Functor

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A functor is a type of map between two categories.

More precisely, a functor $F:\mathcal{C} \to \mathcal{D}$ is a mapping which

sends every object $X$ of $\mathcal{C}$ to and object $F(X)$ of $\mathcal{D}$ .
sends every morphism $f:X\to Y$ of $\mathcal{C}$ to a morphism $F(f):F(X)\to F(Y)$ of $\mathcal{D}$ .

Which satisfies the conditions:

$F(1_X) = 1_{F(X)}$ for all $X\in \text{Ob}(\mathcal{C})$ .
$F(g\circ f) = F(g)\circ F(f)$ for all morphisms $f:X\to Y$ and $g:Y \to Z$ of $\mathcal{C}$ .

A contravariant functor a mapping satisfying the same properties as above, except that $F(f)$ is a morphism from $F(Y)$ to $F(X)$ , and instead of having $F(g\circ f) = F(g)\circ F(f)$ we have $F(g\circ f) = F(f)\circ F(g)$ . Alternatively, we can define a contravariant functor as an ordinary functor $F:\mathcal{C}^{op}\to \mathcal{D}$ , where $\mathcal{C}^{op}$ is the opposite category of $\mathcal{C}$ . We sometimes call our original type of functors covariant functor to distinguish them from contravariant functors.

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Functor

From AoPSWiki