This article is about the mathematical function. For the biological phenomenon, see Epimorphosis.
In category theory, an epimorphism is a morphism f : X → Y that is right-cancellative in the sense that, for all objects Z and all morphisms g1, g2: Y → Z,
Epimorphisms are categorical analogues of onto or surjective functions (and in the category of sets the concept corresponds exactly to the surjective functions), but they may not exactly coincide in all contexts; for example, the inclusion is a ring epimorphism. The dual of an epimorphism is a monomorphism (i.e. an epimorphism in a category C is a monomorphism in the dual category Cop).
Many authors in abstract algebra and universal algebra define an epimorphism simply as an onto or surjective homomorphism. Every epimorphism in this algebraic sense is an epimorphism in the sense of category theory, but the converse is not true in all categories. In this article, the term "epimorphism" will be used in the sense of category theory given above. For more on this, see § Terminology below.
\mathbb {Z} \to \mathbb {Q} } is a ring epimorphism. The dual of an epimorphism is a monomorphism (i.e. an epimorphism in a category C is a monomorphism in...
split epimorphism is always an epimorphism, for both meanings of epimorphism. For sets and vector spaces, every epimorphism is a split epimorphism, but...
is one in which every epimorphism is conormal. A monomorphism is normal if it is the kernel of some morphism, and an epimorphism is conormal if it is the...
morphism f : X → Y is called an epimorphism if g1 ∘ f = g2 ∘ f implies g1 = g2 for all morphisms g1, g2 : Y → Z. An epimorphism can be called an epi for short...
Morphine, formerly also called morphia, is a strong opiate that is found naturally in opium, a dark brown resin produced by drying the latex of opium poppies...
homomorphisms are vastly different from epimorphisms in the category of rings. For example, the inclusion Z ⊆ Q is a ring epimorphism, but not a surjection. However...
commutative diagram are exact and m and p are epimorphisms and q is a monomorphism, then n is an epimorphism. If the rows in the commutative diagram are...
if f ∘ g1 = f ∘ g2 implies g1 = g2 for all morphisms g1, g2 : x → a. epimorphism (or epic) if g1 ∘ f = g2 ∘ f implies g1 = g2 for all morphisms g1, g2 :...
categorical dual of a monomorphism is an epimorphism, that is, a monomorphism in a category C is an epimorphism in the dual category Cop. Every section...
such that ST is the identity map on V. T is said to be surjective or an epimorphism if any of the following equivalent conditions are true: T is onto as...
ἐπί meaning over, above, on. Any morphism with a right inverse is an epimorphism, but the converse is not true in general. A right inverse g of a morphism...
indicates an epimorphism (the map mod 2). This is an exact sequence because the image 2Z of the monomorphism is the kernel of the epimorphism. Essentially...
superfluous epimorphism in Hom(P, X). If R is a ring, then in the category of R-modules, a superfluous epimorphism is then an epimorphism p : P → X {\displaystyle...
is not necessarily surjective. Every coequalizer is an epimorphism. In a topos, every epimorphism is the coequalizer of its kernel pair. In categories with...
pullback, and if f is a regular epimorphism, then g is a regular epimorphism as well. A regular epimorphism is an epimorphism that appears as a coequalizer...
coequalizers, the cokernel q : Y → Q is necessarily an epimorphism. Conversely an epimorphism is called normal (or conormal) if it is the cokernel of...
In the mathematical field of graph theory, a graph homomorphism is a mapping between two graphs that respects their structure. More concretely, it is a...
monomorphism and every epimorphism is normal. This means that every monomorphism is a kernel of some morphism, and every epimorphism is a cokernel of some...
the functions coinciding with their inverses. Adjoint endomorphism Epimorphism (surjective homomorphism) Frobenius endomorphism Monomorphism (injective...
ticker symbol in the United Kingdom prior to 1996 Epic morphism, or epimorphism, a mathematical concept European Prospective Investigation into Cancer...
f:X\rightarrow Y} , then a coimage of f {\displaystyle f} (if it exists) is an epimorphism c : X → C {\displaystyle c:X\rightarrow C} such that there is a map f...
I}R\to M\,} is an epimorphism, then the restriction ϕ : ⨁ i ∈ F R → M {\displaystyle \phi :\bigoplus _{i\in F}R\to M\,} is an epimorphism for some finite...
Let S be a set of generators of G. The natural map φ: F(S) → G is an epimorphism, which proves the claim. Equivalently, G is isomorphic to a quotient...
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