each representation \mu=\varphi\circ\varepsilon, where \varepsilon is an epimorphism, the morphism \varepsilon is automatically an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
. * A monomorphism \mu is said to be immediate if in each representation \mu=\mu'\circ\varepsilon, where \mu' is a monomorphism and \varepsilon is an epimorphism, the morphism \varepsilon is automatically an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
. * A monomorphism \mu:C\to D is said to be strong if for any epimorphism \varepsilon:A\to B and any morphisms \alpha:A\to C and \beta:B\to D such that \beta\circ\varepsilon=\mu\circ\alpha, there exists a morphism \delta:B\to C such that \delta\circ\varepsilon=\alpha and \mu\circ\delta=\beta. * A monomorphism \mu is said to be split if there exists a morphism \varepsilon such that \varepsilon\circ\mu=1 (in this case \varepsilon is called a left-sided inverse for \mu).


Terminology

The companion terms ''monomorphism'' and ''epimorphism'' were originally introduced by
Nicolas Bourbaki Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure (Paris), École normale supérieure - PSL (ENS). Founded in 1934–1935, the Bourbaki group originally in ...
; Bourbaki uses ''monomorphism'' as shorthand for an injective function. Early category theorists believed that the correct generalization of injectivity to the context of categories was the cancellation property given above. While this is not exactly true for monic maps, it is very close, so this has caused little trouble, unlike the case of epimorphisms.
Saunders Mac Lane Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician who co-founded category theory with Samuel Eilenberg. Early life and education Mac Lane was born in Norwich, Connecticut, near where his family lived in Taftvill ...
attempted to make a distinction between what he called ''monomorphisms'', which were maps in a concrete category whose underlying maps of sets were injective, and ''monic maps'', which are monomorphisms in the categorical sense of the word. This distinction never came into general use. Another name for monomorphism is ''
extension Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * E ...
'', although this has other uses too.


See also

*
Embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is gi ...
*
Nodal decomposition In category theory, an abstract mathematical discipline, a nodal decomposition of a morphism \varphi:X\to Y is a representation of \varphi as a product \varphi=\sigma\circ\beta\circ\pi, where \pi is a strong epimorphism, \beta a bimorphism, and \ ...
*
Subobject In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category. The notion is a generalization of concepts such as subsets from set theory, subgroups from group theory,M ...


Notes


References

* * * * *


External links

* *{{nlab, id=strong+monomorphism, title=Strong monomorphism Morphisms Algebraic properties of elements