reflects

Definition

Let $F:\mathcal{C}\to \mathcal{D}$ be a functor. We say $F$ reflects limits of shape $\mathcal{I}$ if for every diagram $D:\mathcal{I}\to \mathcal{C}$, a cone $(L,\varphi )$ over $D$ is a limiting cone in $\mathcal{C}$ whenever its image $(FL,F\varphi )$ is a limiting cone over $F\circ D$ in $\mathcal{D}$.

Reflection does not require that the limit exists in $\mathcal{D}$, nor that every limit in $\mathcal{D}$ must originate from $\mathcal{C}$; it only asserts that if the image of a cone is a limit, the original cone must have been a limit.

Reflection of Colimits

Dually, $F$ reflects colimits of shape $\mathcal{I}$ if a cocone $(C,\psi )$ over $D$ is a colimiting cocone in $\mathcal{C}$ whenever $(FC,F\psi )$ is a colimiting cocone over $F\circ D$ in $\mathcal{D}$.

Properties

1. Every faithful and conservative functor reflects limits and colimits that it preserves.
2. Every fully faithful functor reflects all limits and colimits that exist in its domain.
3. If $G$ is a right adjoint and the unit of the adjunction is an isomorphism, then $G$ reflects limits.

References

• riehl2016-category-theory