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
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 Cop). In Category theory, an abstract branch of Mathematics, the dual category or opposite category C op of a category C is the
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.
Left invertible maps are necessarily monic: if l is a left inverse for f (meaning lf = idX), then f is monic, as
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 is monic if and only if the induced map is injective for all Z.
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,
contradicting q o h = 0, so h(x) = 0 and q is therefore a monomorphism.
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.