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properties of regular and extremal monomorphisms
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(Theorem)
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We will denote the equalizer of $f$ and $g$ by $e=\Eq(f,g)$ .
Proof. Let $f=\Eq(r,s)$ . Let $f\circ g=f\circ h$ . Then $r\circ(f\circ g)=s\circ(f\circ g)$ and by the definition of the equalizer there exists a unique morphism $h$ such that $f\circ g=f\circ h$ , thus $g=h$ . 
Proof. Since $g\circ f$ is a monomorphism, $f$ is a monomorphism too. Let $f=h\circ e$ and $e$ be an epimorphism. Then $g\circ f=g\circ h\circ e$ , but $g\circ f$ is an extremal monomorphism, thus $e$ is an isomorphism.
The second part of the proposition is dual to the first part. 
Proof. (i)
 (ii) straightforward from the definition.
(ii)
(iii) Let $g\circ f=id_A$ , we will show that $f=\Eq(id_B,f\circ g)$ . It holds $(f\circ g)\circ f=f\circ(g\circ f)=f\circ id_A=f=id_B\circ f$ . If $(f\circ g)\circ h=h$ then $h=f\circ(g\circ h)$ and there is unique such morphism, since $f$ is a monomorphism (every section is a monomorphism).
(iii)
(iv) Let $f=\Eq(r,s)$ and $f=g\circ e$ with $e$ an epimorphism. It holds: $(r\circ g)\circ e=r\circ(g\circ e)=r\circ f=s\circ f=s\circ (g\circ e)=(s\circ g)\circ e$ , thus it holds $r\circ g=s\circ g$ as well (since $e$ is an epimorphism). By the universal property in the definition of equalizer there exists a unique morphism $e'$ such that $g=f\circ e'$ . Thus we get $f\circ id_A=f=g\circ e=f\circ e'\circ e$ and $f$ is a monomorphism, hence $e'\circ e=id_A$ , i.e., $e$ is a section. Moreover $id_E\circ e=e=e\circ id_A=e\circ(e'\circ e)=(e\circ e')\circ e$ and $e$ is an epimorphism, hence $id_E=e\circ e'$ , i.e., $e$ is a section. The morphism $e$ is a retraction and a section too, thus $e$ is an isomorphism.
(iv)
(v) Follows easily from the definition. 
The implication retraction
regular epimorphism can be interpreted in the category of topological spaces $\Top$ as the well-known fact that each retraction is a quotient map.
Proposition 4 Let $\Map fAB$ be a morphism. The following conditions are equivalent:
- (i).
- $f$ is an isomorphism
- (ii).
- $f$ is an epimorphism and a section
- (iii).
- $f$ is an epimorphism and an extremal monomorphism
- (iv).
- $f$ is a monomorphism and a retraction
- (v).
- $f$ is a monomorphism and an extremal epimorphism.
Proof. Thanks to the duality principle, it suffices to prove the equivalence of the first three conditions.
(i)
(ii) follows directly from the definition and (ii)
(iii) is an easy consequence of the above proposition. (iii)
(i): $f=id_B\circ f$ and $f$ an epimorphism and extremal monomorphism. This implies that $f$ is an isomorphism. 
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"properties of regular and extremal monomorphisms" is owned by kompik.
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Cross-references: consequence, equivalence, duality principle, equivalent, quotient map, topological spaces, category, implication, universal property, retraction, section, implies, proposition, isomorphism, extremal epimorphism, extremal monomorphism, morphism, regular epimorphism, monomorphism, regular monomorphism, equalizer
There are 2 references to this entry.
This is version 9 of properties of regular and extremal monomorphisms, born on 2006-06-30, modified 2008-09-22.
Object id is 8117, canonical name is PropertiesOfRegularAndExtremalMonomorphisms.
Accessed 1178 times total.
Classification:
| AMS MSC: | 18A05 (Category theory; homological algebra :: General theory of categories and functors :: Definitions, generalizations) |
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Pending Errata and Addenda
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