Monomorphism

For other uses, see Dimorphism (disambiguation) or Polymorphism (disambiguation).

In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules Universal algebra (sometimes called general algebra) is the field of Mathematics that studies Algebraic structures themselves not examples ("models" In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector

In the more general setting of category theory, a monomorphism (also called a monic morphism or a mono) is a left-cancellative morphism, that is, a map $f\colon X \to Y$ such that

$f \circ g_1 = f \circ g_2 \implies g_1 = g_2$ for all morphisms $g_1, g_2 \colon Z \to X$ . In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, the notion of cancellative is a generalization of the notion of Invertible. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and

Monomorphisms are a categorical generalization of injective functions; in some categories the notions coincide, but monomorphisms are more general, as in the examples below.

The dual of a monomorphism is an epimorphism (i. In Category theory, an abstract branch of Mathematics, the dual category or opposite category C op of a category C is the In Category theory an epimorphism (also called an epic morphism or an epi) is a Morphism f: X &rarr Y which e. a monomorphism in a category C is an epimorphism in the dual category C^op). In Category theory, an abstract branch of Mathematics, the dual category or opposite category C op of a category C is the

1 Terminology
2 Relation to invertibility
3 Examples
4 Related concepts
5 See also
6 References

Terminology

The companion terms monomorphism and epimorphism were originally introduced by Bourbaki; Bourbaki uses monomorphism as shorthand for an injective function. In Category theory an epimorphism (also called an epic morphism or an epi) is a Morphism f: X &rarr Y which Nicolas Bourbaki is the collective Pseudonym under which a group of (mainly French) 20th-century Mathematicians wrote a series of books presenting an exposition Early category theorists believed that the correct generalization of injectivity to the context of categories was the 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 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. Saunders Mac Lane ( 4 August 1909, Taftville, Connecticut – 14 April 2005, San Francisco) was an American This distinction never came into general use.

Another name for monomorphism is extension, although this has other uses too.

Relation to invertibility

Left invertible maps are necessarily monic: if l is a left inverse for f (meaning $l f = i d X$ ), then f is monic, as

$f \circ g_1 = f \circ g_2 \implies lfg_1 = lfg_2 \implies g_1 = g_2$

A left invertible map is called a split mono. In Category theory, a branch of Mathematics, a section is a right inverse of a morphism

A map $f\colon X \to Y$ is monic if and only if the induced map $f_*\colon \operatorname{Hom}(Z,X) \to\operatorname{Hom}(Z,Y)$ is injective for all Z.

Examples

Every morphism in a concrete category whose underlying function is injective is a monomorphism. In Mathematics, a concrete category is commonly understood as a category whose objects are structured sets, whose Morphisms are structure-preserving The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In the category of sets, the converse also holds so the monomorphisms are exactly the injective morphisms. In Mathematics, the category of sets, denoted as Set, is the category whose objects are all sets and whose Morphisms are The converse also holds in most naturally occurring categories of algebras because of the existence of a free object on one generator. In Mathematics, the idea of a free object is one of the basic concepts of Abstract algebra. In particular, it is true in the categories of groups and rings, 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

It is not true in general, however, that all monomorphisms must be injective in other categories. For example, in the category Div of divisible abelian groups and group homomorphisms between them there are monomorphisms that are not injective: consider the quotient map q : Q → Q/Z. In Mathematics, especially in the field of Group theory, a divisible group is an Abelian group in which every element can in some sense be divided by An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function This is clearly not an injective map; nevertheless, it is a monomorphism in this category. To see this, note that if q o f = q o g for some morphisms f,g : G → Q where G is some divisible abelian group then q o h = 0 where h = f - g (this makes sense as this is an additive category). In Mathematics, specifically in Category theory, an additive category is a Preadditive category C such that any finitely many objects A This implies that h(x) is an integer if x ∈ G. If h(x) is not 0 then, for instance,

$h\left(\frac{x}{4h(x)}\right) = \frac{1}{4}$

so that

$(q \circ h)\left(\frac{x}{4h(x)}\right) \neq 0$ ,

contradicting q o h = 0, so h(x) = 0 and q is therefore a monomorphism.

Related concepts

There are also useful concepts of regular monomorphism, strong monomorphism, and extremal monomorphism. A regular monomorphism equalizes some parallel pair of morphisms. In Mathematics, an equaliser, or equalizer, is a set of arguments where two or more functions have equal values An extremal monomorphism is a monomorphism that cannot be nontrivially factored through an epimorphism: Precisely, if m=g o e with e an epimorphism, then e is an isomorphism. A strong monomorphism satisfies a certain lifting property with respect to commutative squares involving an epimorphism.

References

Francis Borceaux (1994), Handbook of Categorical Algebra 1, Cambridge University Press. In Category theory an epimorphism (also called an epic morphism or an epi) is a Morphism f: X &rarr Y which In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Category theory, there is a general definition of subobject extending the idea of Subset and Subgroup. ISBN 0-521-44178-1.
George Bergman (1998), An Invitation to General Algebra and Universal Constructions, Henry Helson Publisher, Berkeley. ISBN 0-9655211-4-1.
Jaap van Oosten, Basic Category Theory

Dictionary

-noun

(mathematics) an injective homomorphism
(biology) the absence of sexual dimorphism

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