Functor: Difference between revisions

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Definition

Suppose ${\mathcal {C}},{\mathcal {D}}$ are categories. A functor (also called covariant functor) ${\mathcal {F}}$ from ${\mathcal {C}}$ to ${\mathcal {D}}$ comprises the following data:

Object level mapping: A mapping ${\mathcal {F}}:\operatorname {Ob} {\mathcal {C}}\to \operatorname {Ob} {\mathcal {D}}$ .
Morphism level mapping: For any $A,B\in {\mathcal {C}}$ , a mapping ${\mathcal {F}}:{\mathcal {C}}(A,B)\to {\mathcal {D}}(FA,FB)$ .

satisfying the following condition:

It preserves the identity morphism: For any $A\in {\mathcal {C}}$ , ${\mathcal {F}}(\operatorname {id} _{A})=\operatorname {id} _{F(A)}$ .
It preserves composition: For any $A,B,C\in {\mathcal {C}}$ , and $f\in {\mathcal {C}}(A,B),g\in {\mathcal {C}}(B,C)$ , we have $F(g\circ f)=Fg\circ Ff$ .

There is a related notion of contravariant functor. Note that a contravariant functor between two categories is not a functor between the categories. To emphasize that a functor is a functor and not a contravariant functor, we use the term covariant functor.

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 Suppose <math>\mathcal{C},\mathcal{D}</math> are [[category|categories]]. A '''functor''' (also called '''covariant functor''') <math>\mathcal{F}</math> from <math>\mathcal{C}</math> to <math>\mathcal{D}</math> comprises the following data:
-* A mapping <math>\mathcal{F}: \operatorname{Ob}\mathcal{C} \to \operatorname{Ob}\mathcal{D}</math>.
+* '''Object level mapping''': A mapping <math>\mathcal{F}: \operatorname{Ob}\mathcal{C} \to \operatorname{Ob}\mathcal{D}</math>.
-* For any <math>A,B \in \mathcal{C}</math>, a mapping <math>\mathcal{F}: \mathcal{C}(A,B) \to \mathcal{D}(FA,FB)</math>.
+* '''Morphism level mapping''': For any <math>A,B \in \mathcal{C}</math>, a mapping <math>\mathcal{F}: \mathcal{C}(A,B) \to \mathcal{D}(FA,FB)</math>.
 satisfying the following condition:
-* For any <math>A \in \mathcal{C}</math>, <math>\mathcal{F}(\operatorname{id}_A) = \operatorname{id}_{F(A)}</math>.
+* '''It preserves the identity morphism''': For any <math>A \in \mathcal{C}</math>, <math>\mathcal{F}(\operatorname{id}_A) = \operatorname{id}_{F(A)}</math>.
-* For any <math>A,B,C \in \mathcal{C}</math>, and <math>f \in \mathcal{C}(A,B), g \in \mathcal{C}(B,C)</math>, we have <math>F(g \circ f) = Fg \circ Ff</math>.
+* '''It preserves composition''': For any <math>A,B,C \in \mathcal{C}</math>, and <math>f \in \mathcal{C}(A,B), g \in \mathcal{C}(B,C)</math>, we have <math>F(g \circ f) = Fg \circ Ff</math>.
 There is a related notion of [[contravariant functor]]. Note that a contravariant functor between two categories is ''not'' a functor between the categories. To emphasize that a functor is a functor and not a contravariant functor, we use the term ''covariant functor''.