Definition:Projection Functor

Definition

Let $\mathbf C$ and $\mathbf D$ be metacategories, and let $\mathbf C \times \mathbf D$ be their product.

The first projection functor $\pr_1: \mathbf C \times \mathbf D \to \mathbf C$ is defined by:

$\map {\pr_1} {C, D} := C$ for all objects $\tuple {C, D} \in \operatorname{ob} \mathbf C \times \mathbf D$

$\map {\pr_1} {f, g} := f$ for all morphisms $\tuple {f, g} \in \operatorname{mor} \mathbf C \times \mathbf D$

The second projection functor $\pr_2: \mathbf C \times \mathbf D \to \mathbf D$ is defined by:

$\map {\pr_2} {C, D} := D$ for all objects $\tuple {C, D} \in \operatorname{ob} \mathbf C \times \mathbf D$

$\map {\pr_2} {f, g} := g$ for all morphisms $\tuple {f, g} \in \operatorname{mor} \mathbf C \times \mathbf D$

That these constitute functors is shown on Projection Functor is Functor.

Sources

• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.6.1$