- Μία απεικόνιση
Ορίζεται ως η απεικόνιση:
In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another.
The notion of morphism recurs in much of contemporary mathematics.
- In set theory, morphisms are functions;
- in linear algebra, linear transformations;
- in group theory, group homomorphisms;
- in topology, continuous functions,
- and so on.
In category theory, morphism is a broadly similar idea, but somewhat more abstract:
The study of morphisms and of the structures (called "objects") over which they are defined is central to category theory.
Much of the terminology of morphisms, as well as the intuition underlying them, comes from concrete categories, where the objects are simply sets with some additional structure, and morphisms are structure-preserving functions.
In category theory, morphisms are sometimes also called arrows.
- μορφισμός (morphism)
- μονομορφισμός (monomorphism)
- επιμορφισμός (epimorphism)
- αμφιμορφισμός (bimorphism)
- ισομορφισμός (isomorphism)
- ενδομορφισμός (endomorphism)
- αυτομορφισμός (automorphism)
- Αναμορφισμός (Anamorphism)
- Απομορφισμός (Apomorphism)
- Καταμορφισμός (Catamorphism)
- Υλομορφισμός (Hylomorphism)
Συμβολισμός Επεξήγηση Μονομορφισμός * Monomorphism: f : X → Y is called a monomorphism if f o g1 = f o g2 implies g1 = g2 for all morphisms g1, g2 : Z → X. It is also called a mono or a monic.
- The morphism f has a left inverse if there is a morphism g:Y → X such that g o f = idX. The left inverse g is also called a retraction of f. Morphisms with left inverses are always monomorphisms, but the converse is not always true in every category; a monomorphism may fail to have a left-inverse.
- A split monomorphism h : X → Y is a monomorphism having a left inverse g : Y → X, so that g o h = idX. Thus h o g : Y → Y is idempotent, so that (h o g)2 = h o g.
- In concrete categories, a function that has a left inverse is injective. Thus in concrete categories, monomorphisms are often, but not always, injective. The condition of being an injection is stronger than that of being a monomorphism, but weaker than that of being a split monomorphism.
Επιμορφισμός * Epimorphism: Dually, f : X → Y is called an epimorphism if g1 o f = g2 o f implies g1 = g2 for all morphisms g1, g2 : Y → Z. It is also called an epi or an epic.
- The morphism f has a right-inverse if there is a morphism g : Y → X such that f o g = idY. The right inverse g is also called a section of f. Morphisms having a right inverse are always epimorphisms, but the converse is not always true in every category, as an epimorphism may fail to have a right inverse.
- A split epimorphism is an epimorphism having a right inverse. Note that if a monomorphism f splits with left-inverse g, then g is a split epimorphism with right-inverse f.
- In concrete categories, a function that has a right inverse is surjective. Thus in concrete categories, epimorphisms are often, but not always, surjective. The condition of being a surjection is stronger than that of being an epimorphism, but weaker than that of being a split epimorphism. In the category of sets, every surjection has a section, a result equivalent to the axiom of choice.
Αμφιμορφισμός * A bimorphism is a morphism that is both an epimorphism and a monomorphism. Ισομορφισμός * Isomorphism: f : X → Y is called an isomorphism if there exists a morphism g : Y → X such that f o g = idY and g o f = idX. If a morphism has both left-inverse and right-inverse, then the two inverses are equal, so f is an isomorphism, and g is called simply the inverse of f. Inverse morphisms, if they exist, are unique. The inverse g is also an isomorphism with inverse f. Two objects with an isomorphism between them are said to be isomorphic or equivalent. Note that while every isomorphism is a bimorphism, a bimorphism is not necessarily an isomorphism. For example, in the category of commutative rings the inclusion Z → Q is a bimorphism, which is not an isomorphism. However, any morphism that is both an epimorphism and a split monomorphism, or both a monomorphism and a split epimorphism, must be an isomorphism. A category, such as Set, in which every bimorphism is an isomorphism is known as a balanced category. Ενδομορφισμός * Endomorphism: f : X → X is an endomorphism of X. A split endomorphism is an idempotent endomorphism f if f admits a decomposition f = h o g with g o h = id. In particular, the Karoubi envelope of a category splits every idempotent morphism. Αυτομορφισμός * An automorphism is a morphism that is both an endomorphism and an isomorphism.
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