Projection from Product Category
Jump to navigation
Jump to search
Theorem
Let $\CC, \DD$ be categories.
Let $\CC \times \DD$ be the product category.
The projections:
- $\pi_\CC: \CC \times \DD \to \CC : \tuple {f, g} \mapsto f$
- $\pi_\DD: \CC \times \DD \to \DD : \tuple {f, g} \mapsto g$
where $\tuple {f, g} \in \map {\operatorname{ob} } {\CC \times \DD}$ or $\tuple {f, g} \in \map {\operatorname{mor} } {\CC \times \DD}$ are functors.
Moreover, $\pi_\CC, \pi_\DD$ satisfy the following universal property:
- For any category $\EE$ and any functors $F : \EE \to \CC$, $G : \EE \to \DD$, there exists a unique functor $H : \EE \to \CC \times \DD$ such that $F = \pi_\CC H$ and $G = \pi_\DD H$
That is, the following diagram commutes:
![]() | Although this article appears correct, it's inelegant. There has to be a better way of doing it. In particular: XyJax? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by redesigning it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Improve}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Proof
![]() | This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |