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In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. Types of dimorphism (having two body types) include: Sexual dimorphism, differences in the body appearance of a species based on sex Nuclear dimorphism, when a cells nuclear apparatus is composed of two structurally and functionally differentiated types of nuclei Phenotypic switching, switching between two cell-types. ...
In general, polymorphism describes multiple possible states for a single property (it is said to be polymorphic, or polymorphous). ...
Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ...
Universal algebra (sometimes called General algebra) is the field of mathematics that studies the ideas common to all algebraic structures. ...
In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). ...
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  such that In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...
In mathematics, the notion of cancellative is a generalization of the notion of invertible. ...
In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...
 for all morphisms . -
Monomorphisms are a categorical generalization of injective functions; in some categories the notions coincide, but monomorphisms are more general, as in the examples below. Image File history File links No higher resolution available. ...
An injective function. ...
The dual of a monomorphism is an epimorphism (i.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 of a category C is the category formed by reversing all the morphisms of C. That is, we take Cop to be the category with objects that are those of C, but with the morphisms from X to Y in...
In category theory an epimorphism (also called an epic morphism or an epi) is a morphism f : X → Y which is right-cancellable in the following sense: g1 o f = g2 o f implies g1 = g2 for all morphisms g1, g2 : Y → Z. Epimorphisms are analogues of surjective functions, but...
In category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing all the morphisms of C. That is, we take Cop to be the category with objects that are those of C, but with the morphisms from X to Y in...
Terminology The companion terms monomorphism and epimorphism were originally introduced by Bourbaki; 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 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. This distinction never came into general use. In category theory an epimorphism (also called an epic morphism or an epi) is a morphism f : X → Y which is right-cancellable in the following sense: g1 o f = g2 o f implies g1 = g2 for all morphisms g1, g2 : Y → Z. Epimorphisms are analogues of surjective functions, but...
Nicolas Bourbaki is the pseudonym under which a group of mainly French 20th-century mathematicians wrote a series of books of exposition of modern advanced mathematics, beginning in 1935. ...
Saunders Mac Lane (4 August 1909, Taftville, Connecticut - 14 April 2005, San Francisco) was an American mathematician who cofounded category theory with Samuel Eilenberg. ...
Relation to invertibility 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 the mathematical field of category theory, a section is a morphism which has a left inverse, i. ...
A map is monic if and only if the induced map  is injective for all Z. In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...
Examples Every morphism in a concrete category whose underlying function is injective is a monomorphism. In the category of sets, the converse also holds so the monomorphisms are exactly the injective morphisms. The converse also holds in most naturally occurring categories of algebras because of the existence of a free object on one generator. In particular, it is true in the categories of groups and rings, and in any abelian category. In mathematics, a concrete category is a category in which, roughly speaking, all objects are sets possibly carrying some additional structure, all morphisms are functions between those sets, and the composition of morphisms is the composition of functions. ...
Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...
In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...
In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. ...
In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...
In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. ...
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 nice properties. ...
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. 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). This implies that h(x) is an integer if x ∈ G. If h(x) is not 0 then, for instance, In group theory, a divisible group is an abelian group G such that for any positive integer n and any g in G, there exists y in G such that ny = g. ...
In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ...
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) · h(v) From this property, one can deduce that h maps the identity element...
In mathematics, specifically in category theory, an additive category is a preadditive category C such that any finitely many objects A1,...,An of C have a biproduct A1 ⊕ ··· ⊕ An in C. (Recall that a category C is preadditive if all its morphism sets are Abelian groups and morphism...
 so that  , 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. 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. This article is about equalisers in mathematics. ...
See also An injective function. ...
In category theory an epimorphism (also called an epic morphism or an epi) is a morphism f : X → Y which is right-cancellable in the following sense: g1 o f = g2 o f implies g1 = g2 for all morphisms g1, g2 : Y → Z. Epimorphisms are analogues of surjective functions, but...
In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...
In category theory, there is a general definition of subobject extending the idea of subset and subgroup. ...
References - Francis Borceaux (1994), Handbook of Categorical Algebra 1, Cambridge University Press. 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
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