Creation (Category Theory)

Definition

Let $F:\mathcal{C}\to \mathcal{D}$ be a functor. We say $F$ creates limits of shape $\mathcal{I}$ if for every diagram $D:\mathcal{I}\to \mathcal{C}$, the existence of a limiting cone $(M,\psi )$ over $F\circ D$ in $\mathcal{D}$ implies:

1. There exists a unique cone $(L,\varphi )$ over $D$ in $\mathcal{C}$ such that $(FL,F\varphi )=(M,\psi )$.
2. This cone $(L,\varphi )$ is a limiting cone over $D$.

Strict vs. Non-strict Creation

In the non-strict sense, we say $F$ creates limits if for every limiting cone $(M,\psi )$ in $\mathcal{D}$, there exists a limiting cone $(L,\varphi )$ in $\mathcal{C}$ such that $(FL,F\varphi )\cong (M,\psi )$, and any such lift is unique up to isomorphism.

Properties

• $F$ creates limits $⟹$ $F$ preserves and reflects them.

Examples

• The forgetful functor $U:Group\to Set$ creates all small limits.
• Monadic functors create all limits that their underlying category possesses.

See also

• Reflection (Category Theory)
• Preservation (Category Theory)

References

• riehl2016-category-theory