884: %Category Name% Automorphism

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definition of %category name% automorphism

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Topics

About: structure

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The table of contents of this article

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

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Starting Context

• The reader knows a definition of category.
• The reader knows a definition of %category name% isomorphism.

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Target Context

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

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

Conditions:
$(f_1,f_2)\in \{{\text{ the }}^{\prime }{C}^{\prime }\text{ isomorphisms }\}$
//

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2: Note

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

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

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

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References

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