definition of %category name% automorphism
Topics
About: structure
The table of contents of this article
Starting Context
- The reader knows a definition of category.
- The reader knows a definition of %category name% isomorphism.
Target Context
- The reader will have a definition of %category name% automorphism.
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.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( C\): \(\in \{\text{ the categories }\}\)
\( O\): \(\in Obj (C)\)
\( f_1\): \(\in Mor (O, O)\)
\( f_2\): \(\in Mor (O, O)\)
\(*(f_1, f_2)\):
//
Conditions:
\((f_1, f_2) \in \{\text{ the } 'C' \text{ isomorphisms }\}\)
//
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.
