84: Contravariant Functor

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definition of contravariant functor

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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 contravariant functor.

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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:
$C_1$: $\in \{\text{ the categories }\}$
$C_2$: $\in \{\text{ the categories }\}$
$\ast F$: $:Obj(C_1)\to Obj(C_2),Mor(C_1)\to Mor(C_2)$
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Conditions:
$\mathrm{\forall }{O}_1,O_2,O_3\in Obj(C_1),\mathrm{\forall }{f}_1\in Mor(O_1,O_2),\mathrm{\forall }{f}_2\in Mor(O_2,O_3)$
(
1) $F(f_1)\in Mor(F(O_2),F(O_1))$
2) $F(i{d}_{O_1})=i{d}_{F(O_1)}$
3) $F(f_2\circ f_1)=F(f_1)\circ F(f_2)$
)
//

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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) $F(i{d}_{O_1})=i{d}_{F(O_1)}$; 3) $F(f_2\circ f_1)=F(f_1)\circ F(f_2)$

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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'.

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References

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