118: %Category Name% Isomorphism

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definition of %category name% isomorphism

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Topics

About: category

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Note

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Starting Context

• The reader knows a definition of category.

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Target Context

• The reader will have a definition of %category name% isomorphism.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$C$: $\in \{\text{ the categories }\}$
$O_1$: $\in Obj(C)$
$O_2$: $\in Obj(C)$
$f_1$: $\in Mor(O_1,O_2)$
$f_2$: $\in Mor(O_2,O_1)$
$\ast (f_1,f_2)$:
//

Conditions:
$f_2\circ f_1=i{d}_{O_1}$
$\wedge$
$f_1\circ f_2=i{d}_{O_2}$
//

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2: Natural Language Description

For any category, $C$, any object, $O_1\in Obj(C)$, and any object, $O_2\in Obj(C)$, any pair of morphisms, $f_1\in Mor(O_1,O_2)$ and $f_2\in Mor(O_2,O_1)$, such that $f_2\circ f_1=i{d}_{O_1}$ and $f_1\circ f_2=i{d}_{O_2}$

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3: Note

A frequently seen definition of 'groups - homomorphisms' isomorphism requires only that the group homomorphism is bijective, which is because that guarantees that the inverse is a group homomorphism, but 'isomorphism' is the general concept applied for any category that requires that the inverse is a morphism. If group happens to have the natures that the inverse of any bijective group homomorphism is a group homomorphism, that is just a result. The definition of 'groups - homomorphisms' isomorphism inevitably requires that the inverse is a group homomorphism, and the fact that being bijective guarantees the inverse's being a group homomorphism is a proposition, by virtue of which, the definition should not be deformed.

For example, for the 'topological spaces - continuous maps' category, a bijective continuous map is not necessarily any 'topological spaces - continuous maps' isomorphism.

Often called just "isomorphism" for a pair of maps, but any pair of maps is never just a "isomorphism", but a 'sets - map morphisms' isomorphism, a 'vectors spaces - linear morphisms' isomorphism, etc., depending on what category, the maps are regarded to be morphisms of, which is the reason why the title of this article is "%category name% isomorphism" with "%category name%" as a place holder.

For example, between some 2 vectors spaces, a bijective map with its inverse may be a 'sets - map morphisms' isomorphism, but not a 'vectors spaces - linear morphisms' isomorphism, with the maps not being linear.

Often, one of the pair of the maps is called '%category name%' isomorphism, with the inverse map implicitly supposed.

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References

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