PlanetMath: properties of regular and extremal monomorphisms

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properties of regular and extremal monomorphisms

(Theorem)

We will denote the equalizer of and by $e=\mathrm{Eq}(f,g)$ .

Proposition 1 Every regular monomorphism is a monomorphism. (Every regular epimorphism is an epimorphism.)

Proof. Let $f=\mathrm{Eq}(r,s)$ . 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

such that $f\circ g=f\circ h$ , thus

. $\qedsymbol$

Proposition 2 If $g\circ f$ is an extremal monomorphism, then is an extremal monomorphism.
If $g\circ f$ is an extremal epimorphism, then is an extremal epimorphism.

Proof. Since $g\circ f$ is a monomorphism,

is a monomorphism too. Let $f=h\circ e$ and

be an epimorphism. Then $g\circ f=g\circ h\circ e$ , but $g\circ f$ is an extremal monomorphism, thus

is an isomorphism.

The second part of the proposition is dual to the first part. $\qedsymbol$

Proposition 3 If $f:X\to Y$ is a morphism then each of the following conditions implies the next one:

(i).: is an isomorphism
(ii).: is a section
(iii).: is a regular monomorphism
(iv).: is an extremal monomorphism
(v).: is a monomorphism.

(Dual claim: is an isomorphism $\Rightarrow$ retraction $\Rightarrow$ regular epimorphism $\Rightarrow$ extremal epimorphism $\Rightarrow$ epimorphism.)

Proof. (i) $\Rightarrow$ (ii) straightforward from the definition.

(ii) $\Rightarrow$ (iii) Let $g\circ f=id_A$ , we will show that $f=\mathrm{Eq}(id_B,f\circ g)$ . 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 is a monomorphism (every section is a monomorphism).

(iii) $\Rightarrow$ (iv) Let $f=\mathrm{Eq}(r,s)$ and $f=g\circ e$ with 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 is an epimorphism). By the universal property in the definition of equalizer there exists a unique morphism such that $g=f\circ e'$ . Thus we get $f\circ id_A=f=g\circ e=f\circ e'\circ e$ and is a monomorphism, hence $e'\circ e=id_A$ , i.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 is an epimorphism, hence $id_E=e\circ e'$ , i.e., is a section. The morphism is a retraction and a section too, thus is an isomorphism.

(iv) $\Rightarrow$ (v) Follows easily from the definition. $\qedsymbol$

The implication retraction $\Rightarrow$ regular epimorphism can be interpreted in the category of topological spaces $\mathbf{Top}$ as the well-known fact that each retraction is a quotient map.

Proposition 4 Let $f:A\to B$ 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.

Proof. Thanks to the duality principle, it suffices to prove the equivalence of the first three conditions.

(i) $\Rightarrow$ (ii) follows directly from the definition and (ii) $\Rightarrow$ (iii) is an easy consequence of the above proposition. (iii) $\Rightarrow$ (i): $f=id_B\circ f$ and je an epimorphism and extremal monomorphism. This implies that is an isomorphism. $\qedsymbol$

"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
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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)

Pending Errata and Addenda

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