Get to know Epimorphism better with 10+ real example sentences, the meaning.
Epimorphism meaning
A morphism p such that for any other pair of morphisms f and g, if f∘p=g∘p, then f = g.
Using Epimorphism
- The main meaning on this page is: A morphism p such that for any other pair of morphisms f and g, if f∘p=g∘p, then f = g.
- In the example corpus, epimorphism often appears in combinations such as: an epimorphism, epimorphism in, epimorphism is.
Context around Epimorphism
- Average sentence length in these examples: 19.5 words
- Position in the sentence: 3 start, 8 middle, 9 end
- Sentence types: 20 statements, 0 questions, 0 exclamations
Corpus analysis for Epimorphism
- In this selection, "epimorphism" usually appears near the end of the sentence. The average example has 19.5 words, and this corpus slice is mostly made up of statements.
- Around the word, homological, split, strong, followed and satisfies stand out and add context to how "epimorphism" is used.
- Recognizable usage signals include is an epimorphism and of an epimorphism followed by. That gives this page its own corpus information beyond isolated example sentences.
- By corpus frequency, "epimorphism" sits close to words such as aare, aarti and abl, which helps place it inside the broader word index.
Example types with epimorphism
The same corpus examples are grouped by length and sentence type, making it easier to see the contexts in which the word appears:
Every retraction is an epimorphism. (5 words)
For example, the unique map is an epimorphism. (8 words)
It follows in particular that every cokernel is an epimorphism. (10 words)
If D is a subcategory of C, then every morphism in D which is an epimorphism when considered as a morphism in C is also an epimorphism in D; the converse, however, need not hold; the smaller category can (and often will) have more epimorphisms. (45 words)
For instance, since a function is bijective if and only if it is both injective and surjective, in abstract algebra a homomorphism is an isomorphism if and only if it is both a monomorphism and an epimorphism. (37 words)
A morphism f:A → B of rings is a homological epimorphism if it is an epimorphism and it induces a full and faithfull functor on derived categories: D(f) : D(B) → D(A). (33 words)
Example sentences (20)
A morphism f:A → B of rings is a homological epimorphism if it is an epimorphism and it induces a full and faithfull functor on derived categories: D(f) : D(B) → D(A).
Every epimorphism in this algebraic sense is an epimorphism in the sense of category theory, but the converse is not true in all categories.
If D is a subcategory of C, then every morphism in D which is an epimorphism when considered as a morphism in C is also an epimorphism in D; the converse, however, need not hold; the smaller category can (and often will) have more epimorphisms.
If the composition fg of two morphisms is an epimorphism, then f must be an epimorphism.
Any morphism with a right inverse is an epimorphism, but the converse is not true in general.
A similar argument shows that the natural ring homomorphism from any commutative ring R to any one of its localizations is an epimorphism.
A split epimorphism is a morphism which has a right-sided inverse.
As some of the above examples show, the property of being an epimorphism is not determined by the morphism alone, but also by the category of context.
A strong epimorphism satisfies a certain lifting property with respect to commutative squares involving a monomorphism.
Every retraction is an epimorphism.
Examples Every morphism in a concrete category whose underlying function is surjective is an epimorphism.
For example, the inclusion map Q → R, is a non-surjective epimorphism.
For example, the unique map is an epimorphism.
For instance, since a function is bijective if and only if it is both injective and surjective, in abstract algebra a homomorphism is an isomorphism if and only if it is both a monomorphism and an epimorphism.
In an abelian category, every morphism f can be written as the composition of an epimorphism followed by a monomorphism.
In many categories it is possible to write every morphism as the composition of an epimorphism followed by a monomorphism.
In this article, the term "epimorphism" will be used in the sense of category theory given above.
It follows in particular that every cokernel is an epimorphism.
Terminology The companion terms monomorphism and epimorphism were originally introduced by Nicolas Bourbaki ; Bourbaki uses monomorphism as shorthand for an injective function.
There is also the notion of homological epimorphism in ring theory.
Common combinations with epimorphism
These word pairs occur most frequently in English texts:
- an epimorphism 16×
- epimorphism in 5×
- epimorphism is 4×
- every epimorphism 3×
- homological epimorphism 2×
- epimorphism followed 2×