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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 and by
.
Proof. Let
 . Let
 . Then
 and by the definition of the equalizer there exists a unique morphism  such that
 , thus  . 
Proof. Since  is a monomorphism,  is a monomorphism too. Let
 and  be an epimorphism. Then
 , but  is an extremal monomorphism, thus  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
, we will show that
. It holds
. If
then
and there is unique such morphism, since is a monomorphism (every section is a monomorphism).
(iii)
(iv) Let
and
with an epimorphism. It holds:
, thus it holds
as well (since is an epimorphism). By the universal property in the definition of equalizer there exists a unique morphism such that
. Thus we get
and is a monomorphism, hence
, i.e., is a section. Moreover
and is an epimorphism, hence
, i.e., is a section. The morphism is a retraction and a section too, thus is an isomorphism.
(iv)
(v) Follows easily from the definition. 
The implication retraction
regular epimorphism can be interpreted in the category of topological spaces
as the well-known fact that each retraction is a quotient map.
Proposition 4 Let be a morphism. The following conditions are equivalent:
- (i).
is an isomorphism
- (ii).
is an epimorphism and a section
- (iii).
is an epimorphism and an extremal monomorphism
- (iv).
is a monomorphism and a retraction
- (v).
is a monomorphism and an extremal epimorphism.
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"properties of regular and extremal monomorphisms" is owned by kompik.
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(view preamble)
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 is 1 reference to this entry.
This is version 8 of properties of regular and extremal monomorphisms, born on 2006-06-30, modified 2006-10-15.
Object id is 8117, canonical name is PropertiesOfRegularAndExtremalMonomorphisms.
Accessed 767 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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