definition of %category name% isomorphism
Topics
About: category
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of category.
Target Context
- The reader will have a definition of %category name% isomorphism.
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:
\( C\): \(\in \{\text{ the categories }\}\)
\( O_1\): \(\in Obj (C)\)
\( O_2\): \(\in Obj (C)\)
\( f_1\): \(\in Mor (O_1, O_2)\)
\( f_2\): \(\in Mor (O_2, O_1)\)
\(*(f_1, f_2)\):
//
Conditions:
\(f_2 \circ f_1 = id_{O_1}\)
\(\land\)
\(f_1 \circ f_2 = id_{O_2}\)
//
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 \(f_2 \circ f_1 = id_{O_1}\) and \(f_1 \circ f_2 = id_{O_2}\)
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.
