definition of contravariant functor
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 contravariant functor.
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:
\( C_1\): \(\in \{\text{ the categories }\}\)
\( C_2\): \(\in \{\text{ the categories }\}\)
\(*F\): \(: Obj (C_1) \to Obj (C_2), Mor (C_1) \to Mor (C_2)\)
//
Conditions:
\(\forall O_1, O_2, O_3 \in Obj (C_1), \forall f_1 \in Mor (O_1, O_2), \forall f_2 \in Mor (O_2, O_3)\)
(
1) \(F (f_1) \in Mor (F (O_2), F (O_1))\)
2) \(F (id_{O_1}) = id_{F (O_1)}\)
3) \(F (f_2 \circ f_1) = F (f_1) \circ F (f_2)\)
)
//
2: Natural Language Description
For any categories, \(C_1, C_2\), any map, \(F: Obj (C_1) \to Obj (C_2), Mor (C_1) \to Mor (C_2)\), such that for any objects, \(O_1, O_2, O_3 \in Obj (C_1)\), and any morphisms, \(f_1 \in Mor (O_1, O_2)\) and \(f_2 \in Mor (O_2, O_3)\), 1) \(F (f_1) \in Mor (F (O_2), F (O_1))\); 2) \(F (id_{O_1}) = id_{F (O_1)}\); 3) \(F (f_2 \circ f_1) = F (f_1) \circ F (f_2)\)
3: Note
While \(F: Obj (C_1) \to Obj (C_2), Mor (C_1) \to Mor (C_2)\) is not any usual map expression, \(F\) is really a pair of maps with the same name given.
There is also 'covariant functor'.
