Extreme monomorphism and epimorphism

Extreme monomorphisms and epimorphisms are terms from the mathematical subfield of category theory . It is a tightening of the monomorphisms or epimorphisms .

definition

A morphism in a category is called extreme monomorphism , if ${\displaystyle f}$

• ${\displaystyle f}$ is a monomorphism
• If there is a morphism and an epimorphism , it is an isomorphism . ${\displaystyle f=h\circ g}$ ${\displaystyle h}$ ${\displaystyle g}$ ${\displaystyle g}$

Dual definition is:

A morphism in a category is called extreme epimorphism , if ${\displaystyle f}$

• ${\displaystyle f}$ is an epimorphism
• If there is a morphism and a monomorphism , it is an isomorphism . ${\displaystyle f=g\circ h}$ ${\displaystyle h}$ ${\displaystyle g}$ ${\displaystyle g}$

comment

In the definition of extreme monomorphism, there must be a monomorphism because there is one. Since epimorphism is assumed, there is monomorphism and epimorphism, so a so-called bimorphism , which is weaker than isomorphism. What is special about the above definition is that in this special situation it is not only possible to infer bimorphism, but even isomorphism. The same can of course be said about extreme epimorphisms. ${\displaystyle g}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle g}$

Furthermore, this remark shows that in so-called balanced categories, i.e. those in which every bimorphism is already isomorphism, the above terms do not bring anything new. In such categories the extreme monomorphisms (or epimorphisms) are exactly the ordinary monomorphisms (or epimorphisms). This consideration can even be reversed, i.e. the following statements are equivalent for a category:

• The category is balanced.
• Every epimorphism is extreme epimorphism.
• Every monomorphism is extreme monomorphism.

Examples

• In the category of sets , groups or R modules over a commutative ring R , extreme monomorphisms are precisely the monomorphisms (and precisely the injective mappings or homomorphisms ), because these categories are balanced (see remark above). Exactly the extreme epimorphisms are exactly the epimorphisms (and exactly the surjective mappings or homomorphisms ).

• In the category of the rings with 1 and the ring homomorphisms that map 1 to 1, the inclusion map is an epimorphism that is not extreme, because , without isomorphism. At the same time, this is a monomorphism that is not extreme, in which case one has to consider. In general, any bimorphism that is not an isomorphism is such an example. ${\displaystyle \iota \colon \mathbb {Z} \rightarrow \mathbb {Q} }$ ${\displaystyle \iota =\iota \circ \mathrm {id} _{\mathbb {Z} }}$ ${\displaystyle \iota }$ ${\displaystyle \iota =\mathrm {id} _{\mathbb {Q} }\circ \iota }$

• In the category of topological spaces with continuous mappings , the extreme monomorphisms are exactly the homeomorphisms of to subspaces of . Furthermore, in this category the extreme epimorphisms are precisely the quotient maps . Hence, in this category there are monomorphisms and epimorphisms that are not extreme. For example, if the unit interval is the discrete topology and the unit interval is the Euclidean topology , it is a non-extreme monomorphism and a non-extreme epimorphism. ${\displaystyle X\rightarrow Y}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle [0,1]}$ ${\displaystyle Y}$ ${\displaystyle \mathrm {id} _{[0,1]}\colon X\rightarrow Y}$

• In the category of Banach spaces and continuous linear mappings, the following applies to a morphism : ${\displaystyle f\colon X\rightarrow Y}$

${\displaystyle f}$is extreme monomorphism     There is a constant with for everyone . ${\displaystyle \Leftrightarrow }$ ${\displaystyle m>0}$ ${\displaystyle m\|x\|\leq \|f(x)\|}$ ${\displaystyle x\in X}$

${\displaystyle f}$is extreme epimorphism   is surjective. ${\displaystyle \Leftrightarrow }$   ${\displaystyle f}$

In this category, too, one can easily specify monomorphisms and epimorphisms that are not extreme.

Individual evidence

1. ^ Gerhard Preuss: General Topology , Springer-Verlag 1972, ISBN 978-3-540-06006-2 , definition 1.5.9.
2. ↑ Horst Herrlich: Topologische Reflexionen und Coreflexionen , Lecture Notes in Mathematics 78 (1968), Definition 7.1.1
3. ^ K. Morita, J. Nagata: Topics in General Topology , North Holland 1998, 0-444-70455-8, Chapter 14, Definition 2.8
4. ^ Horst Herrlich, George E. Strecker: Category Theory , Allyn and Bacon Inc. 1973, sentence 17.13
5. ↑ Horst Herrlich, George E. Strecker: Category Theory , Allyn ans Bacon Inc. 1973, Example 17.10 (4)
6. ↑ Lothar Tschampel: Topology 1. General topology. Version 2. Buch-X-Verlag, Berlin 2011, ISBN 978-3-934671-60-7 , sentence 3.062.2
7. ↑ Lothar Tschampel: Topology 1. General topology. Version 2. Buch-X-Verlag, Berlin 2011, ISBN 978-3-934671-60-7 , sentence 3.062.3
8. ↑ Horst Herrlich, George E. Strecker: Category Theory , Allyn ans Bacon Inc. 1973, Example 17.10 (5)